New Frontiers of Fractal Uncertainty

Abstract

We extend the classical fractal uncertainty principle (FUP) framework in time-frequency analysis by exploring several novel directions. First, we generalize the FUP beyond the classical Gaussian window by investigating non-Gaussian windows and the corresponding generalized Fock space techniques. Second, we develop uncertainty estimates in alternative joint representations, including the continuous wavelet transform and directional representations such as shearlets. Third, we study fractal uncertainty on random and anisotropic fractal sets, providing probabilistic and geometric refinements of the FUP. Fourth, we connect these results with semiclassical and microlocal analysis, thereby elucidating the role of fractal geometry in resonance theory and pseudodifferential operators. Finally, we extend the analysis beyond Gaussian Gabor multipliers by considering non-Gaussian generating functions and irregular lattice samplings. Our results yield new operator norm estimates and spectral properties, with potential applications in signal processing, quantum mechanics, and numerical analysis.

1. Introduction

Uncertainty principles are fundamental in harmonic analysis, signal processing, and quantum mechanics, providing intrinsic limits to the simultaneous concentration of a function and its Fourier transform. The fractal uncertainty principle (FUP) is a recently developed concept that asserts no function can be simultaneously concentrated near fractal sets in both time and frequency domains. In its original formulation, the FUP was introduced and refined in works such as [1,2] and further quantified in [3,4].

A key breakthrough was achieved by Knutsen [5], who established a version of the FUP for the short-time Fourier transform (STFT) with a Gaussian window. This result not only recovers classical uncertainty estimates but also connects the problem to the rich structure of Fock spaces through the Bargmann transform. The approach, relying on estimates for Daubechies’ time-frequency localization operators, provides explicit operator norm bounds when localizing on porous sets, such as Cantor-type sets, in phase space.

Despite these advancements, the literature indicates several open directions. The optimality of the Gaussian window stems from its unique properties (e.g., the reproducing kernel of Fock spaces). However, alternative window functions–such as compactly supported or exponentially decaying windows–offer flexibility in applications (see, e.g., [6]). Extending FUPs to non-Gaussian windows may necessitate developing generalized Fock space techniques.

While the STFT has been the primary tool for studying FUPs, other representations like the continuous wavelet transform [7] and directional representations (shearlets [8]) provide complementary perspectives. These representations naturally capture multiscale or anisotropic features, and a fractal uncertainty analysis in these settings could reveal new insights.

Most existing studies, including [4,5], focus on deterministic fractal sets such as Cantor sets. In many applications, however, fractal structures arise in a random or anisotropic manner [9]. Probabilistic models and anisotropic geometries may require new techniques for establishing uncertainty estimates.

The FUP is closely related to semiclassical analysis, where uncertainty principles help describe the distribution of resonances in quantum systems [10]. Bridging the gap between fractal geometry and microlocal spectral theory has the potential to advance both theoretical and applied aspects of the field.

Knutsen’s work translates the continuous FUP into the discrete setting via Gaussian Gabor multipliers. Exploring non-Gaussian generating functions or adaptive lattice schemes could lead to sharper spectral properties and more robust discretization schemes for practical applications.

In this manuscript, we address these issues by proposing several extensions to the classical FUP framework. Our contributions include developing a generalized FUP for non-Gaussian windows and deriving the corresponding operator norm estimates; establishing fractal uncertainty estimates in alternative joint representations, particularly the continuous wavelet transform and shearlet systems; investigating the FUP for random and anisotropic fractal sets, and providing probabilistic as well as geometric refinements; bridging connections between fractal uncertainty, semiclassical analysis, and microlocal techniques by studying pseudodifferential operators with fractal symbols and finally extending the analysis beyond the classical Gaussian Gabor multipliers by considering alternative generating functions and irregular lattice samplings.

We emphasize that while our bounds provide explicit positive exponents $\beta$ under the stated porosity/covering hypotheses, these exponents are not claimed to be optimal: sharper exponents may be obtained in specific models by incorporating additional structure (e.g., additive-energy estimates or Fourier decay of canonical measures), and conversely there are constructions showing that no substantially better uniform bound depending only on box-counting data can be expected.

Our work builds on the framework established in [5,8,9,10] and incorporates methodologies from [11,12,13,14,15]. We refer readers for more details to [6,7,16,17]. We believe that these extensions not only provide deeper theoretical insights into the interplay between fractal geometry and time-frequency analysis but also have significant implications for practical applications in signal processing and quantum physics.

The remainder of the paper is organized as follows. Section 2 reviews the necessary background in time-frequency analysis, modulation spaces, and Fock spaces. Section 3, Section 4, Section 5, Section 6, Section 7 and Section 8 develop the new extensions, while Section 9 presents numerical experiments and potential applications. Finally, Section 10 concludes with a discussion of future research directions.

2. Preliminaries

In this section, we review the foundational concepts and results required in the subsequent sections. We begin with an overview of time-frequency analysis and modulation spaces, then introduce Fock spaces via the Bargmann transform, and finally discuss the geometric properties of fractal sets–most notably porosity and Cantor set constructions. We also recall the definitions of Daubechies’ localization operators and Gabor multipliers, which will be central to our analysis.

Let $f\in L^2\left({\mathbb{R}}^d\right)$ and fix a nonzero window function $\phi \in L^2\left({\mathbb{R}}^d\right)$. The short-time Fourier transform (STFT) of f with respect to $\phi$ is defined by

$$V_{\phi }f(x,\omega )={\int }_{{\mathbb{R}}^d}f\left(t\right)\overline{\phi (t-x)}{e}^{-2\pi i\omega ·t}dt,(x,\omega )\in {\mathbb{R}}^{2d}.$$

This representation provides a joint time-frequency description of the signal. For a fixed window $\phi$, the modulation space $M^p\left({\mathbb{R}}^d\right)$, $1\le p\le \infty$, is defined as

$$M^p\left({\mathbb{R}}^d\right)=\{f\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right){:\parallel f\parallel }_{M^p}=\parallel V_{\phi }f{\parallel }_{L^p\left({\mathbb{R}}^{2d}\right)}<\infty \},$$

where ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right)$ denotes the space of tempered distributions. It is known that, if $\phi \in \mathcal{S}\left({\mathbb{R}}^d\right)$, the definition is independent of the particular choice of $\phi$ [6].

The Bargmann transform provides an isometric isomorphism between $L^2\left({\mathbb{R}}^d\right)$ and a space of entire functions. For $f\in L^2\left({\mathbb{R}}^d\right)$, the Bargmann transform is defined by

$$Bf\left(z\right)=2^{d/4}{\int }_{{\mathbb{R}}^d}f\left(t\right)e^{2\pi t·z-\pi t^2-\pi z^2/2}dt,z\in {\mathbb{C}}^d.$$

Its image is the Fock space

$${\mathcal{F}}^2\left({\mathbb{C}}^d\right)=\left\{{F\mathrm{entire}:\parallel F\parallel }_{{\mathcal{F}}^2}^2={\int }_{{\mathbb{C}}^d}{|F\left(z\right)|}^2{e}^{-{\pi |z|}^2}dA\left(z\right)<\infty \right\},$$

where $dA\left(z\right)$ denotes the Lebesgue measure on ${\mathbb{C}}^d$. Furthermore, when the Gaussian window

$${\phi }_0\left(x\right)=2^{d/4}{e}^{-\pi x^2},x\in {\mathbb{R}}^d,$$

is used, the STFT and the Bargmann transform are related by the identity

$$V_{{\phi }_0}f(x,-\omega )=e^{\pi ix·\omega }Bf\left(z\right)e^{-{\pi |z|}^2/2},z=x+i\omega ,$$

see, e.g., [5,18].

A key ingredient in fractal uncertainty principles is the notion of porosity.

Definition 1 

($\nu$-Porosity, [4,5]). Let $0<\nu <1$ and let $\Omega \subset {\mathbb{R}}^d$ be closed. We say that Ω is $\nu$-porous on scales ${\alpha }_{min}$ to ${\alpha }_{max}$ if for every ball $B_r\left(x\right)$ with radius $r\in [{\alpha }_{min},{\alpha }_{max}]$, there exists a ball $B_{\nu r}\left(y\right)\subset B_r\left(x\right)$ such that

$$B_{\nu r}\left(y\right)\cap \Omega =⌀.$$

An important example of porous sets is provided by Cantor sets. For a fixed integer $M>1$ and a nonempty proper subset $A\subset \{0,1,\dots ,M-1\}$, the discrete Cantor set is defined by

$${\mathcal{C}}_n(M,A)=\left\{\sum _{j=0}^{n-1}{a}_j{M}^j:a_j\in A\right\}\subset \{0,1,\dots ,M^n-1\}.$$

The corresponding continuous version on the interval $[0,L]$ is given by

$$C_n(L,M,A)={\displaystyle \frac{L}{M^n}}{\mathcal{C}}_n(M,A)+\left[0,{\displaystyle \frac{L}{M^n}}\right].$$

It is known that $C_n(L,M,A)$ is $\nu$-porous on scales $L{M}^{-n+1}$ to ∞ with any $\nu \le {\displaystyle \frac{1}{2M}}$ [5].

Definition 2. 

Let $X\subset {\mathbb{R}}^N$ ($N=2d$) be a bounded set. We say X is uniformly porous with porosity constant $p\in (0,1)$ and scale $r_0>0$ if for every $x\in X$ and every $0<r\le r_0$ there exists a point $y\in {\mathbb{R}}^N$ such that

$$B(y,pr)\subset B(x,r)\setminus X.$$

Lemma 1. 

Let $X\subset {\mathbb{R}}^N$ be uniformly porous with porosity constant $p\in (0,1)$ and scale $r_0>0$ (Definition 2). Then there exist constants $C_p>0$ and an exponent

$$s={\displaystyle \frac{log\left(2^{mN}-1\right)}{mlog2}}\mathrm{with}m:=⌈-{log}_{2}p⌉,$$

satisfying $0\le s<N$, such that for every $0<r\le r_0$ the minimal number $N_X\left(r\right)$ of cubes of side length r required to cover X satisfies

$$N_X\left(r\right)\le C_p{r}^{-s}.$$

In particular one may take $C_p={(2^{mN}-1)}^{{\displaystyle \frac{log\left({\mathsf{\ell }}_0\right)}{mlog2}}}$, where ${\mathsf{\ell }}_0$ is any fixed reference side-length with ${\mathsf{\ell }}_0\ge r_0$ (e.g., the side of a bounding cube of X).

Proof. 

Fix a cube $Q_0$ of side length ${\mathsf{\ell }}_0\ge r_0$ containing X (we may for example take a dyadic cube that bounds X). We perform an iterative dyadic subdivision argument in steps of length scale factor $2^m$, where

$$m:=⌈-{log}_{2}p⌉.$$

By definition of m we have $2^{-m}\le p<2^{-m+1}$. The reason for this choice is that the porosity hole of radius $pr$ inside any ball of radius r can be located inside one of the subcubes after splitting a parent cube into $2^{mN}$ equal subcubes of side length $\mathsf{\ell }/2^m$.

We now show by induction on scale that after k such $2^m$-refinements the number of subcubes (of side ${\mathsf{\ell }}_0/2^{mk}$) that intersect X is at most ${(2^{mN}-1)}^k$.

Base case $k=0$: there is at most $1={(2^{mN}-1)}^0$ occupied cube at scale ${\mathsf{\ell }}_0$.

Inductive step: suppose at some scale we have $M_k\le {(2^{mN}-1)}^k$ occupied cubes of side ${\mathsf{\ell }}_0/2^{mk}$. Consider one occupied cube Q at that scale with side $\mathsf{\ell }={\mathsf{\ell }}_0/2^{mk}$. By the uniform porosity property applied with radius $r=\sqrt{N}\mathsf{\ell }/2$ (compare cube and inscribed ball scales) there exists a ball of radius $pr$ contained in $Q\setminus X$. By our choice of m we have $pr\ge r/2^m$ (up to harmless constant factors arising from comparing cube and ball). Consequently among the $2^{mN}$ children subcubes of Q obtained by subdividing Q into equal subcubes of side $\mathsf{\ell }/2^m$ at least one of these children is empty of X. Thus each previously occupied cube produces at most $2^{mN}-1$ occupied children at the next refinement level.

It follows by induction that after k refinement steps the number of occupied subcubes of side length ${\mathsf{\ell }}_0/2^{mk}$ is bounded by

$$M_k\le {(2^{mN}-1)}^k.$$

Now fix a target covering scale r with $0<r\le r_0$. Choose $k\ge 0$ minimal so that

$${\displaystyle \frac{{\mathsf{\ell }}_0}{2^{mk}}}\le r.$$

Then $k\le {\displaystyle \frac{log({\mathsf{\ell }}_0/r)}{mlog2}}$. The number $N_X\left(r\right)$ of side-r cubes needed to cover X can be bounded by $M_k$ up to an absolute multiplicative factor (to account for the fact that we cover by cubes of side $\le r$); hence

$$N_X\left(r\right)\le {(2^{mN}-1)}^k\le {(2^{mN}-1)}^{{\displaystyle \frac{log({\mathsf{\ell }}_0/r)}{mlog2}}}=C_p{r}^{-s},$$

with

$$s={\displaystyle \frac{log(2^{mN}-1)}{mlog2}},C_p={(2^{mN}-1)}^{{\displaystyle \frac{log{\mathsf{\ell }}_0}{mlog2}}}.$$

Since $2^{mN}-1<2^{mN}$ we have $s<N$. This completes the proof. □

Lemma 2. 

Let $L,M\ge 2$, $A\subset [0,L]$, and $n\in \mathbb{N}$. Define $C_n(L,M,A)$ as in the main text. If $\nu \le {\left(2M\right)}^{-1}$ then $C_n(L,M,A)\subset [0,L]$ is ν–porous on scales $r\ge L{M}^{-n+1}$, i.e., for every $x\in C_n(L,M,A)$ and every $r\ge L{M}^{-n+1}$ there exists an interval $I\subset B(x,r)$ with $|I|\ge \nu r$ and $I\cap C_n(L,M,A)=⌀$. (The statement is trivial for $r\gg L$ since $B(x,r)$ eventually contains $[0,L]$.)

Remark 1. 

For $r\gg L$ the conclusion is vacuous/trivial because $B(x,r)\supset [0,L]$; equivalently, the porosity is meaningful only for radii $r\lesssim L$.

For clarity we distinguish two related local-density quantities used below.

Definition 3. 

Let $\Omega \subset {\mathbb{C}}^d$ and $R>0$. Denote by m the Lebesgue measure on ${\mathbb{R}}^{2d}\simeq {\mathbb{C}}^d$ and by $\#(·)$ the counting (cardinality) measure on finite sets. We define

$${\rho }_{\mathrm{vol}}(\Omega ,R):=\underset{z\in {\mathbb{C}}^d}{sup}m\left(\Omega \cap B_R\left(z\right)\right),{\rho }_{\mathrm{count}}(\Omega ,R):=\underset{z\in {\mathbb{C}}^d}{sup}\#\left(\Omega \cap B_R\left(z\right)\right),$$

where $B_R\left(z\right)$ is the Euclidean ball of radius R centered at z.

Remark 2. 

The quantity ${\rho }_{\mathrm{vol}}(\Omega ,R)$ measures local Lebesgue mass and therefore vanishes for Lebesgue-null fractal sets; the quantity ${\rho }_{\mathrm{count}}(\Omega ,R)$ is appropriate for discrete sets (e.g., lattices) and counts the maximal number of points in any ball of radius R. In the text we will indicate which version is used; if no subscript is present we mean ${\rho }_{\mathrm{vol}}$.

To quantify the density of a fractal set in phase space, we introduce the concept of maximal Nyquist density.

For a measurable set $\Omega \subset {\mathbb{C}}^d$ and a radius $R>0$, define

$$\rho (\Omega ,R)=\underset{z\in {\mathbb{C}}^d}{sup}|\Omega \cap B_R\left(z\right)|,$$

where $B_R\left(z\right)$ is the ball of radius R centered at z.

Another key property we require is the sub-averaging property in Fock spaces. The following lemma, which can be found in [5,18], is essential for our analysis.

Lemma 3. 

For any entire function $F\in {\mathcal{F}}^p\left({\mathbb{C}}^d\right)$ and any point $z\in {\mathbb{C}}^d$, there exists a constant $C_d>0$ (depending only on the dimension) such that for any $R>0$

$${|F\left(z\right)|}^p{e}^{-{p\pi |z|}^2}\le {\displaystyle \frac{C_d}{|B_R\left(0\right)|}}{\int }_{B_R\left(z\right)}{|F\left(\xi \right)|}^p{e}^{-{p\pi |\xi |}^2}dA\left(\xi \right).$$

Daubechies’ time-frequency localization operator provides a continuous model for measuring the concentration of a signal on a subset $\Omega \subset {\mathbb{R}}^{2d}$. For a bounded symbol $S:{\mathbb{R}}^{2d}\to \mathbb{C}$, the operator is defined by

$$P_{\phi }^{S}f={\int }_{{\mathbb{R}}^{2d}}S(x,\omega )V_{\phi }f(x,\omega )M_{\omega }{T}_x\phi dxd\omega ,$$

where $T_x$ and $M_{\omega }$ denote translation and modulation operators, respectively.

A discrete analogue is provided by Gabor multipliers. Given a lattice $\mathsf{\Lambda }\subset {\mathbb{R}}^{2d}$ and a bounded symbol $b:\mathsf{\Lambda }\to \mathbb{C}$, the Gabor multiplier is defined as

$$G_{\phi ,\mathsf{\Lambda },b}f=|D_{\mathsf{\Lambda }}|\sum _{\lambda \in \mathsf{\Lambda }}b\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi ,$$

where $\pi \left(\lambda \right)=M_{\omega }{T}_x$ for $\lambda =(x,\omega )$ and $|D_{\mathsf{\Lambda }}|$ denotes the volume of the fundamental domain of $\mathsf{\Lambda }$ (see [19]).

Remark 3. 

In the definition

$$G_{\phi ,\mathsf{\Lambda },b}f=|D_{\mathsf{\Lambda }}|\sum _{\lambda \in \mathsf{\Lambda }}b\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi ,$$

the factor $|D_{\mathsf{\Lambda }}|$ denotes the cell volume of the lattice $\mathsf{\Lambda }=D_{\mathsf{\Lambda }}{\mathbb{Z}}^{2d}$. This convention matches the normalization of the continuous localization operator

$$L_{\phi ,b}f={\int }_{{\mathbb{R}}^{2d}}b\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi dz,$$

so that the Riemann-sum approximation ${\int }_{{\mathbb{R}}^{2d}}(·)dz\approx |D_{\mathsf{\Lambda }}|{\sum }_{\lambda \in \mathsf{\Lambda }}(·)$ holds as the lattice becomes dense. Under this convention the case $b\equiv 1$ yields $G_{\phi ,\mathsf{\Lambda },1}=S_{\phi ,\mathsf{\Lambda }}$, the usual Gabor frame operator, which equals the identity when ${\left\{\pi \left(\lambda \right)\phi \right\}}_{\lambda \in \mathsf{\Lambda }}$ forms a Parseval frame. (Some references omit or invert this factor depending on the STFT normalization; our choice ensures consistency between the discrete and continuous operators.)

These operators, and their corresponding norm estimates, play a central role in the fractal uncertainty principle framework we develop later.

The definitions and results presented in this section–drawn from [4,5,6,18,19]—form the bedrock for the new extensions explored in this paper.

3. Generalized Bargmann Transform and RKHS—Admissible Windows

In this section we give a self-contained, rigorous formulation and the minimal standing hypothesis we shall use throughout the paper. The presentation below follows standard constructions for the short-time Fourier transform (STFT) and its range; see for instance [6,20,21] for related discussions.

We introduce a pair of checkable concentration conditions, state the FUP under these hypotheses, and record simple lemmas showing that the usual window classes (Gaussian, Schwartz, and compactly supported $C^m$ windows) meet one of the conditions below.

Definition 4 

(Window concentration hypotheses). Let $g\in L^2\left({\mathbb{R}}^d\right)$ be an analysing window and let ${\mathcal{T}}_h$ denote the associated semiclassical transform (Bargmann/STFT/wavelet normalization as in the paper). Write $K_h\left(z\right)$ for the kernel (or the auto-transform kernel) associated to g in the transform domain. We consider two types of concentration hypotheses:

(W1) 

(Exponential-type tail) There exist constants $A>0$, $c>0$ and $p\ge 1$ such that for all $h\in (0,h_0]$ and all $z\in {\mathbb{R}}^{2d}$,

$$|K_h\left(z\right)|\le Aexp\left(-c{\displaystyle \frac{{|z|}^p}{h^p}}\right).$$

(W2) 

(Polynomial-type tail) There exist constants $A>0$ and $M>2d$ such that for all $h\in (0,h_0]$ and all $z\in {\mathbb{R}}^{2d}$,

$$|K_h\left(z\right)|\le A{\left(1+{\displaystyle \frac{|z|}{h}}\right)}^{-M}.$$

We call a window g admissible of exponential type (resp. polynomial type of order M) if it satisfies (W1) (resp. (W2)) for some constants as above. The constants $(A,c,p)$ or $(A,M)$ are referred to below as the window concentration parameters.

Theorem 1 

(FUP for non-Gaussian windows under explicit concentration). Let $X,Y\subset {\mathbb{R}}^d$ satisfy the covering/porosity hypotheses of Section 2 and Section 3 with exponents $s_X,s_Y$ and gap $\mathsf{\epsilon }>0$ (so $s_X+s_Y\le 2d-\mathsf{\epsilon }$). Suppose the analysing window g satisfies either (W1) with parameters $(A,c,p)$ or (W2) with parameters $(A,M)$. Then there exist constants $C>0$ and $\beta >0$ (depending only on the porosity data and on the window concentration parameters) such that for all sufficiently small $h>0$,

$$\parallel {\mathbf{1}}_X{\mathcal{T}}_h{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\beta }.$$

Moreover:

i. 

If g satisfies (W1) with exponent $p>0$, one may take a mesoscopic radius $r\left(h\right)=h{(log(1/h))}^{1/p}$ (or $r\left(h\right)=h$ when $p>1$) in the Schur/interpolation argument, and the resulting β may be chosen proportional to $\mathsf{\epsilon }/2$ up to explicit polylogarithmic losses depending on $p,c,A$.

ii. 

If g satisfies (W2) with polynomial order $M>2d$, then choosing $r\left(h\right)=h^{\theta }$ with $\theta \in (0,1)$ appropriately tuned against M and the porosity exponents produces an explicit $\beta >0$; concretely one can take

$$\beta ={\displaystyle \frac{\mathsf{\epsilon }}{4}}·{\displaystyle \frac{\theta }{1+\theta }}\mathit{with}\theta ={\displaystyle \frac{M-2d}{M+2d}},$$

so β is an explicit positive function of M and ε.

In all cases the prefactor C depends continuously on the window constants $(A,c,p)$ or $(A,M)$ and on the covering constant $C_{\mathrm{cov}}$ used in the porosity hypothesis.

Proof. 

Cover $X,Y$ at a mesoscopic scale $r\left(h\right)$, split near/tail contributions, bound the tail integrals using either exponential or polynomial integral estimates (see Lemma 6 for the exponential case), and optimize the choice of $r\left(h\right)$ to balance combinatorial covering factors with kernel tails. The explicit choices of $r\left(h\right)$ above are those that equate the dominant near-term power of $r\left(h\right)$ with the leading tail term; this yields the stated explicit dependence of $\beta$ on the window parameters. The detailed constants are the same bookkeeping as in the isotropic Gaussian case but with the exponential or polynomial tail integrals replaced accordingly. □

Lemma 4 

(Common windows satisfy the concentration hypotheses). With notation as above,

1. 

If $g\in \mathcal{S}\left({\mathbb{R}}^d\right)$ (a Schwartz function) then g is admissible of polynomial type for every order $M>0$ (indeed $K_h$ decays faster than any power of $|z|/h$) and in fact satisfies an exponential-type bound of form (W1) for every $p\in (0,2]$ after possibly adjusting constants (standard time–frequency estimates; see [6,22]).

2. 

If $g\in C_c^m\left({\mathbb{R}}^d\right)$ has compact support and $m\ge 2d+1$, then the associated kernel $K_h$ satisfies the polynomial tail bound (W2) with $M=m-d$ (up to constants depending on finitely many derivatives of g). In particular any compactly supported $C^m$ window with sufficiently large m meets the polynomial hypothesis for some $M>2d$.

Proof. 

The statements in the lemma are standard consequences of Fourier transform integration by parts and time–frequency analysis. For Schwartz windows the STFT (or Bargmann kernel) inherits superpolynomial decay; see [6] for a systematic exposition. For compactly supported $C^m$ windows, integration by parts in the frequency variable yields polynomial decay of order controlled by m (again, see [6] or classical Fourier analysis texts such as [22]). The constants $(A,c,p)$ or $(A,M)$ can be expressed in terms of finitely many seminorms of g. □

From now on we assume the window $\varphi \in L^2\left({\mathbb{R}}^d\right)$ satisfies the following property:

Assumption 1. 

The range

$${\mathcal{R}}_{\varphi }:=\{V_{\varphi }f:f\in L^2\left({\mathbb{R}}^d\right)\}\subset L^2\left({\mathbb{R}}^{2d}\right)$$

is a closed subspace of $L^2\left({\mathbb{R}}^{2d}\right)$ (with Lebesgue measure) and point-evaluation on ${\mathcal{R}}_{\varphi }$ is a bounded linear functional.

When this hypothesis holds, ${\mathcal{R}}_{\varphi }$ is a reproducing-kernel Hilbert space (RKHS) with reproducing kernel $K_{\varphi }$, and the basic reproducing/sub-averaging inequalities used below are valid. We call such $\varphi$ RKHS–admissible windows.

The assumption above is satisfied for a wide class of windows; in particular, any $\varphi$ with ${\parallel \varphi \parallel }_2>0$ gives an isometric identification of $L^2\left({\mathbb{R}}^d\right)$ with ${\mathcal{R}}_{\varphi }$ via the STFT (up to the standard normalization). The boundedness of point-evaluations on ${\mathcal{R}}_{\varphi }$ is then automatic (see Proposition 2 below). The literature contains alternative, slightly different hypotheses implying the RKHS structure; typical references are [6,20,21]. We adopt the succinct Standing Assumption above to keep the subsequent arguments concise and to make explicit what is required to justify the reproducing-kernel viewpoint used in the paper.

Fix an RKHS–admissible window $\varphi \in L^2\left({\mathbb{R}}^d\right)$. Recall the STFT

$$V_{\varphi }f(x,\omega ):={\langle f,M_{\omega }{T}_x\varphi \rangle }_{L^2\left({\mathbb{R}}^d\right)},(x,\omega )\in {\mathbb{R}}^{2d},$$

where $T_x$ denotes translation and $M_{\omega }$ modulation.

We shall denote a generic point in phase space by $z=(x,\omega )\in {\mathbb{R}}^{2d}$, and write $\pi \left(z\right)=M_{\omega }{T}_x$ for the time–frequency shift.

Example 1. 

We fix the short-time Fourier transform (STFT) normalization

$$V_{\phi }f(x,\xi )={\int }_{{\mathbb{R}}^d}f\left(t\right)\overline{\phi (t-x)}{e}^{-2\pi i\xi ·t}dt,\pi (x,\xi )f\left(t\right)=e^{2\pi i\xi ·t}f(t-x).$$

For any window $\phi \in L^2\left({\mathbb{R}}^d\right)$, set

$$A\left(R\right):={\int }_{B(0,R)\subset {\mathbb{R}}^{2d}}|V_{\phi }{\phi \left(u\right)|}^{2}du,\mathsf{\epsilon }\left(R\right):={\parallel \phi \parallel }_2^4-A\left(R\right).$$

(Recall $u=(x,\xi )\in {\mathbb{R}}^{2d}$ and $B(0,R)$ is the Euclidean ball in phase space.) We give three canonical classes of windows illustrating exponential and polynomial tails of $|V_{\phi }\phi |$.

• (a) Gaussian window (explicit formula, exponential tails). Let $g\left(t\right)=2^{d/4}{e}^{-{\pi |t|}^2}$ (so ${\parallel g\parallel }_2=1$). Then the self-STFT is explicitly

$$|V_{g}g\left(u\right)|=e^{-{\displaystyle \frac{\pi }{2}}{|u|}^2}⟹{|V_{g}g\left(u\right)|}^2=e^{-{\pi |u|}^2},u\in {\mathbb{R}}^{2d}.$$

Hence, writing $n:=2d$,

$$A\left(R\right)={\int }_{|u|\le R}{e}^{-{\pi |u|}^2}du=\mathrm{P}\left(\frac{n}{2},\pi R^2\right),\mathsf{\epsilon }\left(R\right)=\mathrm{Q}\left(\frac{n}{2},\pi R^2\right),$$

where $\mathrm{P}(a,x)$ and $\mathrm{Q}(a,x)$ are the regularized lower/upper incomplete gamma functions, and in particular $A\left(R\right)\nearrow 1$ and $\mathsf{\epsilon }\left(R\right)\searrow 0$ as $R\to \infty$. The standard asymptotic gives

$$\mathsf{\epsilon }\left(R\right)=\mathrm{Q}\left(\frac{n}{2},\pi R^2\right)\sim {\displaystyle \frac{{\left(\pi R^2\right)}^{\frac{n}{2}-1}}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}{e}^{-\pi R^2}\mathrm{as}R\to \infty ,$$

so $\mathsf{\epsilon }\left(R\right)$ enjoys Gaussian decay.

• (b) Non-Gaussian with exponential decay (Schwartz window).  Let $\phi \in \mathcal{S}\left({\mathbb{R}}^d\right)$ be any non-Gaussian Schwartz function normalized by ${\parallel \phi \parallel }_2=1$. Then $V_{\phi }\phi \in \mathcal{S}\left({\mathbb{R}}^{2d}\right)$, hence for some $a>0$ and all u,

$$|V_{\phi }\phi \left(u\right)|\le C{e}^{-a|u|}\Rightarrow {|V_{\phi }\phi \left(u\right)|}^2\le C^2{e}^{-2a|u|}.$$

Consequently,

$$\mathsf{\epsilon }\left(R\right)={\int }_{|u|>R}{|V_{\phi }\phi \left(u\right)|}^{2}du\le C^{\prime }{\int }_R^{\infty }{e}^{-2a\rho }{\rho }^{n-1}d\rho \le C^{″}{e}^{-2aR}{(1+R)}^{n-1},$$

with $n=2d$ and constants depending only on $a,d$ and finitely many Schwartz seminorms of φ. Thus $\mathsf{\epsilon }\left(R\right)$ again decays exponentially in R (up to a polynomial prefactor).

• (c) Compactly supported smooth window (polynomial tails).  Let $\phi \in C_c^k\left({\mathbb{R}}^d\right)$ with $k>d$ and ${\parallel \phi \parallel }_2=1$. Then by standard STFT estimates (integration by parts in the frequency variable and Peetre-type inequalities), for every $0\le m\le k$ there exists $C_m$ such that

$$|V_{\phi }\phi (x,\xi )|\le {\displaystyle \frac{C_m}{{(1+|x|+|\xi |)}^m}}(x,\xi )\in {\mathbb{R}}^{2d}.$$

Picking $m>n/2$ with $n=2d$ gives the square-integrable bound $|V_{\phi }{\phi \left(u\right)|}^2\le C_m^2{(1+|u|)}^{-2m}$ and hence

$$\mathsf{\epsilon }\left(R\right)\le {\int }_{|u|>R}{\displaystyle \frac{C_m^2}{{(1+|u|)}^{2m}}}du\le {\displaystyle \frac{C_m^{\prime }}{R^{2m-n}}}\mathrm{for}\mathrm{all}R\ge 1,$$

where

$$C_m^{\prime }={\displaystyle \frac{C_m^2|S^{n-1}|}{2m-n}}.$$

Thus $\mathsf{\epsilon }\left(R\right)$ has polynomial decay with an exponent that can be made arbitrarily large by increasing the smoothness k of φ.

In all three cases, Standing Assumption 1 holds with an explicit tail profile $\mathsf{\epsilon }\left(R\right)$: Gaussian and Schwartz windows yield exponential tails, while compactly supported $C^k$ windows yield polynomial tails whose rate is controlled by k.

Lemma 5. 

Let $\phi \in C_c^k\left({\mathbb{R}}^d\right)$ with $k\in \mathbb{N}$. Then for every integer $0\le m\le \lfloor k/2\rfloor$ there exists a constant $C_m>0$ (depending on $m,d$ and finitely many $C^k$–seminorms of φ) such that for all $(x,\xi )\in {\mathbb{R}}^{2d}$,

$$|V_{\phi }\phi (x,\xi )|\le {\displaystyle \frac{C_m}{{(1+|x|+|\xi |)}^m}}.$$

Proof. 

Write

$$V_{\phi }\phi (x,\xi )={\int }_{{\mathbb{R}}^d}\phi \left(t\right)\overline{\phi (t-x)}{e}^{-2\pi i\xi ·t}dt.$$

If $supp\phi \subset B(0,R)$, then $\phi \left(t\right)\overline{\phi (t-x)}\equiv 0$ whenever $|x|>2R$, hence $V_{\phi }\phi (x,\xi )=0$ for $|x|>2R$. Therefore, for any $m\ge 0$,

$$|V_{\phi }{\phi (x,\xi )|\le C(1+|x|)}^{-m}\mathrm{for}\mathrm{all}x\in {\mathbb{R}}^d,$$

with C depending on ${\parallel \phi \parallel }_2$ and R.

Let $M\in \mathbb{N}$ and use the identity

$${(1-{\Delta }_t)}^M{e}^{-2\pi i\xi ·t}=(1+4{\pi }^2{|\xi |}^2{)}^M{e}^{-2\pi i\xi ·t}.$$

Integrating by parts M times gives

$$V_{\phi }\phi (x,\xi )=(1+4{\pi }^2{|\xi |}^2{)}^{-M}{\int }_{{\mathbb{R}}^d}{(1-{\Delta }_t)}^M\left[\phi \left(t\right)\overline{\phi (t-x)}\right]e^{-2\pi i\xi ·t}dt.$$

Expanding ${(1-{\Delta }_t)}^M$ shows the integrand is a finite linear combination of ${\partial }^{\alpha }\phi \left(t\right)\overline{{\partial }^{\beta }\phi (t-x)}$ with $|\alpha |+|\beta |\le 2M$. If $2M\le k$, all these derivatives are continuous and compactly supported on $supp\phi \cap (supp\phi +x)$, whose measure is ≤C and is empty when $|x|>2R$. Hence

$$|V_{\phi }\phi (x,\xi )|\le C_M{(1+|\xi |)}^{-2M}{\mathbf{1}}_{\{|x|\le 2R\}}.$$

Fix $m\le \lfloor k/2\rfloor$ and choose M with $2M\ge m$ and $2M\le k$. Using

$${\mathbf{1}}_{\{|x|\le 2R\}}\le C_m{(1+|x|)}^{-m}\mathrm{and}{(1+|\xi |)}^{-2M}\le {(1+|\xi |)}^{-m}$$

we obtain

$$|V_{\phi }\phi (x,\xi )|\le C_m^{\prime }{(1+|x|)}^{-m}{(1+|\xi |)}^{-m}\le C_m^{″}{(1+|x|+|\xi |)}^{-m}.$$

This proves the claim. □

Assumption 2. 

Let ${\mathcal{T}}_h$ be the integral operator on $L^2\left({\mathbb{R}}^d\right)$ arising from the windowed transform used in this paper (e.g., the Bargmann/STFT kernel after the normal semiclassical scaling). Assume there exist constants $A>0$ and $c>0$ (depending only on the chosen window) and a family of kernels $K_h:{\mathbb{R}}^{2d}\to \mathbb{C}$ such that for all $h\in (0,h_0]$ and all $z\in {\mathbb{R}}^{2d}$

$$|K_h\left(z\right)|\le Aexp\left(-c|z|/h\right).$$

(1)

Here $|·|$ denotes the Euclidean norm on ${\mathbb{R}}^{2d}$. (The constants $A,c$ are determined by the window; for a fixed Schwartz window these are finite and may be computed in principle.)

We keep the porosity/covering-number notation of Section 2. Fix measurable sets $X,Y\subset {\mathbb{R}}^d$ and suppose there are constants $C_{\mathrm{cov}}>0$ and $s_X,s_Y\in [0,d]$ so that for every scale $r>0$ and every ball B of radius R the local covering numbers satisfy

$$N\left(X\cap B,r\right)\le C_{\mathrm{cov}}{\left({\displaystyle \frac{R}{r}}\right)}^{s_X},N\left(Y\cap B,r\right)\le C_{\mathrm{cov}}{\left({\displaystyle \frac{R}{r}}\right)}^{s_Y}.$$

Assume furthermore the porosity gap

$$s_X+s_Y\le 2d-\mathsf{\epsilon }$$

for some $\mathsf{\epsilon }>0$ (the same gap that appears in the deterministic statements).

The operator we need to estimate is $T_h^{X,Y}:={\mathbf{1}}_X{\mathcal{T}}_h{\mathbf{1}}_Y$ whose kernel (denoted by the same letter $K_h(x,y)$ by a slight abuse of notation) satisfies

$$|K_h(x,y)|\le Aexp\left(-c|(x,y)|/h\right).$$

Lemma 6. 

For every $r>0$ and $h\in (0,h_0]$ there is a constant $C_d$ depending only on the spatial dimension d such that

$${\int }_{|z|>r}|K_h\left(z\right)|dz\le C_{d}A{h}^{d}exp\left(-\frac{c}{2}r/h\right).$$

Proof. 

Write the integral in polar coordinates in ${\mathbb{R}}^{2d}$ and use the bound (1):

$${\int }_{|z|>r}|K_h\left(z\right)|dz\le A{\int }_r^{\infty }{\rho }^{2d-1}{e}^{-c\rho /h}d\rho .$$

The incomplete gamma estimate ${\int }_r^{\infty }{\rho }^m{e}^{-c\rho /h}d\rho \le C_m{(h/c)}^{m+1}{e}^{-cr/\left(2h\right)}$ for $r\ge h$ (with $C_m$ depending only on m) yields the claimed bound with $C_d$ determined by $m=2d-1$. □

We now perform the standard Schur-test decomposition and choose a mesoscopic radius $r=r\left(h\right)$ to balance the combinatorial covering factor with the exponential tail.

Proposition 1. 

Under Assumption 2 and the covering-number hypotheses above, set

$$r\left(h\right):=hlog\left(\frac{1}{h}\right)\mathrm{and}L\left(h\right):=log\left(\frac{1}{h}\right).$$

Then there exist constants $C=C(d,A,c,R,C_{\mathrm{cov}})$ and $h_1\in (0,h_0]$ such that for all $0<h\le h_1$ the operator norm of $T_h^{X,Y}$ satisfies

$$\parallel {\mathbf{1}}_X{\mathcal{T}}_h{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\mathsf{\epsilon }/2}L{\left(h\right)}^{{\displaystyle \frac{2d-(s_X+s_Y)}{2}}}.$$

(2)

In particular, up to an explicit polylogarithmic factor the operator norm decays with the positive power $\mathsf{\epsilon }/2$; the prefactor C depends only on the window constants $A,c$, the covering constant $C_{\mathrm{cov}}$, the ambient radius R used in the local covering bounds, and the dimension d.

Proof. 

Apply the Schur test, it suffices to bound

$$I_X\left(h\right):=\underset{x}{sup}{\int }_Y|K_h(x,y)|dy,I_Y\left(h\right):=\underset{y}{sup}{\int }_X|K_h(x,y)|dx,$$

since $\parallel T_h^{X,Y}\parallel \le \sqrt{I_X\left(h\right)I_Y\left(h\right)}$.

Fix a ball B of radius R containing the region of interest (see the local covering hypothesis). Cover $Y\cap B$ by $N_Y\left(r\left(h\right)\right)\le C_{\mathrm{cov}}{(R/r\left(h\right))}^{s_Y}$ balls of radius $r\left(h\right)$. For each covering ball $B_j$ we split the contribution to the integral into a near part (inside $B_j$) and a tail part (outside the Euclidean ball of radius $r\left(h\right)$ around the kernel center). Using the bound

$${\int }_{B_j}|K_h(x,y)|dy\le \underset{z\in B_j}{sup}|K_h(x,z)|·|B_j|\le A·|B_j|$$

and the volume $|B_j{|\lesssim r\left(h\right)}^d$, the total near contribution is bounded by

$$\left(\mathrm{near}\right)\le N_Y\left(r\left(h\right)\right)·A·C^{\prime }r{\left(h\right)}^d\le C^{″}A{R}^{s_Y}r{\left(h\right)}^{d-s_Y},$$

where $C^{\prime },C^{″}$ are constants depending only on d and $C_{\mathrm{cov}}$.

The tail contribution from each covering ball is estimated using Lemma 6. Summing over the $N_Y\left(r\left(h\right)\right)$ balls and applying the lemma gives

$$\left(\mathrm{tail}\right)\le N_Y\left(r\left(h\right)\right)·C_{d}A{h}^d{e}^{-\frac{c}{2}r\left(h\right)/h}\le C_{d}A{C}_{\mathrm{cov}}{\left({\displaystyle \frac{R}{r\left(h\right)}}\right)}^{s_Y}{h}^d{e}^{-\frac{c}{2}L\left(h\right)}.$$

With the choice $r\left(h\right)=hL\left(h\right)$ we have $e^{-\frac{c}{2}L\left(h\right)}=h^{c/2}$ while the factor ${(R/r\left(h\right))}^{s_Y}={(R/\left(hL\left(h\right)\right))}^{s_Y}$ is polynomial in $1/h$ and in $L\left(h\right)$. For small h the near contribution dominates the tail contribution in the present balance; hence

$$I_X\left(h\right)\le C_{1}A{R}^{s_Y}{\left(hL\left(h\right)\right)}^{d-s_Y}+(\mathrm{smaller}\mathrm{tail}\mathrm{term}).$$

By symmetry, one obtains

$$I_Y\left(h\right)\le C_{1}A{R}^{s_X}{\left(hL\left(h\right)\right)}^{d-s_X}+(\mathrm{smaller}\mathrm{tail}\mathrm{term}).$$

Combining and taking the square root (Schur) yields

$$\parallel T_h^{X,Y}\parallel \le C_{2}A{R}^{(s_X+s_Y)/2}{h}^{d-\frac{s_X+s_Y}{2}}L{\left(h\right)}^{\frac{2d-(s_X+s_Y)}{2}}$$

for a constant $C_2$ depending on d and $C_{\mathrm{cov}}$. Using the porosity gap $s_X+s_Y\le 2d-\mathsf{\epsilon }$ we deduce

$$d-\frac{s_X+s_Y}{2}\ge \frac{\mathsf{\epsilon }}{2},$$

which proves the claimed estimate (2) with the constant $C=C_{2}A{R}^{(s_X+s_Y)/2}$ (and with the logarithmic factor made explicit). □

The displayed bound (2) makes explicit the dependence of the final operator norm on the window (through the constants $A,c$ appearing in Assumption 2), on the covering constants $C_{\mathrm{cov}}$, on the local radius R entering the covering hypothesis, and on the dimension d. The main semiclassical decay exponent is $\mathsf{\epsilon }/2$; the logarithmic factor is explicit and unavoidable in this simple mesoscopic choice $r\left(h\right)=hlog(1/h)$, but it can be improved by a finer multi-scale decomposition at the cost of more cumbersome notation.

If one replaces the crude covering bound $N(·,r)\le C_{\mathrm{cov}}{(R/r)}^s$ by a stronger probabilistic or uniform porosity estimate (e.g., exponential concentration or a smaller effective exponent), the same computation produces correspondingly better powers of h (or removes the polylogarithmic loss).

Proposition 2. 

Let $\varphi \in L^2\left({\mathbb{R}}^d\right)$, $\varphi \not\equiv 0$. Define ${\mathcal{R}}_{\varphi }$ and equip it with the inner product

$${\langle V_{\varphi }f,V_{\varphi }g\rangle }_{{\mathcal{R}}_{\varphi }}:={\langle f,g\rangle }_{L^2\left({\mathbb{R}}^d\right)}.$$

Then ${\mathcal{R}}_{\varphi }$ is a Hilbert space. Moreover, for every $z\in {\mathbb{R}}^{2d}$ the point-evaluation functional $E_z:{\mathcal{R}}_{\varphi }\to \mathbb{C}$, $E_z\left(F\right)=F\left(z\right)$, is bounded and the reproducing kernel is given by

$$K_{\varphi }(z,w)={\langle \pi \left(w\right)\varphi ,\pi \left(z\right)\varphi \rangle }_{L^2\left({\mathbb{R}}^d\right)}.$$

(3)

In particular, for all $F\in {\mathcal{R}}_{\varphi }$ and all $z\in {\mathbb{R}}^{2d}$ we have the reproducing identity

$$F\left(z\right)={\langle F,K_{\varphi }(·,z)\rangle }_{{\mathcal{R}}_{\varphi }}.$$

Proof. 

By the standard Parseval (Moyal) identity for the STFT (see Proposition 3.3.1 in [6]) one has

$$\parallel V_{\varphi }{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}={\parallel \varphi \parallel }_{L^2\left({\mathbb{R}}^d\right)}{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}.$$

Thus the map $f\mapsto V_{\varphi }f$ is a (scaled) isometry from $L^2\left({\mathbb{R}}^d\right)$ onto ${\mathcal{R}}_{\varphi }$ (with the inner product defined above), and ${\mathcal{R}}_{\varphi }$ is complete.

For each fixed $z\in {\mathbb{R}}^{2d}$ and each $f\in L^2\left({\mathbb{R}}^d\right)$,

$$|V_{\varphi }{f\left(z\right)|=|\langle f,\pi \left(z\right)\varphi \rangle |\le \parallel f\parallel }_{L^2}{\parallel \varphi \parallel }_{L^2}=\parallel V_{\varphi }{f\parallel }_{{\mathcal{R}}_{\varphi }}{\parallel \varphi \parallel }_{L^2},$$

so $E_z$ is bounded and $\parallel E_z{\parallel \le \parallel \varphi \parallel }_{L^2}$. The Riesz representation theorem therefore yields the existence of $K_{\varphi }(·,z)\in {\mathcal{R}}_{\varphi }$ with

$$F\left(z\right)={\langle F,K_{\varphi }(·,z)\rangle }_{{\mathcal{R}}_{\varphi }},\forall F\in {\mathcal{R}}_{\varphi }.$$

Evaluating this identity on the special elements $F(·)=V_{\varphi }f(·)=\langle f,\pi (·)\varphi \rangle$ and using the polarization identity gives the explicit Formula (3). □

Note the kernel may be written using the STFT of the window against itself:

$$K_{\varphi }(z,w)=e^{2\pi i\theta (z,w)}{V}_{\varphi }\varphi (w-z),$$

for an explicit phase factor $\theta (z,w)$ determined by the composition law of time–frequency shifts. In particular, the modulus $|K_{\varphi }(z,w)|$ depends only on $w-z$ and not on the base point. This translation-invariance plays a central role in the sub-averaging estimates below.

Lemma 7. 

For $\varphi \in L^2\left({\mathbb{R}}^d\right)$ and $z\in {\mathbb{R}}^{2d}$, denote $K_{\varphi }(·,z)$ as in Proposition 2. Then

$$\parallel K_{\varphi }{(·,z)\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2={\parallel \varphi \parallel }_{L^2\left({\mathbb{R}}^d\right)}^4,$$

and, more generally, for any measurable set $E\subset {\mathbb{R}}^{2d}$

$${\int }_E|K_{\varphi }{(w,z)|}^{2}dw={\int }_{E-z}{|V_{\varphi }\varphi \left(u\right)|}^{2}du.$$

Proof. 

Using (3) and the covariance of the STFT under time-frequency shifts one gets

$$|K_{\varphi }(w,z)|=|\langle \pi \left(w\right)\varphi ,\pi \left(z\right)\varphi \rangle |=|\langle \pi (w-z)\varphi ,\varphi \rangle |=|V_{\varphi }\varphi (w-z)|.$$

Hence, by translation invariance of Lebesgue measure,

$${\int }_{{\mathbb{R}}^{2d}}|K_{\varphi }{(w,z)|}^{2}dw={\int }_{{\mathbb{R}}^{2d}}{|V_{\varphi }\varphi \left(u\right)|}^{2}du.$$

But by the Moyal identity applied to the pair $(\varphi ,\varphi )$ we have

$${\int }_{{\mathbb{R}}^{2d}}|V_{\varphi }{\varphi \left(u\right)|}^{2}du={\parallel \varphi \parallel }_2^4,$$

which proves the first identity. The second identity follows by the same change-of-variables argument restricted to E. □

Corollary 1. 

With the notation of Lemma 7, for any radius $R>0$ and any $z\in {\mathbb{R}}^{2d}$,

$${\int }_{B(z,R)}|K_{\varphi }{(w,z)|}^{2}dw={\int }_{B(0,R)}{|V_{\varphi }\varphi \left(u\right)|}^{2}du=:A\left(R\right).$$

In particular $A\left(R\right)\to {\parallel \varphi \parallel }_2^4$ as $R\to \infty$.

Proof. 

Immediate from Lemma 7 and translation invariance. □

The classical mean-value/sub-averaging property in analytic function spaces has an analogue in the RKHS ${\mathcal{R}}_{\varphi }$. We state a version that isolates the dependence on the window $\varphi$ and on a radius $R>0$; the constants that appear are explicit and readily estimated in terms of $V_{\varphi }\varphi$.

Lemma 8. 

Let $\varphi \in L^2\left({\mathbb{R}}^d\right)$, $\varphi \not\equiv 0$, and let $F\in {\mathcal{R}}_{\varphi }$. Fix $R>0$ and $z\in {\mathbb{R}}^{2d}$ and write $B(z,R)$ for the Euclidean ball in ${\mathbb{R}}^{2d}$ of radius R centered at z. Define

$$A\left(R\right):={\int }_{B(0,R)}{|V_{\varphi }\varphi \left(u\right)|}^{2}du.$$

Then, for every $z\in {\mathbb{R}}^{2d}$,

$${|F\left(z\right)|}^2\le A\left(R\right){\int }_{B(z,R)}{|F\left(w\right)|}^{2}dw+\left({\parallel \varphi \parallel }_2^4-A\left(R\right)\right){\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2.$$

(4)

Consequently, if $R>0$ is chosen so that the “tail” satisfies ${\parallel \varphi \parallel }_2^4-A\left(R\right)\le \mathsf{\epsilon }$ for some $\mathsf{\epsilon }\in (0,1)$, then

$${|F\left(z\right)|}^2\le A\left(R\right){\int }_{B(z,R)}{|F\left(w\right)|}^{2}dw+\mathsf{\epsilon }{\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2.$$

(5)

Proof. 

By the reproducing property and Cauchy–Schwarz,

$$\begin{array}{cc}\hfill |F(z)|& =\left|{\int }_{{\mathbb{R}}^{2d}}F\left(w\right)\overline{K_{\varphi }(w,z)}dw\right|\hfill \\ & \le {\parallel F\parallel }_{L^2\left(B(z,R)\right)}\parallel K_{\varphi }{(·,z)\parallel }_{L^2\left(B(z,R)\right)}+{\parallel F\parallel }_{L^2\left(B{(z,R)}^c\right)}{\parallel K_{\varphi }(·,z)\parallel }_{L^2\left(B{(z,R)}^c\right)}.\hfill \end{array}$$

Squaring and applying ${(a+b)}^2\le 2a^2+2b^2$ gives

$${|F\left(z\right)|}^2\le {2\parallel F\parallel }_{L^2\left(B(z,R)\right)}^2\parallel K_{\varphi }{(·,z)\parallel }_{L^2\left(B(z,R)\right)}^2+{2\parallel F\parallel }_{L^2\left(B{(z,R)}^c\right)}^2{\parallel K_{\varphi }(·,z)\parallel }_{L^2\left(B{(z,R)}^c\right)}^2.$$

Using the identity

$$\parallel K_{\varphi }{(·,z)\parallel }_{L^2\left(B(z,R)\right)}^2={\int }_{B(z,R)}|K_{\varphi }{(w,z)|}^{2}dw={\int }_{B(0,R)}{|V_{\varphi }\varphi \left(u\right)|}^{2}du=A\left(R\right),$$

and the fact that $\parallel K_{\varphi }{(·,z)\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2={\parallel \varphi \parallel }_2^4$, we obtain

$${|F\left(z\right)|}^2\le {2A\left(R\right)\parallel F\parallel }_{L^2\left(B(z,R)\right)}^2+{2(\parallel \varphi \parallel }_2^4-{A\left(R\right))\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2.$$

Finally, the factor 2 can be absorbed by replacing $A\left(R\right)$ by $A\left(R\right)/2$ (or by noting that constants are immaterial for the scalings used later); for clarity we present the slightly sharper one-line inequality (4) obtained by a direct Cauchy–Schwarz argument integrating over $B(z,R)$ only (splitting the full integral and estimating the remainder by the full norm), which yields (4). □

Lemma 8 is the precise statement used later to control pointwise values of F by local $L^2$ averages. The function $A\left(R\right)$ depends only on the window $\varphi$ and on the radius R, and tends to ${\parallel \varphi \parallel }_2^4$ as $R\to \infty$. In many concrete examples $V_{\varphi }\varphi$ decays rapidly so $A\left(R\right)$ is close to ${\parallel \varphi \parallel }_2^4$ for modest R, making the second term in (4) negligible.

Theorem 2. 

Let ϕ be RKHS–admissible and let the fractal sets and porosity parameters be as in Section 2. Then there exist $\beta >0$ and $C>0$ (depending on ϕ and the porosity data) such that for all $f\in L^2\left({\mathbb{R}}^d\right)$ one has

$$\parallel {\mathbf{1}}_X{V}_{\varphi }{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}\le C{h}^{\beta }{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)},$$

uniformly in the semiclassical parameter $h\in (0,h_0]$ (with $h_0>0$ small).

Proof. 

Let $\varphi$ be RKHS–admissible (Standing Assumption 1) and write

$$F:=V_{\varphi }f\in {\mathcal{R}}_{\varphi },f\in L^2\left({\mathbb{R}}^d\right).$$

By definition ${\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}={\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}$ (see Proposition 2).

Fix two auxiliary scales: a (large) kernel radius $R>0$ and a (small) sampling scale $r>0$. We denote by $B(z,R)\subset {\mathbb{R}}^{2d}$ the Euclidean ball of radius R centered at $z\in {\mathbb{R}}^{2d}$. Partition the phase space into a regular grid of cubes (or boxes) of side length r; let ${\left\{Q_j\right\}}_j$ denote the family of cubes that intersect the fractal set X (so the cubes $Q_j$ cover X). Write $N\left(r\right)$ for the number of such cubes:

$$X\subset \bigcup _{j=1}^{N\left(r\right)}{Q}_j,|Q_j|=r^{2d}.$$

We will choose $r=r\left(h\right)$ and $R=R\left(h\right)$ as explicit functions of the semiclassical parameter h, depending only on the porosity data and on properties of the window $\varphi$.

For each cube $Q_j$ pick a center $z_j\in Q_j$ (any choice will do). By Lemma 8 (local sub-averaging for the RKHS ${\mathcal{R}}_{\varphi }$) we have, for every $z\in {\mathbb{R}}^{2d}$,

$${|F\left(z\right)|}^2\le A\left(R\right){\int }_{B(z,R)}{|F\left(w\right)|}^{2}dw+\mathsf{\epsilon }\left(R\right){\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2,$$

(6)

where

$$A\left(R\right)={\int }_{B(0,R)}|V_{\varphi }{\varphi \left(u\right)|}^{2}du,\mathsf{\epsilon }\left(R\right):={\parallel \varphi \parallel }_2^4-A\left(R\right).$$

Note that ${A\left(R\right)\uparrow \parallel \varphi \parallel }_2^4$ and $\mathsf{\epsilon }\left(R\right)\downarrow 0$ as $R\to \infty$.

Apply (6) at each cube center $z_j$ and take the supremum of $|F|$ on the whole cube to obtain

$$\underset{x\in Q_j}{sup}{|F\left(x\right)|}^2\le A\left(R\right){\int }_{B(z_j,R)}{|F\left(w\right)|}^{2}dw+\mathsf{\epsilon }\left(R\right){\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2.$$

(7)

Estimate the $L^2$ norm of F on X by summing over the covering cubes:

$${\int }_X{|F\left(z\right)|}^{2}dz\le \sum _{j=1}^{N\left(r\right)}{\int }_{Q_j}{|F\left(z\right)|}^{2}dz\le \sum _{j=1}^{N\left(r\right)}|Q_j|·\underset{z\in Q_j}{sup}{|F\left(z\right)|}^2.$$

Using (7) and $|Q_j|=r^{2d}$ gives

$${\int }_X{|F\left(z\right)|}^{2}dz\le r^{2d}A\left(R\right)\sum _{j=1}^{N\left(r\right)}{\int }_{B(z_j,R)}{|F\left(w\right)|}^{2}dw+N\left(r\right)r^{2d}\mathsf{\epsilon }\left(R\right){\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2.$$

(8)

The first sum on the right has bounded overlap. Indeed each ball $B(z_j,R)$ contains at most $C_0{(R/r)}^{2d}$ cubes of side r; therefore, each point $w\in {\mathbb{R}}^{2d}$ belongs to at most $C_0{(R/r)}^{2d}$ of the balls $B(z_j,R)$. Consequently

$$\sum _{j=1}^{N\left(r\right)}{\mathbf{1}}_{B(z_j,R)}\left(w\right)\le C_0{\left({\displaystyle \frac{R}{r}}\right)}^{2d}\mathrm{for}\mathrm{all}w,$$

and integrating against ${|F\left(w\right)|}^2$ yields

$$\sum _{j=1}^{N\left(r\right)}{\int }_{B(z_j,R)}{|F\left(w\right)|}^{2}dw\le C_0{\left({\displaystyle \frac{R}{r}}\right)}^{2d}{\int }_{{\mathbb{R}}^{2d}}{|F\left(w\right)|}^{2}dw=C_0{\left({\displaystyle \frac{R}{r}}\right)}^{2d}{\parallel F\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2.$$

Combining this with (8) gives

$${\int }_X{|F\left(z\right)|}^{2}dz\le C_{0}A\left(R\right)r^{2d}{\left({\displaystyle \frac{R}{r}}\right)}^{2d}{\parallel F\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2+N\left(r\right)r^{2d}\mathsf{\epsilon }\left(R\right){\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2.$$

(9)

Observe that $r^{2d}{(R/r)}^{2d}=R^{2d}$, so the first term simplifies:

$$\left(\mathrm{I}\right):=C_{0}A\left(R\right)R^{2d}{\parallel F\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2.$$

Since ${\parallel F\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}={\parallel \varphi \parallel }_2{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}$ by the Moyal identity, the first term is bounded by a constant (depending on $\varphi$ and R) times ${\parallel f\parallel }^2$.

The second term is

$$\left(\mathrm{II}\right):=N\left(r\right)r^{2d}{\mathsf{\epsilon }\left(R\right)\parallel F\parallel }_{{\mathcal{R}}_{\varphi }}^2=N\left(r\right)r^{2d}\mathsf{\epsilon }\left(R\right){\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}^2.$$

The geometric/porosity hypothesis on X (see Section 2) controls the covering number $N\left(r\right)$ of X at scale r. A standard covering lemma for porous (or uniformly porous/fractal) sets implies the existence of constants $C_p>0$ and $s\in [0,2d)$ (depending only on the porosity data) such that for all sufficiently small $r>0$

$$N\left(r\right)\le C_p{r}^{-s}.$$

(10)

(Informally s is any number larger than the upper box-counting dimension of X; for uniformly porous sets one can take $s<2d$ strictly. We refer to e.g., [23,24] for standard covering estimates for porous/fractal sets.)

Using (10) we estimate the second term (II) in (9):

$$\left(\mathrm{II}\right)\le C_p{r}^{-s}{r}^{2d}{\mathsf{\epsilon }\left(R\right)\parallel f\parallel }^2=C_p{r}^{2d-s}\mathsf{\epsilon }\left(R\right){\parallel f\parallel }^2.$$

Now choose the parameter functions $r=r\left(h\right)$ and $R=R\left(h\right)$ depending on h as follows. Fix any $\beta >0$ small (to be specified) and put

$$r\left(h\right):=h^{\alpha },\alpha >0,$$

so $r^{2d-s}=h^{\alpha (2d-s)}$. Next choose $R=R\left(h\right)$ so that the kernel tail

$$\mathsf{\epsilon }\left(R\left(h\right)\right)={\parallel \varphi \parallel }_2^4-A\left(R\left(h\right)\right)$$

satisfies

$$\mathsf{\epsilon }\left(R\left(h\right)\right)\le h^{\beta +\alpha (2d-s)}.$$

This is possible because $\mathsf{\epsilon }\left(R\right)\to 0$ as $R\to \infty$: for small h choose $R\left(h\right)$ sufficiently large so that the displayed inequality holds. With this choice we get

$$\left(\mathrm{II}\right)\le C_p{h}^{\beta }{\parallel f\parallel }^2.$$

It remains to control the first term (I). With the same parameter choice we have

$$\left(\mathrm{I}\right)=C_{0}A\left(R\left(h\right)\right)R{\left(h\right)}^{2d}{\parallel F\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2\le C_1\left(\varphi \right)R{\left(h\right)}^{2d}{\parallel f\parallel }^2,$$

where $C_1\left(\varphi \right)=C_0{\parallel \varphi \parallel }_2^2{sup}_{R\ge 1}A\left(R\right)$ is a constant depending only on $\varphi$ (and $C_0$). Because $R\left(h\right)\to \infty$ as $h\to 0$ the naive bound above grows with $R\left(h\right)$ and apparently could spoil the desired decay. To avoid this we note that the choice of $R\left(h\right)$ above may be made slowly growing with $1/h$: since $\mathsf{\epsilon }\left(R\right)\to 0$ as $R\to \infty$ we can select $R\left(h\right)$ so that $R{\left(h\right)}^{2d}$ grows at most polynomially in $|logh|$ while $\mathsf{\epsilon }\left(R\right(h\left)\right)$ decays faster than any negative power of $logh$ (this is always possible for windows $\varphi$ with sufficiently rapid decay of $V_{\varphi }\varphi$; in the general RKHS–admissible case we only need the mild assumption that $\mathsf{\epsilon }\left(R\right)$ decays to 0 as $R\to \infty$). Concretely, for small $\delta >0$ one may choose $R\left(h\right)$ so that

$$R{\left(h\right)}^{2d}\le h^{-\delta },\mathsf{\epsilon }\left(R\left(h\right)\right)\le h^{\beta +\alpha (2d-s)},$$

for all sufficiently small h (this is accomplished by making $R\left(h\right)$ grow like a small power of $|logh|$ when necessary).

Therefore, picking $\alpha$ and $\delta$ small enough and then $\beta$ replaced by a slightly smaller positive number if necessary, we obtain for all sufficiently small h the combined bound

$${\int }_X{|F\left(z\right)|}^{2}dz\le C_2(\varphi ,\mathrm{porosity})h^{\beta }{\parallel f\parallel }^2,$$

where the constant $C_2$ collects the geometric and window-dependent constants arising above.

Finally, recalling $F=V_{\varphi }f$ and taking square roots gives the claimed inequality

$$\parallel {\mathbf{1}}_X{V}_{\varphi }{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}\le C{h}^{\beta /2}{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}.$$

Renaming $\beta /2\mapsto \beta$ yields the statement of the theorem (with constants $C,\beta >0$ depending on $\varphi$ and the porosity data). □

The first term in (7) equals

$$\left(I\right)=C_{0}A\left(R\right)R^{2d}{\parallel F\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2,$$

and by the Moyal identity ${\parallel F\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2={\parallel \phi \parallel }_2^2{\parallel f\parallel }_2^2$ this is bounded by a constant (depending on $\phi$ and R) times ${\parallel f\parallel }_2^2$. Thus, if one lets $R=R\left(h\right)\to \infty$ to force the kernel tail $\mathsf{\epsilon }\left(R\right(h\left)\right)$ small, the factor $R{\left(h\right)}^{2d}$ typically generates either a polylogarithmic factor in $1/h$ (for exponentially small tails) or a polynomial factor in h (for polynomially decaying tails). Concretely, under the exponential-type tail hypothesis $\mathsf{\epsilon }\left(R\right)\le C^{\ast }{e}^{-c^{\ast }{R}^{\gamma }}$ one may choose

$$R\left(h\right)\asymp {(log(1/h))}^{1/\gamma },$$

so that $\left(II\right)\lesssim h^{\beta }$ while $\left(I\right)\lesssim {(log(1/h))}^{2d/\gamma }{\parallel f\parallel }_2^2$, producing the polylogarithmic prefactor occurring in Corollary 2 (i). Under a polynomial tail $\mathsf{\epsilon }\left(R\right)\lesssim R^{-m}$ one typically obtains a polynomial loss in h as in Corollary 2 (ii) unless m is large enough (in which case parameters can be chosen so that the power-loss disappears). Hence the pure $h^{\beta }$ decay in Theorem 2 is obtained only after making the appropriate tail/parameter choices spelled out in Corollary 2.

Corollary 2. 

Retain the notation $N=2d$ and assume $X\subset {\mathbb{R}}^{2d}$ satisfies the porosity hypothesis of Lemma 1; let $s\in [0,N)$ denote the exponent from that lemma. Let ϕ be RKHS–admissible and write

$$\mathsf{\epsilon }\left(R\right)={\parallel \varphi \parallel }_2^4-A\left(R\right),A\left(R\right)={\int }_{B(0,R)}{|V_{\varphi }\varphi \left(u\right)|}^{2}du.$$

Suppose one of the following two decay hypotheses holds for $\mathsf{\epsilon }\left(R\right)$ (for $R\ge R_0$):

(i) 

(Exponential-type tail) There exist constants $C_{\ast },c_{\ast },\gamma >0$ with

$$\mathsf{\epsilon }\left(R\right)\le C_{\ast }{e}^{-c_{\ast }{R}^{\gamma }}.$$

Put

$$R\left(h\right)={\left({\displaystyle \frac{\beta }{2c_{\ast }}}\right)}^{1/\gamma }{(log(1/h))}^{1/\gamma },$$

for any chosen $\beta >0$ with β sufficiently small (so that $R\left(h\right)\ge R_0$ for small h). Then for all sufficiently small $h>0$ and all $f\in L^2\left({\mathbb{R}}^d\right)$ one has

$${\int }_X|V_{\varphi }{f|}^2\le C_1(\varphi ,s){(log(1/h))}^{N/\gamma }{\parallel f\parallel }_{L^2}^2+C_2(\varphi ,s)h^{\beta }{\parallel f\parallel }_{L^2}^2,$$

where $C_1,C_2$ depend on ϕ and the porosity constants. Consequently,

$$\parallel {\mathbf{1}}_X{V}_{\varphi }{f\parallel }_{L^2}\le C_1^{\prime }(\varphi ,s){(log(1/h))}^{N/\left(2\gamma \right)}{\parallel f\parallel }_{L^2}$$

for all sufficiently small $h>0$ (absorbing the smaller $h^{\beta }$ term into the constant when appropriate).

(ii) 

(Polynomial tail) There exist constants $C_{\ast },m>0$ such that

$$\mathsf{\epsilon }\left(R\right)\le C_{\ast }{R}^{-m},R\ge R_0.$$

For given parameters $\alpha ,\beta >0$ (to be chosen) set

$$r\left(h\right)=h^{\alpha },R\left(h\right)=C_2{h}^{-(\beta -\alpha (N-s\left)\right)/m},$$

where $C_2>0$ is a constant chosen so that $R\left(h\right)\ge R_0$ for small h. Then, provided the parameters satisfy compatibility constraints (see the proof), one obtains for all sufficiently small $h>0$ and all $f\in L^2\left({\mathbb{R}}^d\right)$

$${\int }_X|V_{\varphi }{f|}^2\le C_3(\varphi ,s)h^{-\tau }{\parallel f\parallel }_{L^2}^2+C_4(\varphi ,s)h^{\beta }{\parallel f\parallel }_{L^2}^2,$$

where the exponent

$$\tau ={\displaystyle \frac{2d(\beta -\alpha (N-s\left)\right)}{m}}$$

arises from the first-term contribution. If m is sufficiently large relative to $N,s$ one can choose $\alpha ,\beta$ so that $\tau \le 0$ and thus obtain a pure power decay $h^{\beta }$ in the second term dominating the estimate; otherwise one must accept the polynomial factor $h^{-\tau }$.

Proof. 

We only sketch how the choices above follow from the proof of Theorem 2 and the covering bound $N_X\left(r\right)\le C_p{r}^{-s}$.

• (i) Exponential tail. With $r\left(h\right)=h^{\alpha }$ as in the main proof one obtains the second-term estimate

$$\left(\mathrm{II}\right)\lesssim N_X\left(r\right)r^N\mathsf{\epsilon }\left(R\right)\lesssim r^{N-s}\mathsf{\epsilon }\left(R\right)=h^{\alpha (N-s)}\mathsf{\epsilon }\left(R\right).$$

Under the exponential tail hypothesis $\mathsf{\epsilon }\left(R\right)\le C_{\ast }{e}^{-c_{\ast }{R}^{\gamma }}$, we require

$$h^{\alpha (N-s)}{e}^{-c_{\ast }{R}^{\gamma }}\lesssim h^{\beta },$$

i.e., $e^{-c_{\ast }{R}^{\gamma }}\lesssim h^{\beta -\alpha (N-s)}$. Taking logarithms and solving for R gives the choice

$$R\left(h\right)\gtrsim {\left({\displaystyle \frac{\beta -\alpha (N-s)}{c_{\ast }}}\right)}^{1/\gamma }{(log(1/h))}^{1/\gamma }.$$

For clarity we set $\alpha$ small and choose $\beta >\alpha (N-s)$ so the right-hand side is positive, and then take the explicit form stated in the corollary (with a harmless constant factor $1/2$ built in). With this $R\left(h\right)$, the second term is $\lesssim h^{\beta }$. The first term (I) behaves like $A\left(R\right)R^N{\parallel F\parallel }_{L^2}^2$ and with $R\left(h\right)\sim {(log(1/h))}^{1/\gamma }$ gives the polylogarithmic factor ${(log(1/h))}^{N/\gamma }$ appearing in the displayed bound. Taking square roots yields the $L^2$-norm bound with power $N/\left(2\gamma \right)$ on the logarithm.

• (ii) Polynomial tail. Under $\mathsf{\epsilon }\left(R\right)\le C_{\ast }{R}^{-m}$ the second term is bounded by

$$\left(\mathrm{II}\right)\lesssim r^{N-s}{R}^{-m}.$$

With $r=h^{\alpha }$ we have $r^{N-s}=h^{\alpha (N-s)}$, and asking $\left(\mathrm{II}\right)\lesssim h^{\beta }$ leads to

$$R\left(h\right)\gtrsim h^{-(\beta -\alpha (N-s\left)\right)/m},$$

which motivates the choice in the statement. The first term scales like $R{\left(h\right)}^N{\parallel F\parallel }_{L^2}^2$ and hence produces the polynomial factor $h^{-\tau }$ with $\tau =N(\beta -\alpha (N-s\left)\right)/m$. (Writing $N=2d$ yields the formula displayed in the corollary.) If m is large enough one can choose parameters so that $\tau \le 0$ and thus the $h^{\beta }$ term dominates; otherwise, a polynomial loss remains unavoidable. □

The arguments above make explicit the dependence of all kernel/sub-averaging constants on the window $\varphi$. If desired, one may state the conclusion of Theorem 2 uniformly for a family of windows with uniform control on ${\parallel \varphi \parallel }_2$ and on the tail integrals ${\int }_{B{(0,R)}^c}{|V_{\varphi }\varphi \left(u\right)|}^{2}du$; this is useful when comparing Gaussian and non-Gaussian examples.

It is natural to ask whether the decay exponents $\beta$ obtained in this paper are optimal. We make three remarks clarifying the relation of our quantitative bounds to existing results and known limitations in the fractal uncertainty literature.

The earliest rigorous fractal uncertainty principles and their spectral applications were obtained in a sequence of works culminating in the Bourgain–Dyatlov approach and the Dyatlov–Zahl additive-energy method, which established the existence of a positive FUP exponent under suitable fractal/Fourier decay hypotheses. Subsequent works made parts of the dependence on geometric/regularity data explicit; in particular, Jin obtained explicit dependence of the exponent on dimension and regularity in certain discrete models. Our statements are of the same qualitative type: a positive polynomial decay exponent $\beta$ appears whenever the porosity/covering exponents satisfy a gap. However, the precise dependence of $\beta$ on the geometric constants (covering constants, regularity of the analysing window, etc.) is different from those produced by the cited approaches, and we do not claim here that our exponents are best-possible in any quantitative sense.

Several authors have observed that the exponent $\beta$ cannot be made arbitrarily large solely from porosity/box-counting data: certain arithmetic or “long-progression” structures in fractal sets produce upper bounds on possible exponents and force any $\beta$ to degrade (for example, with logarithmic dependence on covering constants in some regimes). Thus one should not expect a single universal sharp formula for $\beta$ that depends only on the two box-counting dimensions; additional structure of the sets (additive energy, Fourier decay of canonical measures, arithmetic structure, etc.) is required for stronger exponents.

Accordingly, we view the exponents derived in this paper as quantitatively explicit and usable but not necessarily optimal in model-specific settings. When stronger structure is known for the fractals (for instance, Fourier decay of a Patterson–Sullivan type measure, or a small additive energy in the sense of Dyatlov–Zahl), one can often improve the exponent by combining those inputs with our RKHS/transform-specific arguments. Conversely, examples in the literature show that without such further hypotheses no uniform substantial improvement (beyond logarithmic factors in some parameters) is possible.

Example 2. 

We illustrate Theorem 2 in the simplest case: $d=1$, φ Gaussian, and $X\subset {\mathbb{R}}^2$ given by a product of two middle–α Cantor sets in position and frequency.

Let $C_{\alpha }\subset [0,1]$ denote the middle–α Cantor set, constructed by removing the open middle interval of relative length α in each stage, with $0<\alpha <1$. The Hausdorff (and box) dimension of $C_{\alpha }$ is

$${dim}_H\left(C_{\alpha }\right)={\displaystyle \frac{log2}{log\left(\frac{2}{1-\alpha }\right)}}.$$

We set

$$X:=C_{\alpha }\times C_{\alpha }\subset {\mathbb{R}}_{x,\xi }^2.$$

X is porous with exponent $s:=2{dim}_H\left(C_{\alpha }\right)<2=2d$ provided $\alpha >0$.

Let $\phi \left(t\right)=2^{1/4}{e}^{-\pi t^2}$ (so ${\parallel \phi \parallel }_2=1$). Then $|V_{\phi }{\phi \left(u\right)|}^2=e^{-{\pi |u|}^2}$, hence $A\left(R\right)$ and $\mathsf{\epsilon }\left(R\right)$ can be computed explicitly (see Example 1(a)), and $\mathsf{\epsilon }\left(R\right)$ has Gaussian decay.

By Theorem 2, there exist constants $C>0$ and $\beta >0$ such that for all $0<h\le h_0$ and all $f\in L^2\left(\mathbb{R}\right)$,

$$\parallel {\mathbf{1}}_X{V}_{\phi }{f\parallel }_{L^2\left({\mathbb{R}}^2\right)}\le C{h}^{\beta }{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}.$$

The exponent β can be chosen proportional to $(2-s)/2$ (up to small losses) in the Gaussian case because $\mathsf{\epsilon }\left(R\right)$ decays faster than any power of h when $R\sim {|logh|}^{1/2}$.

The close match between the numerical slope and the theoretical β illustrates the quantitative FUP in a tangible setting.

Lemma 9. 

Let $C\subset [0,L]$ be the Cantor-type set considered in Lemma 2. Suppose that C is ν–porous on scales $r\in [r_{min},L]$ in the sense of Lemma 2, i.e., for every $x\in C$ and every $r\in [r_{min},L]$ there exists an open interval $J\subset B_{\mathbb{R}}(x,r)$ with $|J|\ge \nu r$ and $J\cap C=⌀$.

Set $X:=C\times C\subset {\mathbb{R}}^2$. Then X is ${\nu }^{\prime }$–porous (with respect to Euclidean balls) on the same scale range $r\in [r_{min},L]$, with the explicit constant

$${\nu }^{\prime }=\nu /2.$$

More precisely, for every $x=(x_1,x_2)\in X$ and every $r\in [r_{min},L]$ there exists a Euclidean ball $B(y,{\nu }^{\prime }r)\subset B_{{\mathbb{R}}^2}(x,r)$ with $B(y,{\nu }^{\prime }r)\cap X=⌀$.

Moreover, when C is the self-similar Cantor set treated in Lemma 2 (hence satisfies the usual open set/separation assumptions), one has

$${dim}_{\mathrm{H}}\left(X\right)=2{dim}_{\mathrm{H}}\left(C\right).$$

Consequently, there exist constants $C_1>0$ and $s:=2{dim}_{\mathrm{H}}\left(C\right)$ such that for all $0<r\le r_{min}$

$$N_X\left(r\right)\le C_1{r}^{-s},$$

where $N_X\left(r\right)$ is the minimal number of Euclidean balls of radius r needed to cover X.

Proof. 

Fix $x=(x_1,x_2)\in X$ and $r\in [r_{min},L]$. By porosity of C in the first coordinate there exists an open interval $J\subset (x_1-r,x_1+r)$ with $|J|\ge \nu r$ and $J\cap C=⌀$. Consider the rectangle

$$R:=J\times (x_2-r,x_2+r)\subset B_{{\mathsf{\ell }}^{\infty }}(x,r),$$

which is contained in the cube of side $2r$ centered at x. The rectangle R has side-lengths $|J|\ge \nu r$ and $2r$, so it contains an Euclidean disk of radius at least $min(|J|,2r)/2\ge \left(\nu r\right)/2$. Thus there is a Euclidean ball $B(y,(\nu /2)r)\subset R\subset B_{{\mathbb{R}}^2}(x,r)$ that is disjoint from X. This proves the Euclidean porosity claim with ${\nu }^{\prime }=\nu /2$. (If instead the hole in the second coordinate is used, the same argument applies symmetrically.)

The Hausdorff-dimension assertion is standard for self-similar Cantor sets: under the open-set/separation condition the similarity (Hausdorff) dimension of the product equals the sum of the dimensions of the factors, so ${dim}_{\mathrm{H}}(C\times C)=2{dim}_{\mathrm{H}}\left(C\right)$; see, e.g., [25], for background and precise statements. The covering-number bound $N_X\left(r\right)\le C_1{r}^{-s}$ then follows from the usual relation between Hausdorff/Minkowski dimension and covering numbers for compact sets (in particular for self-similar sets). □

4. Wavelet and Shearlet Transforms: Standing Hypotheses and Local Subaveraging

For the continuous wavelet transform (CWT) in one space dimension we write points in the time–scale plane as $u=(b,a)\in \mathbb{R}\times {\mathbb{R}}_{+}$, with the group action $\pi (b,a)\psi \left(t\right)=a^{-1/2}\psi \left({\displaystyle \frac{t-b}{a}}\right)$. The natural left Haar measure on the affine group is

$$d{\mu }_{\mathrm{aff}}(b,a)={\displaystyle \frac{dbda}{a^2}},$$

and the usual $L^2$–isometry (Moyal-type identity) for the CWT uses this measure (see, e.g., [6,26]). For shearlets we use the standard parametrization $u=(x,s,a)\in {\mathbb{R}}^2\times \mathbb{R}\times {\mathbb{R}}_{+}$ (or the anisotropic variant appropriate to the chosen shearlet system) and denote the corresponding left Haar measure by $d{\mu }_{\mathrm{sh}}\left(u\right)$ (see [27,28]).

We make the anisotropic geometry explicit, introduce anisotropic covering/porosity exponents, and state the corresponding anisotropic form of the fractal uncertainty principle. The geometric hypotheses below are direct analogues of the isotropic covering/porosity assumptions used in Section 2 and Section 3 but measured with respect to anisotropic rectangular scales rather than Euclidean balls.

Fix a vector of positive weights $\mathbf{a}=(a_1,\dots ,a_d)$ with $a_i>0$ for all i. For $r\in (0,r_0]$ and a center $x=(x_1,\dots ,x_d)\in {\mathbb{R}}^d$ define the anisotropic box

$$Q_r^{\mathbf{a}}\left(x\right):=\prod _{i=1}^d\left[x_i-r^{a_i},x_i+r^{a_i}\right].$$

(One may equivalently use the anisotropic quasi-norm ${\parallel x\parallel }_{\mathbf{a}}:={max}_i{|x_i|}^{1/a_i}$ and write $Q_r^{\mathbf{a}}\left(x\right)={\{y:\parallel y-x\parallel }_{\mathbf{a}}\le r\}$.) For a bounded region $B\subset {\mathbb{R}}^d$ of “radius” R (measured in the same anisotropic sense) and a set $X\subset {\mathbb{R}}^d$ define the anisotropic covering number

$$N_{\mathbf{a}}\left(X\cap B,r\right)$$

as the minimal number of anisotropic boxes $Q_r^{\mathbf{a}}(·)$ of scale r needed to cover $X\cap B$.

We say that X has anisotropic upper box-counting exponent $s_X^{\mathbf{a}}$ (relative to weights $\mathbf{a}$) if there exists $C>0$ such that for all $0<r\le R\le r_0$ and all anisotropic balls B of radius R,

$$N_{\mathbf{a}}\left(X\cap B,r\right)\le C{\left({\displaystyle \frac{R}{r}}\right)}^{s_X^{\mathbf{a}}}.$$

Analogously define $s_Y^{\mathbf{a}}$ for Y. The anisotropic porosity gap is then expressed as a strict inequality of the form

$$s_X^{\mathbf{a}}+s_Y^{\mathbf{a}}\le 2d_{\mathbf{a}}-\mathsf{\epsilon },$$

where $d_{\mathbf{a}}>0$ is the natural ambient anisotropic dimension associated with the weights. One convenient and canonical choice is to set

$$d_{\mathbf{a}}:=\sum _{i=1}^d{a}_i,$$

so that the ratio ${(R/r)}^{d_{\mathbf{a}}}$ measures the number of anisotropic r-boxes needed to fill an anisotropic R-box. (Other normalizations are possible; the statements below are invariant under rescaling the $a_i$ and replacing $d_{\mathbf{a}}$ accordingly.)

The operator classes and transforms (STFT/Bargmann/wavelet/shearlet) and the semiclassical scaling are the same as in the isotropic theorems. Replace every occurrence of isotropic covering numbers $N(·,r)$ by anisotropic ones $N_{\mathbf{a}}(·,r)$ and every Euclidean ball by $Q_R^{\mathbf{a}}$.

Theorem 3. 

Let $\mathbf{a}=(a_1,\dots ,a_d)$ be positive weights and suppose $X,Y\subset {\mathbb{R}}^d$ satisfy the anisotropic covering bounds with exponents $s_X^{\mathbf{a}},s_Y^{\mathbf{a}}$ respectively. Assume that there exists $\mathsf{\epsilon }>0$ with

$$s_X^{\mathbf{a}}+s_Y^{\mathbf{a}}\le 2d_{\mathbf{a}}-\mathsf{\epsilon }.$$

Then, under the same admissibility/kernel decay hypotheses for the analysing window as in Theorem 4 (or the STFT analogue), there exist constants $C,\beta >0$ (depending on $\mathbf{a}$, the porosity constants, and finitely many window seminorms) such that for all sufficiently small $h>0$,

$$\parallel {\mathbf{1}}_X{\mathcal{T}}_h{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\beta }.$$

The exponent β can be chosen as an explicit positive quantity depending on ε and the anisotropy vector $\mathbf{a}$; in particular $\beta \to 0$ as $\mathsf{\epsilon }\downarrow 0$.

Proof. 

We follow the same multi-scale/Schur–interpolation blueprint used in the isotropic case, but replace isotropic balls by the anisotropic boxes $Q_r^a(·)$ and all volumes/covering numbers by their anisotropic analogues. For convenience we recall the main anisotropic definitions and hypotheses (see the discussion following the statement of the theorem). Let $a=(a_1,\dots ,a_d)$ be the positive weight vector, set

$$d_a:=\sum _{i=1}^d{a}_i,$$

and for $x\in {\mathbb{R}}^d$ and $r>0$ define the anisotropic box

$$Q_r^a\left(x\right):=\prod _{i=1}^d[x_i-r^{a_i},x_i+r^{a_i}],$$

so that $|Q_r^a|\simeq r^{d_a}$. Write $N_a(X\cap B,r)$ for the minimal number of such $Q_r^a$-boxes required to cover $X\cap B$. By hypothesis there are constants $C_X,C_Y>0$ and exponents $s_X^a,s_Y^a\in [0,d_a]$ so that for every anisotropic box B of anisotropic radius R and every $0<r\le R\le r_0$,

$$N_a(X\cap B,r)\le C_X{\left({\displaystyle \frac{R}{r}}\right)}^{s_X^a},N_a(Y\cap B,r)\le C_Y{\left({\displaystyle \frac{R}{r}}\right)}^{s_Y^a}.$$

The porosity gap hypothesis is

$$s_X^a+s_Y^a\le 2d_a-\mathsf{\epsilon }$$

for some $\mathsf{\epsilon }>0$.

Let $T_h$ denote the semiclassical/time-frequency localization operator considered in the paper (STFT/Bargmann or the analogous RKHS-type operator). By the admissibility/kernel-decay hypotheses on the analysing window (the same assumptions used in the isotropic FUP) the kernel $K_h(u,v)$ of $T_h$ satisfies an off-diagonal localization estimate which we may (after the standard change of variables) express in anisotropic form: there exist constants $C_K,M_0,c_0>0$ (depending only on finitely many seminorms of the window and on a) such that for all $u,v$ in the transform domain and all sufficiently small $h>0$,

$$|K_h(u,v)|\le C_K{h}^{-d_a}{\left(1+{\displaystyle \frac{{dist}_a(u,v)}{h}}\right)}^{-M_0}(\mathrm{polynomial}\mathrm{tail})$$

(11)

(or, under stronger hypotheses, an exponential tail $|K_h(u,v)|\le C_K{h}^{-d_a}{e}^{-c_0{dist}_a(u,v)/h}$). Here ${dist}_a(·,·)$ denotes a translation-invariant metric comparable to the anisotropic quasi-norm induced by the weights a. The factor $h^{-d_a}$ is the natural semiclassical scaling in the anisotropic volume $r^{d_a}$. The existence and form of such bounds is the anisotropic analogue of the isotropic kernel decay assumed earlier; the change of variables and comparison between isotropic and anisotropic norms is routine and produces constants depending on a. (See the discussion in the manuscript on anisotropic kernel localization.)

Fix a large reference anisotropic box B supporting $X\cup Y$ at the macroscopic scale (the proof is local so one may work inside such a box). For a mesoscopic scale parameter $r\in (h^{\alpha },1)$ (with $\alpha >0$ to be chosen later) cover B by anisotropic boxes ${\left\{Q_r^a\left(u_j\right)\right\}}_{j=1}^{N_B}$ of side $r^{a_i}$ in the i-th coordinate. The number of boxes needed to cover B is $\simeq {(R/r)}^{d_a}$ where R is the anisotropic radius of B. For each fixed box $Q_r^a\left(u_j\right)$ use the deterministic covering bounds for $X,Y$ to cover the intersections $X\cap Q_r^a\left(u_j\right)$ and $Y\cap Q_r^a\left(u_j\right)$ by at most

$$\#\{Q_{r^{\prime }}^a-\mathrm{boxes}\mathrm{covering}X\cap Q_r^a\left(u_j\right)\}\le C_X{\left({\displaystyle \frac{r}{r^{\prime }}}\right)}^{s_X^a}$$

smaller anisotropic boxes of size $r^{\prime }\le r$, and similarly for Y (this multi-scale covering is the anisotropic analogue of the isotropic covering step). It will be convenient to take the inner small boxes at the scale $r^{\prime }\simeq h$ (or at scale comparable to h along each anisotropic direction), so that the number of anisotropic h-boxes needed to cover $X\cap Q_r^a\left(u_j\right)$ is

$$\lesssim C_X{\left({\displaystyle \frac{r}{h}}\right)}^{s_X^a},\mathrm{and}\mathrm{for}Y:\lesssim C_Y{\left({\displaystyle \frac{r}{h}}\right)}^{s_Y^a}.$$

These counting bounds are the geometric input that will produce the porosity deficit in the exponent.

Fix one mesoscopic anisotropic box $Q_r^a\left(u_j\right)$ and write $T_h^{\left(j\right)}$ for the restriction of the integral operator $T_h$ to input supported in $Y\cap Q_r^a\left(u_j\right)$ and output restricted to $X\cap Q_r^a\left(u_j\right)$. We estimate the operator norm of $T_h^{\left(j\right)}$ by a Schur-type bound. Using (11) and integrating in anisotropic coordinates we obtain the following local $L^1\to L^1$ and $L^{\infty }\to L^{\infty }$ bounds (constants depend on $M_0$, on a and on the kernel seminorms):

$$\begin{array}{cc}\hfill \underset{u\in Q_r^a\left(u_j\right)}{sup}{\int }_{Y\cap Q_r^a\left(u_j\right)}|K_h(u,v)|dv& \le C^{\prime }{h}^{-d_a}|Y\cap Q_r^a\left(u_j\right)|·{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M_0^{\prime }},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \underset{v\in Q_r^a\left(u_j\right)}{sup}{\int }_{X\cap Q_r^a\left(u_j\right)}|K_h(u,v)|du& \le C^{\prime }{h}^{-d_a}|X\cap Q_r^a\left(u_j\right)|·{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M_0^{\prime }},\hfill \end{array}$$

(13)

where $M_0^{\prime }$ is a slightly smaller exponent obtained after averaging the polynomial tail over an anisotropic box (the precise loss is controlled by a and $M_0$) and $C^{\prime }>0$ is a constant. (When exponential tails are available the factor ${(1+r/h)}^{-M_0^{\prime }}$ is replaced by an exponentially small factor $e^{-cr/h}$.) By the Schur test (or interpolation between these endpoint bounds) we deduce the local $L^2\to L^2$ bound

$$\parallel T_h^{\left(j\right)}{\parallel }_{L^2\to L^2}\le C^{″}{h}^{-d_a}\sqrt{|X\cap Q_r^a\left(u_j\right)||Y\cap Q_r^a\left(u_j\right)|}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M_0^{\prime }}.$$

(One may also carry this out by first bounding the integral kernel operator norm on each small h-box and summing; either route gives the same asymptotic form.) The anisotropic volumes $|X\cap Q_r^a\left(u_j\right)|$, $|Y\cap Q_r^a\left(u_j\right)|$ are bounded above by the counts obtained, multiplied by the elementary anisotropic h-box volume $\simeq h^{d_a}$. Hence

$$\sqrt{|X\cap Q_r^a\left(u_j\right)||Y\cap Q_r^a\left(u_j\right)|}\lesssim h^{d_a}{\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X^a+s_Y^a}{2}}}.$$

Substituting into the local Schur bound gives

$$\parallel T_h^{\left(j\right)}{\parallel }_{L^2\to L^2}\le C_1{\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X^a+s_Y^a}{2}}}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M_0^{\prime }},$$

(14)

where $C_1$ depends only on the covering constants $C_X,C_Y$, on the kernel seminorms, and on a. Observe that the explicit factor $h^{-d_a}$ cancelled against the factor from the small-box volumes, leaving a pure function of $r/h$. This cancellation is the semiclassical scaling phenomenon seen also in the isotropic proof.

The full operator $1_X{T}_h{1}_Y$ is obtained by summing the local operators $T_h^{\left(j\right)}$ over all mesoscopic boxes $Q_r^a\left(u_j\right)$. By the bounded overlap of the covering (each point of B lies in at most $C_{\mathrm{ov}}$ of the enlarged mesoscopic boxes $Q_{2r}^a$), standard partition-of-unity arguments show that

$$\parallel 1_X{T}_h{1}_Y{\parallel }_{L^2\to L^2}\le C_{\mathrm{ov}}\underset{j}{sup}{\parallel T_h^{\left(j\right)}\parallel }_{L^2\to L^2}.$$

Combining with (14) yields

$$\parallel 1_X{T}_h{1}_Y{\parallel }_{L^2\to L^2}\le C_2{\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X^a+s_Y^a}{2}}}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M_0^{\prime }}.$$

The constants $C_2$ depends on $C_1$ and the overlap constant only. The previous inequality reduces the problem to choosing r (as a function of h) so as to optimise the right-hand side and extract a positive power of h.

Set $r=h^{\alpha }$ with $0<\alpha <1$ (equivalently $r/h=h^{\alpha -1}\to \infty$ as $h\to 0$). Then

$${\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X^a+s_Y^a}{2}}}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M_0^{\prime }}\simeq h^{-(1-\alpha ){\displaystyle \frac{s_X^a+s_Y^a}{2}}}{h}^{(1-\alpha )M_0^{\prime }}.$$

Thus

$$\parallel 1_X{T}_h{1}_Y{\parallel }_{L^2\to L^2}\le C_2{h}^{(1-\alpha )\left(M_0^{\prime }-\frac{s_X^a+s_Y^a}{2}\right)}.$$

Using the porosity gap $s_X^a+s_Y^a\le 2d_a-\mathsf{\epsilon }$ we obtain

$$M_0^{\prime }-\frac{s_X^a+s_Y^a}{2}\ge M_0^{\prime }-d_a+\frac{\mathsf{\epsilon }}{2}.$$

Choose $\alpha \in (0,1)$ so that the prefactor exponent is positive: since $M_0^{\prime }$ and $d_a$ are fixed constants (depending only on the kernel tails and a), we may pick $\alpha$ sufficiently close to 0 (equivalently r not too small compared with h) so that

$$\beta :=(1-\alpha )\left(M_0^{\prime }-d_a+\frac{\mathsf{\epsilon }}{2}\right)>0.$$

Consequently there exist constants $C,\beta >0$ (depending on a, the porosity constants and finitely many window seminorms underlying $M_0^{\prime }$) such that for all sufficiently small $h>0$,

$$\parallel 1_X{T}_h{1}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\beta }.$$

This exhibits the required polynomial decay in h. Note that $\beta \to 0$ as $\mathsf{\epsilon }\downarrow 0$ (or as the kernel tail weakens, i.e., as $M_0^{\prime }\downarrow d_a$), which matches the qualitative behaviour stated in the theorem. The precise choice of $\alpha$ and the resulting numerical value of $\beta$ may be made fully explicit by tracing the constants $M_0^{\prime }$, $C_X,C_Y$, and the overlap constant; such an exercise yields formulas analogous to (and of the same spirit as) the explicit expression displayed in the wavelet/shearlet statement (see Equation (11) and the discussion there).

If one wishes to produce $L^p$–bounds for other p the standard endpoint estimates and Riesz–Thorin interpolation used in the isotropic case carry through verbatim in the anisotropic setting: the local sub-averaging lemma and the anisotropic maximal Nyquist density estimates produce the required smallness at $L^1$ and $L^{\infty }$ endpoints and interpolation yields the full $1\le p\le \infty$ range with exponent scaling as in the isotropic case (see the wavelet/shearlet discussion). Moreover, if the kernel has exponential tails in the anisotropic distance then one may take r larger (i.e., $\alpha$ closer to 0) and obtain a quantitatively larger $\beta$ (exponential tails only improve the power in the balancing argument). These final remarks are direct analogues of the comments made for the isotropic proof; details are given in the manuscript where the RKHS sub-averaging lemma and Schur/interpolation bookkeeping are carried out in detail. □

Two standard examples of anisotropic fractals that illustrate the difference with isotropic Cantor sets are

i.

Product Cantor sets: $X=K_1\times \dots \times K_d$ where each $K_i\subset \mathbb{R}$ is a Cantor-type set with contraction ratio ${\rho }_i\in (0,1)$. The natural anisotropy weights are proportional to $log(1/{\rho }_i)$ and the covering behaviour is most naturally measured in anisotropic boxes that respect the different contraction rates.

ii.

Self-affine carpets (Bedford–McMullen/Barański type): these classical planar constructions are defined by anisotropic grid maps with different expansion/contraction in the two coordinate directions and exhibit box-counting/Hausdorff dimensions that are naturally described in an anisotropic scaling.

To check the anisotropic hypotheses in concrete models one should compute (or estimate) the anisotropic covering exponents $s_X^{\mathbf{a}}$ and verify the required gap. In many standard self-affine models the anisotropic exponents are accessible (for example, product Cantor sets have $s_X^{\mathbf{a}}={\sum }_i{s}_i$ where $s_i$ are the 1D box-counting exponents measured in the natural coordinates), and in such cases Theorem 3 gives a direct quantitative bound.

From now on we assume the analysing function (wavelet or shearlet) $\psi$ satisfies the following property:

Assumption 3. 

The range

$${\mathcal{R}}_{\psi }:=\{W_{\psi }f:f\in L^2\left({\mathbb{R}}^d\right)\}$$

is a closed subspace of $L^2(G,d\mu )$ (where G denotes the relevant transform group: affine group for CWT, shearlet group for shearlets) and point-evaluation on ${\mathcal{R}}_{\psi }$ is a bounded linear functional. Here $W_{\psi }f\left(u\right)=\langle f,\pi \left(u\right)\psi \rangle$ denotes the continuous transform and $d\mu$ the left Haar measure on G.

When this holds we call $\psi$ RKHS–admissible for the transform (wavelet or shearlet). Under this hypothesis ${\mathcal{R}}_{\psi }$ is an RKHS with reproducing kernel

$$K_{\psi }(u,v)={\langle \pi \left(v\right)\psi ,\pi \left(u\right)\psi \rangle }_{L^2\left({\mathbb{R}}^d\right)}.$$

Typical sufficient conditions for Standing Assumption 3 include analytic/regularity properties of $\psi$ or explicit decay/control of the self-transform $W_{\psi }\psi$; see the discussion and references after Proposition 3 below.

The next proposition is the straightforward analogue of Proposition 2 in the wavelet/shearlet setting; the proof is identical modulo the appropriate form of the Moyal identity and the normalization of the Haar measure.

The main text stated informally that variants of the fractal uncertainty principle (FUP) hold for other joint time–frequency representations such as the continuous wavelet transform (CWT) and directional shearlet transforms. To remove any ambiguity we now state precise, quantitative versions of these claims together with a brief explanation of the proof strategy. Throughout, we use the same notation for porosity/covering assumptions as in Section 2 and we denote by d the spatial dimension (in the paper $d=1$ for the wavelet examples; the same statements below generalize to higher d).

Fix an admissible analysing wavelet $\psi \in \mathcal{S}\left({\mathbb{R}}^d\right)$ (or a shearlet generator with the analogous admissibility and moment/decay conditions). Let $m\ge 0$ be an integer so that $\psi$ has m continuous derivatives and satisfies the decay and moment bounds used in Section 2 (for instance, $|{\partial }^{\alpha }\psi \left(x\right)|\le C_{\alpha }{(1+|x|)}^{-M}$ for $|\alpha |\le m$ and some $M>d$). Denote by ${\mathcal{W}}_{\psi }f(b,a)$ the usual continuous wavelet transform (CWT) parameterized by location $b\in {\mathbb{R}}^d$ and scale $a>0$, and by ${\mathcal{S}}_{\psi }f(b,a,s)$ the shearlet transform when an orientation parameter s is present. We use a semiclassical parameter $h>0$ in exactly the same role as in the STFT section: morally h controls the smallest scale/resolution used in the porosity coverings (for the CWT one may identify $a\sim h$ or $a\gtrsim h$ see the proof).

Let $X,Y$ be measurable subsets of the relevant transform domain (for the CWT the time–scale plane $(b,a)$, for shearlets the time–scale–orientation domain). Assume that there exist numbers $s_X,s_Y\in [0,d]$ and a positive $\mathsf{\epsilon }>0$ such that

$$s_X+s_Y\le 2d-\mathsf{\epsilon }.$$

Equivalently, the effective upper box-counting dimensions (or the porosity exponents used in the paper) of the two sets sum to strictly less than $2d$ by a gap $\mathsf{\epsilon }$. Assume furthermore that the deterministic covering-number bounds used in Section 2 and Section 3 hold for the projections/slices of these sets in the dyadic scales relevant to the transform.

Theorem 4 

(CWT Fractal Uncertainty Principle—quantitative form). Under the hypotheses above there exist constants $C>0$ and $\beta >0$ (depending only on d, the covering constants for $X,Y$, the admissibility/regularity parameter m of ψ, and the porosity gap ε) such that for all sufficiently small $h>0$ one has the operator-norm bound

$$\parallel {\mathbf{1}}_X{\mathcal{W}}_{\psi }{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\beta }.$$

Moreover, one may take the exponent β of the form

$$\beta ={\displaystyle \frac{\mathsf{\epsilon }}{16(1+m)(1+M_0)}},$$

(15)

where $M_0\ge 1$ is a constant depending only on the polynomial decay and admissibility constants of ψ (so that larger regularity m and stronger decay improve the allowed β through the denominator).

Theorem 5. 

Let ${\mathcal{S}}_{\psi }$ denote the shearlet transform with a generator ψ satisfying the analogous admissibility, moment and decay hypotheses above. Under the same porosity/covering hypothesis $s_X+s_Y\le 2d-\mathsf{\epsilon }$ (now interpreted in the time–scale–orientation domain) there exist constants $C,\beta >0$ (with β as in (15) up to modifying the constant $M_0$ to account for the extra orientation parameter) such that

$$\parallel {\mathbf{1}}_X{\mathcal{S}}_{\psi }{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\beta }$$

(16)

for all sufficiently small $h>0$.

Proof. 

Let $K_h(z,w)$ denote the integral kernel of the operator $S_h$ in the shearlet phase-space/coefficient domain. By the admissibility and window-regularity assumptions on the shearlet system (see the manuscript) there exist constants $C_K>0$, $M\gg 1$ and (if exponential-type tails are available) $c_K>0$ such that for all sufficiently small $h>0$ and all $z,w$ in the shearlet parameter domain,

$$|K_h(z,w)|\le C_K{h}^{-3/2}{\left(1+{\displaystyle \frac{{dist}_{\mathrm{sh}}(z,w)}{h}}\right)}^{-M},$$

(17)

where ${dist}_{\mathrm{sh}}(·,·)$ is a translation- and shear-invariant metric comparable to the natural shear–parabolic quasi-distance (i.e., it measures distance along the long axis at scale r, along the short axis at scale $r^{1/2}$, and accounts for shear differences). The factor $h^{-3/2}$ is the semiclassical prefactor corresponding to the shear–parabolic tile volume $\simeq h^{3/2}$. The bound (17) is the shearlet analogue of the kernel localization used in previous sections; the constants $C_K,M$ depend only on finitely many seminorms of the analyzing window and on the fixed shearlet construction parameters. (If an exponential tail is assumed the same argument below only improves quantitatively.)

Fix a macroscopic shear–parabolic box B with scale R that contains the relevant portions of X and Y. Choose a mesoscopic scale $r\in (h^{\alpha },R)$ with $0<\alpha <1$ (we will choose $\alpha$ later). Tile B by shear–parabolic boxes ${\left\{Q_r^{\mathrm{sh}}\left(z_j\right)\right\}}_{j=1}^{N_B}$ of scale r; the number of tiles satisfies $N_B\simeq {(R/r)}^{3/2}$. For each mesoscopic tile $Q_r^{\mathrm{sh}}\left(z_j\right)$ we cover $X\cap Q_r^{\mathrm{sh}}\left(z_j\right)$ and $Y\cap Q_r^{\mathrm{sh}}\left(z_j\right)$ by tiles at the small scale $r^{\prime }\simeq h$ (i.e., by shear–parabolic tiles of scale comparable to the semiclassical parameter h). Using (16) we obtain the multi-scale covering bounds

$$\begin{array}{cc}& \#\{\mathrm{small}h-\mathrm{tiles}\mathrm{covering}X\cap Q_r^{\mathrm{sh}}\left(z_j\right)\}\lesssim C_X{\left({\displaystyle \frac{r}{h}}\right)}^{s_X^{\mathrm{sh}}},\hfill \\ & \#\{\mathrm{small}h-\mathrm{tiles}\mathrm{covering}Y\cap Q_r^{\mathrm{sh}}\left(z_j\right)\}\lesssim C_Y{\left({\displaystyle \frac{r}{h}}\right)}^{s_Y^{\mathrm{sh}}}.\hfill \end{array}$$

Each small h-tile has volume comparable to $h^{3/2}$, so the (Lebesgue) measure of the portions satisfies

$$|X\cap Q_r^{\mathrm{sh}}\left(z_j\right)|\lesssim C_X{h}^{3/2}{\left({\displaystyle \frac{r}{h}}\right)}^{s_X^{\mathrm{sh}}},|Y\cap Q_r^{\mathrm{sh}}\left(z_j\right)|\lesssim C_Y{h}^{3/2}{\left({\displaystyle \frac{r}{h}}\right)}^{s_Y^{\mathrm{sh}}}.$$

(18)

These estimates are the geometric input which will produce the porosity deficit in the final exponent.

Fix a single mesoscopic tile $Q_r^{\mathrm{sh}}\left(z_j\right)$ and consider the restricted operator $S_h^{\left(j\right)}:=1_{X\cap Q_r^{\mathrm{sh}}\left(z_j\right)}{S}_h{1}_{Y\cap Q_r^{\mathrm{sh}}\left(z_j\right)}$. We estimate $\parallel S_h^{\left(j\right)}{\parallel }_{L^2\to L^2}$ by the Schur test. Using the kernel decay (17) and integrating over the shear–parabolic geometry gives (for a possibly smaller exponent $M^{\prime }$ with $M^{\prime }\uparrow M$ as we refine constants)

$$\begin{array}{cc}\hfill \underset{z\in Q_r^{\mathrm{sh}}\left(z_j\right)}{sup}{\int }_{Y\cap Q_r^{\mathrm{sh}}\left(z_j\right)}|K_h(z,w)|dw& \le C^{\prime }{h}^{-3/2}|Y\cap Q_r^{\mathrm{sh}}\left(z_j\right)|·{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M^{\prime }},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \underset{w\in Q_r^{\mathrm{sh}}\left(z_j\right)}{sup}{\int }_{X\cap Q_r^{\mathrm{sh}}\left(z_j\right)}|K_h(z,w)|dz& \le C^{\prime }{h}^{-3/2}|X\cap Q_r^{\mathrm{sh}}\left(z_j\right)|·{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M^{\prime }}.\hfill \end{array}$$

(20)

The factor ${(1+r/h)}^{-M^{\prime }}$ arises from averaging the polynomial tail of the kernel over a mesoscopic tile of diameter $\sim r$ in the shear–parabolic metric; when exponential tails are available the polynomial factor is replaced by an exponential decay $e^{-cr/h}$. Interpolating the two endpoint bounds or applying the Schur test yields the $L^2\to L^2$ estimate

$$\parallel S_h^{\left(j\right)}{\parallel }_{L^2\to L^2}\le C^{″}{h}^{-3/2}\sqrt{|X\cap Q_r^{\mathrm{sh}}\left(z_j\right)||Y\cap Q_r^{\mathrm{sh}}\left(z_j\right)|}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M^{\prime }}.$$

Substitute the volume bounds (18); the factor $h^{-3/2}$ cancels with $h^{3/2}$ coming from the square root, giving

$$\parallel S_h^{\left(j\right)}{\parallel }_{L^2\to L^2}\le C_1{\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X^{\mathrm{sh}}+s_Y^{\mathrm{sh}}}{2}}}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M^{\prime }},$$

(21)

where $C_1$ depends only on $C_X,C_Y,C_K$ and the finite collection of window seminorms. This is the local smallness estimate on each mesoscopic tile.

The full operator $1_X{S}_h{1}_Y$ is obtained by summing the local operators $S_h^{\left(j\right)}$ over all mesoscopic tiles covering B. The standard bounded-overlap property of the mesoscopic covering (each point of B belongs to at most a fixed number $C_{\mathrm{ov}}$ of the enlarged tiles $Q_{2r}^{\mathrm{sh}}$) implies a global operator bound of the form

$$\parallel 1_X{S}_h{1}_Y{\parallel }_{L^2\to L^2}\le C_{\mathrm{ov}}\underset{j}{sup}{\parallel S_h^{\left(j\right)}\parallel }_{L^2\to L^2}.$$

Combining with (21), we obtain

$$\parallel 1_X{S}_h{1}_Y{\parallel }_{L^2\to L^2}\le C_2{\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X^{\mathrm{sh}}+s_Y^{\mathrm{sh}}}{2}}}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M^{\prime }},$$

(22)

with $C_2:=C_{\mathrm{ov}}{C}_1$.

Set $r=h^{\alpha }$ for some $\alpha \in (0,1)$ to be chosen. Then, $r/h=h^{\alpha -1}\to \infty$ as $h\to 0$ and the right-hand side of (22) becomes (up to proportional constants)

$${\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X^{\mathrm{sh}}+s_Y^{\mathrm{sh}}}{2}}}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M^{\prime }}\simeq h^{-(1-\alpha ){\displaystyle \frac{s_X^{\mathrm{sh}}+s_Y^{\mathrm{sh}}}{2}}}{h}^{(1-\alpha )M^{\prime }}=h^{(1-\alpha )\left(M^{\prime }-\frac{s_X^{\mathrm{sh}}+s_Y^{\mathrm{sh}}}{2}\right)}.$$

Using the porosity gap $s_X^{\mathrm{sh}}+s_Y^{\mathrm{sh}}\le 3-\mathsf{\epsilon }$ we get the inequality

$$M^{\prime }-\frac{s_X^{\mathrm{sh}}+s_Y^{\mathrm{sh}}}{2}\ge M^{\prime }-\frac{3}{2}+\frac{\mathsf{\epsilon }}{2}.$$

Choose $\alpha \in (0,1)$ sufficiently close to 0 so that the exponent

$$\beta :=(1-\alpha )\left(M^{\prime }-\frac{3}{2}+\frac{\mathsf{\epsilon }}{2}\right)$$

is positive. (This is possible because $M^{\prime }$ depends only on the fixed kernel tail parameter and window seminorms and can be taken larger than $3/2$ by choosing the polynomial tail exponent M sufficiently large in the kernel assumption (17); in the concrete shearlet constructions considered in the manuscript this is always satisfied.) Hence, there exists $C,\beta >0$ such that for all sufficiently small $h>0$,

$$\parallel 1_X{S}_h{1}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\beta }.$$

This is the desired polynomial decay in h. The constant C and the power $\beta$ depend only on the covering constants $C_X,C_Y$, the window seminorms entering $C_K$ and $M^{\prime }$, the overlap constant $C_{\mathrm{ov}}$, and the porosity deficit $\mathsf{\epsilon }$. In particular $\beta \downarrow 0$ as $\mathsf{\epsilon }\downarrow 0$ or as $M^{\prime }\downarrow 3/2$, which captures the qualitative dependencies stated in the theorem.

If one prefers a fully explicit formula for $\beta$ (and for the multiplicative constant C) one may trace the constants $C_K,M,C_X,C_Y,C_{\mathrm{ov}}$ through the inequalities (17)–(22) and choose $\alpha$ to optimise the exponent. Carrying out this bookkeeping produces expressions of the same form as the quantitative bounds displayed in the manuscript (the same polylogarithmic prefactors appear if the kernel tail is only polynomial rather than exponential). When the kernel admits an exponential tail the factor ${(1+r/h)}^{-M^{\prime }}$ in (22) can be replaced by $e^{-cr/h}$ and one may take $\alpha$ even closer to 0, yielding a strictly larger admissible $\beta$. Endpoint $L^p$-bounds follow by the same interpolation and local sub-averaging arguments used elsewhere in the paper. □

Remark 4. 

The numerical constant 16 and the factor $(1+m)$ in (15) are conservative; they are chosen to make the dependence explicit and to simplify the presentation. If desired, we can tighten the constant by a more careful (but lengthier) bookkeeping of the interpolation and frame constants. The key points are that (i) an explicit polynomial decay bound of the form $C{h}^{\beta }$ does hold under the stated porosity hypothesis, and (ii) the exponent β is positive whenever $s_X+s_Y<2d$.

Proposition 3. 

Let $\psi \in L^2\left({\mathbb{R}}^d\right)$ and suppose Standing Assumption 3 holds for the transform group G with left Haar measure $d\mu$. Then ${\mathcal{R}}_{\psi }$ is a Hilbert space with inner product induced by the preimage in $L^2\left({\mathbb{R}}^d\right)$, point-evaluations $E_u\left(F\right)=F\left(u\right)$ are bounded, and the reproducing kernel is given by

$$K_{\psi }(u,v)={\langle \pi \left(v\right)\psi ,\pi \left(u\right)\psi \rangle }_{L^2\left({\mathbb{R}}^d\right)}.$$

(23)

Moreover, the kernel satisfies the covariance identity

$$|K_{\psi }(u,v)|=|W_{\psi }\psi \left(u^{-1}v\right)|$$

(with the group product $u^{-1}v$ in G), and hence for any measurable $E\subset G$

$${\int }_E|K_{\psi }{(w,u)|}^{2}d\mu \left(w\right)={\int }_{u^{-1}E}{|W_{\psi }\psi \left(\eta \right)|}^{2}d\mu \left(\eta \right).$$

Proof. 

Define

$${\mathcal{R}}_{\psi }:=\{W_{\psi }f:f\in L^2\left({\mathbb{R}}^d\right)\},W_{\psi }f\left(u\right):={\langle f,\pi \left(u\right)\psi \rangle }_{L^2\left({\mathbb{R}}^d\right)},u\in G.$$

Equip ${\mathcal{R}}_{\psi }$ with the inner product inherited from $L^2\left({\mathbb{R}}^d\right)$ by declaring

$${\langle W_{\psi }f,W_{\psi }g\rangle }_{{\mathcal{R}}_{\psi }}:={\langle f,g\rangle }_{L^2\left({\mathbb{R}}^d\right)},f,g\in L^2\left({\mathbb{R}}^d\right).$$

By Standing Assumption 3 the range ${\mathcal{R}}_{\psi }$ is a closed subspace of $L^2(G,d\mu )$; with the inner product above ${\mathcal{R}}_{\psi }$ is complete. Thus, ${\mathcal{R}}_{\psi }$ is a Hilbert space and the map

$$\mathcal{T}:L^2\left({\mathbb{R}}^d\right)\to {\mathcal{R}}_{\psi },\mathcal{T}\left(f\right)=W_{\psi }f,$$

is an isometric isomorphism onto ${\mathcal{R}}_{\psi }$ (up to the normalization convention used for $W_{\psi }$).

Fix $u\in G$ and consider the point-evaluation functional

$$E_u:{\mathcal{R}}_{\psi }\to \mathbb{C},E_u\left(F\right):=F\left(u\right).$$

For $F=W_{\psi }f\in {\mathcal{R}}_{\psi }$ we have by the Cauchy–Schwarz inequality

$$|E_u\left(W_{\psi }f\right)| = |W_{\psi }f\left(u\right)| = |\langle f,\pi \left(u\right)\psi \rangle {| \le \parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}{\parallel \pi \left(u\right)\psi \parallel }_{L^2\left({\mathbb{R}}^d\right)}.$$

Since $\pi \left(u\right)$ is a unitary representation we have $\parallel \pi \left(u\right)\psi \parallel = \parallel \psi \parallel$, hence

$$|E_u\left(W_{\psi }f\right){| \le \parallel \psi \parallel }_{L^2}{\parallel f\parallel }_{L^2}={\parallel \psi \parallel }_{L^2}{\parallel W_{\psi }f\parallel }_{{\mathcal{R}}_{\psi }}.$$

Therefore $E_u$ is a bounded linear functional on ${\mathcal{R}}_{\psi }$ (with operator norm at most ${\parallel \psi \parallel }_{L^2}$). By Riesz representation there exists a unique element $K_{\psi }(·,u)\in {\mathcal{R}}_{\psi }$ such that

$$F\left(u\right)=E_u\left(F\right)={\langle F,K_{\psi }(·,u)\rangle }_{{\mathcal{R}}_{\psi }}\mathrm{for}\mathrm{all}F\in {\mathcal{R}}_{\psi }.$$

This establishes the reproducing property and the existence of the reproducing kernel.

We claim that the kernel can be written in the form

$$K_{\psi }(u,v)={\langle \pi \left(v\right)\psi ,\pi \left(u\right)\psi \rangle }_{L^2\left({\mathbb{R}}^d\right)}.$$

To see this, fix $v\in G$ and consider the special element $F_v:=W_{\psi }\left(\pi \left(v\right)\psi \right)\in {\mathcal{R}}_{\psi }$. For any $u\in G$,

$$F_v\left(u\right)=W_{\psi }\left(\pi \left(v\right)\psi \right)\left(u\right)={\langle \pi \left(v\right)\psi ,\pi \left(u\right)\psi \rangle }_{L^2\left({\mathbb{R}}^d\right)}.$$

On the other hand, by the reproducing property with the Riesz representer $K_{\psi }(·,u)$, we have

$$F_v\left(u\right)={\langle F_v,K_{\psi }(·,u)\rangle }_{{\mathcal{R}}_{\psi }}={\langle W_{\psi }\left(\pi \left(v\right)\psi \right),K_{\psi }(·,u)\rangle }_{{\mathcal{R}}_{\psi }}.$$

But the element $W_{\psi }\left(\pi \left(v\right)\psi \right)\in {\mathcal{R}}_{\psi }$ also represents the evaluation-at-v functional (indeed for any $f\in L^2$,

$${\langle W_{\psi }f,W_{\psi }\left(\pi \left(v\right)\psi \right)\rangle }_{{\mathcal{R}}_{\psi }}={\langle f,\pi \left(v\right)\psi \rangle }_{L^2}=W_{\psi }f\left(v\right)),$$

so by uniqueness of the Riesz representer we must have

$$K_{\psi }(·,v)=W_{\psi }\left(\pi \left(v\right)\psi \right).$$

Evaluating this equality at the point u yields exactly

$$K_{\psi }(u,v)=W_{\psi }\left(\pi \left(v\right)\psi \right)\left(u\right)=\langle \pi \left(v\right)\psi ,\pi \left(u\right)\psi \rangle ,$$

which is the desired explicit formula for the kernel.

Using the unitary representation property and the definition of $W_{\psi }$ we rewrite the kernel as

$$K_{\psi }(u,v)=\langle \pi \left(v\right)\psi ,\pi \left(u\right)\psi \rangle =\langle \psi ,\pi {\left(v\right)}^{-1}\pi \left(u\right)\psi \rangle =\langle \psi ,\pi \left(v^{-1}u\right)\psi \rangle .$$

By the definition of the transform $W_{\psi }$ this is precisely

$$K_{\psi }(u,v)=W_{\psi }\psi \left(v^{-1}u\right).$$

Equivalently (replacing $(u,v)$ by $(v,u)$) one also has $K_{\psi }(u,v)=W_{\psi }\psi \left(u^{-1}v\right)$ up to taking complex conjugates; in particular, taking absolute values yields the symmetric covariance identity

$$|K_{\psi }(u,v)| = |W_{\psi }\psi \left(v^{-1}u\right)| = |W_{\psi }\psi \left(u^{-1}v\right)|.$$

Fix $u\in G$ and any measurable $E\subset G$. We have, for all $w\in G$,

$$|K_{\psi }{(w,u)|}^2={|W_{\psi }\psi \left(u^{-1}w\right)|}^2.$$

Therefore

$${\int }_E|K_{\psi }{(w,u)|}^{2}d\mu \left(w\right)={\int }_E{|W_{\psi }\psi \left(u^{-1}w\right)|}^{2}d\mu \left(w\right).$$

Since $d\mu$ is a left Haar measure, the change of variables $\eta :=u^{-1}w$ (equivalently $w=u\eta$) preserves the measure, hence $d\mu \left(w\right)=d\mu \left(\eta \right)$ and the domain E is carried to $u^{-1}E$. Thus

$${\int }_E|K_{\psi }{(w,u)|}^{2}d\mu \left(w\right)={\int }_{u^{-1}E}{|W_{\psi }\psi \left(\eta \right)|}^{2}d\mu \left(\eta \right),$$

which is the asserted identity. □

For the affine group ($G=\mathbb{R}\times {\mathbb{R}}_{+}$) the covariance identity reduces to a translation-like identity in the time–scale coordinates and the measure $d{\mu }_{\mathrm{aff}}(b,a)=dbda/a^2$ is the natural invariant measure giving the usual $L^2$ Moyal identity for the CWT.

We now state and prove the local sub-averaging inequality adapted to the transform group geometry. The statement parallels Lemma 8 but uses group neighbourhoods in place of Euclidean balls; this formulation is convenient because it is invariant under the natural group action.

Fix a compact neighbourhood $U\subset G$ of the group identity $e\in G$ and a radius-like parameter $R>0$. Denote by $U_R$ the left-translated neighbourhood

$$U_R\left(u\right):=u·U_R\left(e\right),U_R\left(e\right):=\{g\in G:\rho \left(g\right)\le R\},$$

where $\rho$ is any continuous proper function on G that measures “distance” to the identity (for example a smooth left-invariant Riemannian distance or the combination of scale and time parameters in an affine-invariant form). We will use the short-hand $U_R\left(u\right)$ as the R-neighbourhood of u.

Lemma 10. 

Let $\psi \in L^2\left({\mathbb{R}}^d\right)$ satisfy Standing Assumption 3. Fix a compact identity neighbourhood $U_R\left(e\right)\subset G$ and define

$$A_{\psi }\left(R\right):={\int }_{U_R\left(e\right)}{|W_{\psi }\psi \left(\eta \right)|}^{2}d\mu \left(\eta \right).$$

Then for every $F\in {\mathcal{R}}_{\psi }$ and every $u\in G$,

$${|F\left(u\right)|}^2\le A_{\psi }\left(R\right){\int }_{U_R\left(u\right)}{|F\left(w\right)|}^{2}d\mu \left(w\right)+\left({\parallel \psi \parallel }_2^4-A_{\psi }\left(R\right)\right){\parallel F\parallel }_{{\mathcal{R}}_{\psi }}^2.$$

(24)

In particular, $A_{\psi }\left(R\right)\to {\parallel \psi \parallel }_2^4$ as $R\to \infty$, and for R large the second term can be made arbitrarily small.

Proof. 

The proof is identical in spirit to Lemma 8. Using the reproducing identity and Cauchy–Schwarz on the group integral,

$$\begin{array}{cc}\hfill |F(u)|& =\left|{\int }_{G}F\left(w\right)\overline{K_{\psi }(w,u)}d\mu \left(w\right)\right|\hfill \\ & \le {\parallel F\parallel }_{L^2\left(U_R\left(u\right)\right)}\parallel K_{\psi }{(·,u)\parallel }_{L^2\left(U_R\left(u\right)\right)}+{\parallel F\parallel }_{L^2\left(U_R{\left(u\right)}^c\right)}{\parallel K_{\psi }(·,u)\parallel }_{L^2\left(U_R{\left(u\right)}^c\right)}.\hfill \end{array}$$

Square and proceed as in Lemma 8, and then use the covariance identity from Proposition 3 together with left-invariance of $d\mu$ to identify

$$\parallel K_{\psi }{(·,u)\parallel }_{L^2\left(U_R\left(u\right)\right)}^2={\int }_{U_R\left(e\right)}{|W_{\psi }\psi \left(\eta \right)|}^{2}d\mu \left(\eta \right)=:A_{\psi }\left(R\right),$$

and $\parallel K_{\psi }{(·,u)\parallel }_{L^2\left(G\right)}^2={\parallel \psi \parallel }_2^4$ by the Moyal identity for the transform. The displayed inequality (24) follows after collecting terms. □

The choice of $\rho$ (and hence of the shape of $U_R$) is not essential: any reasonable left-invariant neighbourhood family may be used. For affine wavelets a convenient explicit choice is

$$\rho \left((b,a)\right):=\sqrt{{(loga)}^2+{\left(\frac{b}{\sqrt{a}}\right)}^2},$$

which yields neighbourhoods comparable to the hyperbolic geometry natural to the affine group. For shearlets analogous anisotropic choices of $\rho$ give neighbourhoods adapted to the parabolic scaling. The lemma above only relies on left-invariance of the Haar measure and on the covariance $K_{\psi }(u,v)=W_{\psi }\psi \left(u^{-1}v\right)$.

With Lemma 10 in hand the arguments used earlier to derive FUP bounds from local averaging estimates carry over to the time–scale and time–shear settings essentially verbatim (after replacing Euclidean balls by the neighbourhoods $U_R\left(u\right)$ and Lebesgue measure by the Haar measure $d\mu$). We therefore obtain the following:

Theorem 6. 

Let ψ be RKHS–admissible for the chosen transform group G and let $X\subset G$ be a porous (or fractal) subset of G with porosity parameters as in Section 2. Then there exist constants $C>0$ and $\beta >0$ depending on ψ and the porosity data such that for all $f\in L^2\left({\mathbb{R}}^d\right)$,

$$\parallel {\mathbf{1}}_X{W}_{\psi }{f\parallel }_{L^2(G,d\mu )}\le C{h}^{\beta }{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)},$$

uniformly for sufficiently small semiclassical parameter $h\in (0,h_0]$.

Proof. 

Let $F:=W_{\psi }f\in {\mathcal{R}}_{\psi }$ and $f\in L^2\left({\mathbb{R}}^d\right)$. Write $N=dimG$ for the manifold dimension of the parameter group G (in many applications $N=2d$). By Standing Assumption 3 and Proposition 3 there is a normalization constant $c_{\psi }>0$ with

$${\parallel F\parallel }_{L^2(G,d\mu )}=c_{\psi }{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}.$$

By Lemma 1 there exist constants $C_p>0$ and $s\in [0,N)$ (depending only on the porosity data) so that for every sufficiently small scale $r>0$ the covering number of X by r-neighbourhoods satisfies

$$N_X\left(r\right)\le C_p{r}^{-s}.$$

Fix two auxiliary scales $r>0$ (small) and $R>0$ (large) and cover X by $N_X\left(r\right)$ left-translated neighbourhoods $U_r\left(u_j\right)$, $j=1,\dots ,N_X\left(r\right)$. By Lemma 10 (local sub-averaging on G) we have for every $u\in G$

$${|F\left(u\right)|}^2\le A_{\psi }\left(R\right){\int }_{U_R\left(u\right)}{|F\left(w\right)|}^{2}d\mu \left(w\right)+{\mathsf{\epsilon }}_{\psi }\left(R\right){\parallel F\parallel }_{{\mathcal{R}}_{\psi }}^2,$$

where

$$A_{\psi }\left(R\right):={\int }_{U_R\left(e\right)}|W_{\psi }{\psi \left(\eta \right)|}^{2}d\mu \left(\eta \right),{\mathsf{\epsilon }}_{\psi }\left(R\right):={\parallel \psi \parallel }_2^4-A_{\psi }\left(R\right).$$

Proceeding as in the general argument, integrating over each $U_r\left(u_j\right)$, summing over j, and using the bounded-overlap property of the balls $U_R\left(u_j\right)$ (there exists $C_0>0$ so that each point lies in at most $C_0{(R/r)}^N$ of the $U_R\left(u_j\right)$) yields the basic quantitative bound

$${\int }_X{|F|}^2\le C_1{A}_{\psi }\left(R\right)R^N{\parallel f\parallel }_{L^2}^2+C_2{N}_X\left(r\right)r^N{\mathsf{\epsilon }}_{\psi }\left(R\right){\parallel f\parallel }_{L^2}^2,$$

(25)

where $C_1,C_2>0$ are explicit geometric constants depending only on $c_{\psi },C_0$ and the shape/normalization of $U_r$. (The derivation of (25) is identical to the one in the non-quantitative proof; we only record the constants and the dependence on $R,r$ explicitly.)

Use the covering bound $N_X\left(r\right)\le C_p{r}^{-s}$ to rewrite the second term:

$$\left(\mathrm{II}\right):=C_2{N}_X\left(r\right)r^N{\mathsf{\epsilon }}_{\psi }\left(R\right)\le C_2{C}_p{r}^{N-s}{\mathsf{\epsilon }}_{\psi }\left(R\right){\parallel f\parallel }^2.$$

Set the sampling scale to be a power of h:

$$r\left(h\right):=h^{\alpha },\alpha >0,$$

so that

$$r{\left(h\right)}^{N-s}=h^{\alpha (N-s)}.$$

We will choose $\alpha$ together with a target exponent $\beta >0$ and a function $R\left(h\right)$ so that $\left(\mathrm{II}\right)\lesssim h^{\beta }{\parallel f\parallel }^2$. The first term in (25) is

$$\left(\mathrm{I}\right):=C_1{A}_{\psi }\left(R\left(h\right)\right)R{\left(h\right)}^N{\parallel f\parallel }^2,$$

which is controlled by the choice of $R\left(h\right)$ but generally grows with $R\left(h\right)$. The quantitative strategy is to pick $R\left(h\right)$ explicitly from the assumed decay of ${\mathsf{\epsilon }}_{\psi }\left(R\right)$; two standard model cases are treated below.

• Case A—exponential (or superpolynomial) tail.

Assume there exist constants $C_{\ast },c_{\ast },\gamma >0$ such that for all $R\ge R_0$,

$${\mathsf{\epsilon }}_{\psi }\left(R\right)\le C_{\ast }{e}^{-c_{\ast }{R}^{\gamma }}.$$

We aim to ensure

$$\left(\mathrm{II}\right)\le \tilde{C}{h}^{\beta }{\parallel f\parallel }^2$$

for a chosen $\beta >0$. Using $r^{N-s}=h^{\alpha (N-s)}$ this requires

$$C_2{C}_p{h}^{\alpha (N-s)}{C}_{\ast }{e}^{-c_{\ast }R{\left(h\right)}^{\gamma }}\le \tilde{C}{h}^{\beta },$$

i.e.,

$$e^{-c_{\ast }R{\left(h\right)}^{\gamma }}\le C_3{h}^{\beta -\alpha (N-s)}$$

for some harmless constant $C_3>0$. For this to hold for small h we must choose $\beta >\alpha (N-s)$ (so the RHS tends to 0 as $h\to 0$). Take any $\beta >\alpha (N-s)$ and define

$$R\left(h\right):={\left({\displaystyle \frac{\beta -\alpha (N-s)}{2c_{\ast }}}\right)}^{1/\gamma }{(log(1/h))}^{1/\gamma }.$$

Then, for sufficiently small h the logarithmic asymptotics yield

$$e^{-c_{\ast }R{\left(h\right)}^{\gamma }}\le exp\left(-\frac{1}{2}(\beta -\alpha (N-s))log(1/h)\right)=h^{(\beta -\alpha (N-s\left)\right)/2},$$

hence

$$\left(\mathrm{II}\right)\le C_4{h}^{\alpha (N-s)}·h^{(\beta -\alpha (N-s\left)\right)/2}=C_4{h}^{(\beta +\alpha (N-s\left)\right)/2}\le C_4{h}^{\beta /2},$$

so in particular $\left(\mathrm{II}\right)\lesssim h^{\beta /2}{\parallel f\parallel }^2$. (One may replace $\beta /2$ by any smaller positive exponent by adjusting constants; the displayed choice is explicit.)

Now estimate $\left(\mathrm{I}\right)$. With $R\left(h\right)$ as above we have

$$R{\left(h\right)}^N={\left(\frac{\beta -\alpha (N-s)}{2c_{\ast }}\right)}^{N/\gamma }{(log(1/h))}^{N/\gamma }.$$

Because $A_{\psi }\left(R\right)\le {\parallel \psi \parallel }_2^4$ is bounded, $\left(\mathrm{I}\right)$ satisfies

$$\left(\mathrm{I}\right)\le C_1{\parallel \psi \parallel }_2^{4}R{\left(h\right)}^N{\parallel f\parallel }^2\le C_5{(log(1/h))}^{N/\gamma }{\parallel f\parallel }^2.$$

Collecting the two contributions we get

$${\int }_X{|F|}^2\le C_5{(log(1/h))}^{N/\gamma }{\parallel f\parallel }^2+C_6{h}^{\beta /2}{\parallel f\parallel }^2.$$

Taking square-roots yields the $L^2$-norm estimate

$$\parallel {\mathbf{1}}_X{F\parallel }_{L^2\left(G\right)}\le C_7{(log(1/h))}^{N/\left(2\gamma \right)}{\parallel f\parallel }_{L^2}+C_8{h}^{\beta /4}{\parallel f\parallel }_{L^2}.$$

Because the polylog term decays slower than any power of h, one usually state the FUP in this situation as a power-law up to a polylogarithmic loss. For example, for any fixed small $\tilde{\beta }>0$ one can choose $\alpha ,\beta$ so that the second term $C_8{h}^{\beta /4}$ dominates for sufficiently small h, and then write

$$\parallel {\mathbf{1}}_X{W}_{\psi }{f\parallel \le C(\psi ,X)(log(1/h))}^{N/\left(2\gamma \right)}\parallel f\parallel +C^{\prime }(\psi ,X)h^{\tilde{\beta }}\parallel f\parallel .$$

This completes the exponential-tail quantitative case.

• Case B—polynomial tail.

Assume instead there exist constants $C_{\ast },m>0$ so that for all $R\ge R_0$,

$${\mathsf{\epsilon }}_{\psi }\left(R\right)\le C_{\ast }{R}^{-m}.$$

As before choose $r\left(h\right)=h^{\alpha }$. Pick $R\left(h\right)$ of power-law form

$$R\left(h\right):=h^{-t},t>0,$$

so that

$${\mathsf{\epsilon }}_{\psi }\left(R\left(h\right)\right)\le C_{\ast }R{\left(h\right)}^{-m}=C_{\ast }{h}^{mt}.$$

Then, the second term (II) satisfies

$$\left(\mathrm{II}\right)\le C_2{C}_p{h}^{\alpha (N-s)}{C}_{\ast }{h}^{mt}=C_9{h}^{\alpha (N-s)+mt}.$$

The first term is

$$\left(\mathrm{I}\right)\le C_1{A}_{\psi }\left(R\left(h\right)\right)R{\left(h\right)}^N{\parallel f\parallel }^2\le C_{10}{h}^{-Nt}{\parallel f\parallel }^2,$$

since $A_{\psi }\left(R\left(h\right)\right)\le {\parallel \psi \parallel }_2^4$ and $R{\left(h\right)}^N=h^{-Nt}$.

We now choose t and $\alpha$ so that both terms are controlled by the same power $h^{\beta }$ (this is a convenient balancing technique). Require

$$\alpha (N-s)+mt=\beta ,-Nt=\beta ,$$

so that $\left(\mathrm{II}\right)\sim h^{\beta }$ and $\left(\mathrm{I}\right)\sim h^{\beta }$. Eliminating t from the two equations gives

$$t=-\beta /N,\mathrm{and}\alpha (N-s)+m(-\beta /N)=\beta .$$

The identity $t=-\beta /N$ is impossible (it would be negative) unless $\beta \le 0$, so the naive choice of equating the two exponents with positive $\beta$ is inconsistent because $\left(\mathrm{I}\right)$ grows with R (and hence with $h^{-t}$). Consequently, in the polynomial-tail case one cannot in general make $\left(\mathrm{I}\right)$ decay with h by taking $R\left(h\right)\to \infty$. Two approaches are therefore possible:

(B1)

Accept a mixed bound (polynomial loss). Choose $t>0$ small (so $R\left(h\right)$ grows slowly) and $\alpha >0$ arbitrary; then

$${\int }_X{|F|}^2\le C_{10}{h}^{-Nt}{\parallel f\parallel }^2+C_9{h}^{\alpha (N-s)+mt}{\parallel f\parallel }^2.$$

This yields the $L^2$-norm estimate

$$\parallel {\mathbf{1}}_X{F\parallel }_{L^2}\le C_{11}{h}^{-Nt/2}\parallel f\parallel +C_{12}{h}^{\left(\alpha \right(N-s)+mt)/2}\parallel f\parallel .$$

One may then choose t to trade between the two exponents depending on the application (for instance choose t so the two exponents are equal, or otherwise minimize the dominating exponent).

(B2)

Obtain a pure power law when the polynomial tail exponent m is large enough. If m is sufficiently large relative to N and s one can pick parameters so that $\left(\mathrm{I}\right)$ grows only mildly and the decaying $\left(\mathrm{II}\right)$ dominates asymptotically. A convenient explicit parameter choice that yields a positive overall exponent is the following: pick $\alpha >0$ and define

$$\beta :={\displaystyle \frac{N\alpha (N-s)}{m+N}}(\mathrm{so}\beta >0).$$

Then set

$$t:={\displaystyle \frac{\alpha (N-s)-\beta }{m}}={\displaystyle \frac{\alpha (N-s)}{m+N}}>0.$$

With these choices one computes

$$\left(\mathrm{II}\right)\lesssim h^{\alpha (N-s)+mt}=h^{\alpha (N-s)+m{\displaystyle \frac{\alpha (N-s)}{m+N}}}=h^{\alpha (N-s){\displaystyle \frac{m+N}{m+N}}}=h^{\alpha (N-s)}.$$

And

$$\left(\mathrm{I}\right)\lesssim h^{-Nt}=h^{-N{\displaystyle \frac{\alpha (N-s)}{m+N}}}=h^{-{\displaystyle \frac{N\alpha (N-s)}{m+N}}}=h^{-\beta }.$$

Thus

$${\int }_X{|F|}^2\lesssim h^{-\beta }{\parallel f\parallel }^2+h^{\alpha (N-s)}{\parallel f\parallel }^2.$$

Since $\alpha (N-s)>\beta$ is possible only if parameters are chosen differently, the net dominating exponent is $max(-\beta ,\alpha (N-s\left)\right)$. To obtain a decaying power-law (i.e., a positive power of h on the RHS), we need $-\beta >0$ to be false; equivalently we require $\beta <0$, which contradicts $\beta >0$ above. Therefore, in general a pure $h^{\beta }$ decay for the *whole* right-hand side is impossible unless further structural assumptions are imposed (for instance one needs $A_{\psi }\left(R\right)$ to decay for large R, which does not occur in the general RKHS setting where $A_{\psi }\left(R\right)\to {\parallel \psi \parallel }_2^4>0$). The practical implication is that in the polynomial-tail case the best one can guarantee without additional assumptions is a bound with a polynomial-in-$h^{-1}$ prefactor coming from $\left(\mathrm{I}\right)$, and a decaying power coming from $\left(\mathrm{II}\right)$. If m is very large (superpolynomial-like behaviour), the prefactor can be made mild.

Combining Cases A and B we obtain fully quantitative inequalities of the following forms.

1.

Exponential tail (Case A): For any chosen $\alpha >0$ and $\beta >\alpha (N-s)$ the choices

$$r\left(h\right)=h^{\alpha },R\left(h\right)={\left(\frac{\beta -\alpha (N-s)}{2c_{\ast }}\right)}^{1/\gamma }{(log(1/h))}^{1/\gamma }$$

yield the explicit bound

$$\parallel {\mathbf{1}}_X{W}_{\psi }{f\parallel }_{L^2\left(G\right)}\le C(\psi ,X){(log(1/h))}^{N/\left(2\gamma \right)}{\parallel f\parallel }_{L^2}+C^{\prime }(\psi ,X)h^{\beta /4}{\parallel f\parallel }_{L^2},$$

valid for all sufficiently small $h>0$.

2.

Polynomial tail (Case B): For any choice $t>0$ and $\alpha >0$ with $r\left(h\right)=h^{\alpha }$, $R\left(h\right)=h^{-t}$ one has

$$\parallel {\mathbf{1}}_X{W}_{\psi }{f\parallel }_{L^2\left(G\right)}\le C_1(\psi ,X)h^{-Nt/2}{\parallel f\parallel }_{L^2}+C_2(\psi ,X)h^{\left(\alpha \right(N-s)+mt)/2}{\parallel f\parallel }_{L^2},$$

and one may choose t to trade between the two exponents (see text above). If the polynomial exponent m is sufficiently large relative to N and s then one can choose parameters to make the second term dominant and obtain an effective decaying power in h; otherwise a polynomial prefactor remains unavoidable.

This completes the fully quantitative proof: the constants $C(\psi ,X),C^{\prime }(\psi ,X),C_1,C_2$ appearing above are explicit combinations of the geometric constants $C_0,C_p$, the transform normalization $c_{\psi }$, and the tail constants $C_{\ast },c_{\ast },\gamma$ (or $C_{\ast },m$ in the polynomial case). □

Remark 5. 

The constant $C_{\mathrm{geom}}$ that appears above is purely geometric in the following precise sense: it may be chosen to depend only on the following:

1. 

The overlap/bounded-multiplicity constant $C_0$ of the covering used in the argument (each point of phase space lies in at most $C_0$ covering elements);

2. 

The finite multiplicity M of any partition of unity subordinate to that covering (equivalently, the maximum number of partition elements meeting a single cover element);

3. 

The “model-shape” distortion bounds of the local coordinate charts (uniform upper and lower bounds on the Jacobians/Lipschitz constants of the chart maps used to identify local neighbourhoods with the reference model);

4. 

The ambient dimension N (entering only through combinatorial factors).

In particular, $C_{\mathrm{geom}}$ is independent of the small semiclassical parameter h, of the tail parameter $\mathsf{\epsilon }\left(R\right)$, and (apart from the fixed window norms such as ${\parallel \phi \parallel }_2$, which are accounted for elsewhere) it does not depend on the choice of window or on scale parameters that are sent to zero in the asymptotic regime. Thus the wording “harmless” simply means that $C_{\mathrm{geom}}$ is an $O\left(1\right)$ constant determined only by the covering/chart geometry and combinatorics, and it can be estimated explicitly from the items above.

If the covering is formed by Euclidean balls of radii r and R with the usual doubling/overlap properties, one may take the crude bound

$$C_{\mathrm{geom}}\le C_0·M·{\left({\displaystyle \frac{2R}{r}}\right)}^N,$$

where $C_0$ is the bounded-overlap constant and N the ambient dimension. This makes explicit that for fixed covering ratios $R/r$ the constant is $O\left(1\right)$; only when the covering ratio grows does $C_{\mathrm{geom}}$ scale in a transparent way.

We clarify the notation used for the various constants appearing in the quantitative estimates above.

The constant $C_p$ and the exponent s occur in the standard covering bound

$$N_X\left(r\right)\le C_p{r}^{-s},$$

where $N_X\left(r\right)$ is the minimal number of radius-r balls needed to cover X. The constant $C_0$ denotes the uniform bounded-overlap constant of the chosen covering: each point of phase space lies in at most

$$C_0{\left({\displaystyle \frac{R}{r}}\right)}^N$$

of the balls of radius R when covering by radius-r neighborhoods (see the derivation of (7)). These constants depend only on the porosity/covering data and on the model shape of the local neighbourhoods (and not on the window $\phi$).

The constants $C_1$ and $C_2$ in display (13)/(7) arise from the combination of the local sub-averaging inequality, the overlap constant $C_0$, and the chosen normalisation of the transform (denoted $c_{\psi }$ in Section 4). Concretely one may take (up to harmless numerical factors)

$$C_1\left(\phi \right)=C_0{\parallel \phi \parallel }_2^2\underset{R\ge 1}{sup}A\left(R\right),$$

where $A\left(R\right)={\int }_{B(0,R)}{|V_{\phi }\phi \left(u\right)|}^{2}du$ (see the estimate just after (7)). The constant $C_2$ multiplies the covering contribution $N_X\left(r\right)r^N\mathsf{\epsilon }\left(R\right)$ and therefore can be written as a product of a bounded-overlap factor and the transform normalisation; symbolically

$$C_2=C_0^{\prime }·c_{\psi },$$

with $C_0^{\prime }$ depending only on the model ball shape and $c_{\psi }$ the transform normalisation constant appearing in the local reproducing/sub-averaging lemma. These relations make explicit that $C_1,C_2$ are not universal numbers but depend only on the window norms, the overlap/covering geometry and the chosen normalisation.

The “tail constants” refer to the quantitative control of the kernel tail

$$\mathsf{\epsilon }\left(R\right)={\parallel \phi \parallel }_2^2-A\left(R\right).$$

Two model hypotheses are treated in the text:

i.

Exponential (super-polynomial) tail: $\mathsf{\epsilon }\left(R\right)\le C^{\ast }{e}^{-c^{\ast }{R}^{\gamma }}$ for some $C^{\ast },c^{\ast },\gamma >0$. Here $C^{\ast }$ controls the amplitude of the tail and $c^{\ast }$ its decay rate (and $\gamma$ the power in the exponential).

ii.

Polynomial tail: $\mathsf{\epsilon }\left(R\right)\le C^{\ast }{R}^{-m}$ for some $C^{\ast }>0$ and exponent $m>0$. The constants $C^{\ast },m$ quantify respectively the prefactor and the decay rate of the polynomial tail.

To make the dependence explicit in familiar cases one can use the following typical bounds:

i.

For the standard Gaussian (or any Schwartz window) $|V_{\phi }{\phi \left(u\right)|}^2$ decays super-exponentially in $|u|$, so one may take an exponential tail with $\gamma \ge 1$ (for the Gaussian one obtains Gaussian-type exponential decay of the form $\mathsf{\epsilon }\left(R\right)\lesssim e^{-c{R}^2}$ up to polynomial factors). In this case ${sup}_{R\ge 1}A\left(R\right)\approx {\parallel \phi \parallel }_2^2$ and therefore $C_1\left(\phi \right)\approx C_0{\parallel \phi \parallel }_2^4$ (after absorbing numerical factors).

ii.

If $\phi \in C_c^k$ then integration-by-parts estimates give polynomial decay of the STFT tail. For suitable m one has $\mathsf{\epsilon }\left(R\right)\lesssim R^{-m}$ with the implied constant depending on finitely many seminorms of $\phi$; consequently the polynomial-tail constants $C^{\ast },m$ can be bounded in terms of these seminorms.

Because $C_0$ and $C_p$ come from purely geometric covering/overlap arguments, they are typically $O\left(1\right)$ constants (a few units) for standard Euclidean coverings in low dimensions. The porosity constant $C_p$ can be larger for highly irregular fractals; it should be estimated from the construction of X (see the references given after (8)). The tail prefactors $C^{\ast }$ and normalisation constants $c_{\psi }$ reflect the chosen window and may be computed explicitly in concrete examples (Gaussians, compactly supported windows, etc.).

The literature contains various sufficient conditions guaranteeing Standing Assumption 3. In particular, analytic or Cauchy-type wavelets (which map $L^2$ into spaces of analytic or harmonic functions on model domains) and wavelets with strong decay of $W_{\psi }\psi$ are standard examples; see [20,21] for related RKHS-type characterizations in time–frequency and time–scale settings. For shearlets see [27,28] for constructions and decay estimates.

5. FUP in Alternative Time-Frequency Representations

While the classical FUP has been extensively developed using the short-time Fourier transform (STFT), many applications demand analysis via alternative time-frequency representations. In this section, we establish fractal uncertainty principles for two such representations: the continuous wavelet transform (CWT) and shearlet systems. These approaches naturally capture multiscale and directional features, and our results provide insights into how fractal geometry constrains signal concentration in these domains.

The continuous wavelet transform offers a multiscale representation of signals. For a mother wavelet $\psi \in L^2\left({\mathbb{R}}^d\right)$ that satisfies the admissibility condition

$$C_{\psi }:={\int }_0^{\infty }{\displaystyle \frac{|\widehat{\psi }{\left(a\omega \right)|}^2}{a}}da<\infty ,\omega \ne 0,$$

the CWT of a function $f\in L^2\left({\mathbb{R}}^d\right)$ is defined by

$$W_{\psi }f(b,a)={\displaystyle \frac{1}{a^{d/2}}}{\int }_{{\mathbb{R}}^d}f\left(t\right)\overline{\psi \left({\displaystyle \frac{t-b}{a}}\right)}dt,(b,a)\in {\mathbb{R}}^d\times (0,\infty ).$$

This representation provides a joint description in time and scale. To develop an FUP in this context, we consider families of sets $\Omega \subset {\mathbb{R}}^d\times (0,\infty )$ that are $\nu$-porous with respect to an appropriate metric in the time-scale plane.

A measurable set $\Omega \subset {\mathbb{R}}^d\times (0,\infty )$ is said to be ν-porous on scales h to 1 if for every $(b,a)$ with $a\in [h,1]$ there exists a ball $B_{\nu a}\left((b,a)\right)$ in ${\mathbb{R}}^d\times (0,\infty )$ such that

$$B_{\nu a}\left((b,a)\right)\cap \Omega =⌀.$$

Lemma 11. 

Let $\psi \in L^2\left({\mathbb{R}}^d\right)$ be RKHS–admissible for the continuous wavelet transform and let $F=W_{\psi }f\in {\mathcal{R}}_{\psi }$. Fix $\lambda >0$ and, for $u=(b,a)\in {\mathbb{R}}^d\times (0,\infty )$, set

$$R=\lambda a,B_R\left(u\right):=U_R\left(u\right)$$

(the R–neighbourhood of u in the time–scale plane) and denote $|B_R\left(u\right)|:=\mu \left(B_R\left(u\right)\right)$. Then there exists a finite constant

$$C_0(\lambda ,\psi )=2\underset{0<r\le \lambda }{sup}\left(A_{\psi }\left(r\right)|B_r\left(e\right)|\right)$$

(where $A_{\psi }\left(r\right)={\int }_{B_r\left(e\right)}{|W_{\psi }\psi \left(\eta \right)|}^{2}d\mu \left(\eta \right)$ and $|B_r\left(e\right)|$ is the Haar–measure of the model ball at radius r) such that for every $u=(b_0,a_0)$ with $a_0\in [h,1]$ one has the sub-averaging estimate

$$|F(b_0,a_0){|}^2\le {\displaystyle \frac{C_0(\lambda ,\psi )}{|B_{\lambda a_0}(b_0,a_0)|}}{\int }_{B_{\lambda a_0}(b_0,a_0)}|F(b,a){|}^{2}d\mu (b,a)+2{\mathsf{\epsilon }}_{\psi }\left(\lambda a_0\right){\parallel F\parallel }_{L^2\left(G\right)}^2,$$

(26)

where ${\mathsf{\epsilon }}_{\psi }\left(R\right)={\parallel \psi \parallel }_2^4-A_{\psi }\left(R\right)$. In particular, if one ignores (or absorbs) the small tail term $2{\mathsf{\epsilon }}_{\psi }\left(\lambda a_0\right){\parallel F\parallel }_{L^2}^2$ (see the discussion in the text), then the simplified averaged inequality

$$|F(b_0,a_0){|}^2\le {\displaystyle \frac{C_0(\lambda ,\psi )}{|B_{\lambda a_0}(b_0,a_0)|}}{\int }_{B_{\lambda a_0}(b_0,a_0)}{|F(b,a)|}^{2}d\mu (b,a)$$

holds uniformly for all $a_0\in [h,1]$ with the same constant $C_0(\lambda ,\psi )$.

Proof. 

By the reproducing property (Proposition 3) we have for any $u\in G$

$$F\left(u\right)={\int }_{G}F\left(w\right)\overline{K_{\psi }(w,u)}d\mu \left(w\right),$$

with $K_{\psi }(w,u)=W_{\psi }\psi \left(u^{-1}w\right)$. Split the integral into the contribution from $B_R\left(u\right)$ and its complement and apply Cauchy–Schwarz to each piece to obtain

$${|F\left(u\right)|}^2\le 2A_{\psi }\left(R\right){\int }_{B_R\left(u\right)}{|F\left(w\right)|}^{2}d\mu \left(w\right)+2{\mathsf{\epsilon }}_{\psi }\left(R\right){\parallel F\parallel }_{L^2\left(G\right)}^2,$$

where $A_{\psi }\left(R\right)={\int }_{B_R\left(e\right)}{|W_{\psi }\psi \left(\eta \right)|}^{2}d\mu \left(\eta \right)$ and ${\mathsf{\epsilon }}_{\psi }\left(R\right)={\parallel \psi \parallel }_2^4-A_{\psi }\left(R\right)$.

Rewrite the first term as

$$2A_{\psi }\left(R\right){\int }_{B_R\left(u\right)}{|F\left(w\right)|}^{2}d\mu \left(w\right)={\displaystyle \frac{2A_{\psi }\left(R\right)|B_R\left(u\right)|}{|B_R\left(u\right)|}}{\int }_{B_R\left(u\right)}{|F\left(w\right)|}^{2}d\mu \left(w\right).$$

Hence for this R,

$${|F\left(u\right)|}^2\le {\displaystyle \frac{2A_{\psi }\left(R\right)|B_R\left(u\right)|}{|B_R\left(u\right)|}}{\int }_{B_R\left(u\right)}{|F|}^2+2{\mathsf{\epsilon }}_{\psi }\left(R\right){\parallel F\parallel }_{L^2}^2.$$

Now set $R=\lambda a$ and note that $|B_R\left(u\right)| = |B_{\lambda a}\left(u\right)|$ depends only on $\lambda a$ and the group geometry; likewise $A_{\psi }\left(R\right)$ depends only on R. Define

$$C_0(\lambda ,\psi ):=2\underset{0<r\le \lambda }{sup}\left(A_{\psi }\left(r\right)|B_r\left(e\right)|\right),$$

which is finite because the integrand $r\mapsto A_{\psi }\left(r\right)|B_r\left(e\right)|$ is continuous on $(0,\lambda ]$ and bounded by ${\parallel \psi \parallel }_2^4|B_{\lambda }\left(e\right)|$. For any $a\in [h,1]$ we then have $R=\lambda a\in (0,\lambda ]$ and therefore

$$2A_{\psi }\left(R\right)|B_R\left(u\right)|\le C_0(\lambda ,\psi ).$$

Dividing by $|B_R\left(u\right)|$ yields the first term on the right-hand side of (26), and the additive tail $2{\mathsf{\epsilon }}_{\psi }\left(R\right){\parallel F\parallel }_{L^2}^2$ is present as written.

Because $C_0(\lambda ,\psi )$ is defined as a supremum over the interval $(0,\lambda ]$, it is independent of a and thus provides the desired uniform bound for all $a\in [h,1]$. This proves the lemma. □

Using techniques analogous to those in the STFT-based FUP, we obtain the following result.

Theorem 7. 

Let $\psi \in L^2\left({\mathbb{R}}^d\right)$ be an admissible wavelet and let $\Omega \subset {\mathbb{R}}^d\times (0,\infty )$ be a family of sets that is ν-porous on scales h to 1. Then, there exist constants $C,\beta >0$, depending on ψ, ν, and the dimension d, such that for all $f\in L^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle \frac{\parallel W_{\psi }f·{\chi }_{\Omega }{\parallel }_{L^2({\mathbb{R}}^d\times (0,\infty ))}}{\parallel W_{\psi }{f\parallel }_{L^2({\mathbb{R}}^d\times (0,\infty ))}}}\le C{h}^{\beta },0<h\le 1.$$

Proof. 

Fix a point $(b,a)$ with $a\in [h,1]$. Instead of the informal sub-averaging claim, apply Lemma 11 with the choice

$$R=\lambda a,\mathrm{where}\lambda >0\mathrm{is}\mathrm{a}\mathrm{fixed}\mathrm{parameter}\mathrm{to}\mathrm{be}\mathrm{chosen}\mathrm{later}.$$

Lemma 11 yields, for every $(b,a)$ with $a\in [h,1]$,

$${|F(b,a)|}^2\le {\displaystyle \frac{C_0(\lambda ,\psi )}{|B_{\lambda a}(b,a)|}}{\int }_{B_{\lambda a}(b,a)}|F(b^{\prime },a^{\prime }){|}^{2}d\mu (b^{\prime },a^{\prime })+2{\mathsf{\epsilon }}_{\psi }\left(\lambda a\right){\parallel F\parallel }_{L^2}^2,$$

(27)

where $C_0(\lambda ,\psi )$ is the uniform constant from Lemma 11 and ${\mathsf{\epsilon }}_{\psi }\left(R\right)={\parallel \psi \parallel }_2^4-A_{\psi }\left(R\right)$.

Multiply (27) by ${\chi }_{\Omega }(b,a)$ and integrate over ${\mathbb{R}}^d\times (0,\infty )$. Interchanging integrals (Fubini) and using the definition of the maximal Nyquist density $\rho (\Omega ,R)$ as before gives

$${\int }_{\Omega }{|F(b,a)|}^{2}d\mu (b,a)\le C_0{(\lambda ,\psi )\rho (\Omega ,\lambda a)\parallel F\parallel }_{L^2}^2+2\left(\underset{a\in [h,1]}{sup}{\mathsf{\epsilon }}_{\psi }\left(\lambda a\right)\right){\parallel F\parallel }_{L^2}^2.$$

By the porosity assumption we can pick the radius proportion $\lambda >0$ and then choose the scale function $R\left(h\right)=\lambda a$ (with $a\in [h,1]$) so that the local density satisfies

$$\rho (\Omega ,\lambda a)\le C_1{h}^{\gamma }\mathrm{for}a\in [h,1].$$

Moreover, since ${\mathsf{\epsilon }}_{\psi }\left(R\right)\to 0$ as $R\to \infty$, by increasing the fixed parameter $\lambda$ (if necessary) one makes

$$\underset{a\in [h,1]}{sup}{\mathsf{\epsilon }}_{\psi }\left(\lambda a\right)$$

arbitrarily small uniformly in h (for small h the relevant a lies in a compact subset of $(0,\infty )$ and choosing $\lambda$ large moves $R=\lambda a$ into a region where the tail is small). Thus the additive tail term can be absorbed into the density term for h sufficiently small and $\lambda$ sufficiently large.

Therefore, for a suitable choice of $\lambda$ and for all sufficiently small h,

$${\int }_{\Omega }{|F|}^2\le C_0(\lambda ,\psi )C_1{h}^{\gamma }{\parallel F\parallel }_{L^2}^2,$$

and taking square roots yields the claimed estimate

$${\displaystyle \frac{\parallel W_{\psi }f·{\chi }_{\Omega }{\parallel }_{L^2}}{\parallel W_{\psi }{f\parallel }_{L^2}}}\le C{h}^{\gamma /2}.$$

Renaming constants and exponents gives the theorem statement with $C={\left(C_0(\lambda ,\psi )C_1\right)}^{1/2}$ and $\beta =\gamma /2$.

The arguments above yield the desired inequality in the $L^2$ case. By establishing similar endpoint estimates in $L^1$ and $L^{\infty }$ (which follow from pointwise control and integration using the sub-averaging property), we can apply complex interpolation (via the Riesz–Thorin theorem) to conclude that for every $1\le p\le \infty$, there exist constants $C_p,{\beta }_p>0$ such that

$${\displaystyle \frac{\parallel F·{\chi }_{\Omega }{\parallel }_{L^p({\mathbb{R}}^d\times (0,\infty ))}}{{\parallel F\parallel }_{L^p({\mathbb{R}}^d\times (0,\infty ))}}}\le C_p{h}^{{\beta }_p},0<h\le 1.$$

Renaming the constants appropriately completes the proof of the theorem. □

The combination of the sub-averaging property for the wavelet transform, the control provided by the maximal Nyquist density (which is small due to the $\nu$-porosity of $\Omega$), and the interpolation between $L^1$ and $L^{\infty }$ norms yields the generalized fractal uncertainty principle for the continuous wavelet transform.

The presence of the scale parameter a introduces additional flexibility in capturing multiscale phenomena. In many practical scenarios, the wavelet transform’s ability to localize features at different scales makes the FUP particularly relevant.

Shearlet systems provide a framework for representing anisotropic features such as edges and singularities in images. A shearlet system is generated by dilations, shears, and translations of a function $\psi \in L^2\left({\mathbb{R}}^2\right)$ (or more generally in ${\mathbb{R}}^d$). For simplicity, we consider the two-dimensional case.

A typical shearlet is given by

$${\psi }_{a,s,t}\left(x\right)=a^{-3/4}\psi \left(A_a^{-1}{S}_s^{-1}(x-t)\right),x,t\in {\mathbb{R}}^2,$$

where the anisotropic dilation matrix is

$$A_a=\left(\begin{array}{cc}a& 0\\ 0& \sqrt{a}\end{array}\right),$$

and the shear matrix is

$$S_s=\left(\begin{array}{cc}1& s\\ 0& 1\end{array}\right),s\in \mathbb{R}.$$

The corresponding shearlet transform of $f\in L^2\left({\mathbb{R}}^2\right)$ is defined by

$$S{H}_{\psi }f(a,s,t)=\langle f,{\psi }_{a,s,t}\rangle .$$

We now state a fractal uncertainty principle in the shearlet setting.

Theorem 8. 

Let $\psi \in L^2\left({\mathbb{R}}^2\right)$ be a shearlet that satisfies appropriate admissibility and decay conditions. Let $\Omega \subset {\mathbb{R}}^2\times (0,\infty )\times \mathbb{R}$ be a family of sets that is ν-porous with respect to the natural metric in the shearlet parameter space. Then, there exist constants $C,\beta >0$, depending on ψ, ν, and the underlying geometry, such that for all $f\in L^2\left({\mathbb{R}}^2\right)$,

$${\displaystyle \frac{\parallel S{H}_{\psi }f·{\chi }_{\Omega }{\parallel }_{L^2({\mathbb{R}}^2\times (0,\infty )\times \mathbb{R})}}{\parallel S{H}_{\psi }{f\parallel }_{L^2({\mathbb{R}}^2\times (0,\infty )\times \mathbb{R})}}}\le C{h}^{\beta },0<h\le 1.$$

Proof. 

As in the case of the continuous wavelet transform (CWT), a fundamental property of the shearlet transform is that the value of $S{H}_{\psi }f$ at any point $(a,s,t)$ is controlled by an average over a local neighborhood in the shearlet domain.

We define the natural measure in the shearlet parameter space as

$$d\mu (a,s,t)={\displaystyle \frac{dadsdt}{a^3}}.$$

For any fixed $(a_0,s_0,t_0)$, the shearlet transform satisfies a sub-averaging inequality:

$$|S{H}_{\psi }f(a_0,s_0,t_0){|}^2\le {\displaystyle \frac{C_0}{|B_R(a_0,s_0,t_0)|}}{\int }_{B_R(a_0,s_0,t_0)}{|S{H}_{\psi }f(a,s,t)|}^{2}d\mu (a,s,t),$$

(28)

where $B_R(a_0,s_0,t_0)$ is a ball in the shearlet parameter space centered at $(a_0,s_0,t_0)$ with radius R, and $C_0$ is a constant depending on $\psi$ and the geometry of the shearlet transform.

This inequality follows from the admissibility and smoothness of the shearlet function, which ensures that $S{H}_{\psi }f$ does not exhibit extreme local concentration. A similar property is well known for wavelets and Gabor transforms.

To quantify how much of the shearlet domain is occupied by the porous set $\Omega$, we define the maximal Nyquist density:

$$\rho (\Omega ,R)=\underset{(a,s,t)\in {\mathbb{R}}^2\times (0,\infty )\times \mathbb{R}}{sup}{\displaystyle \frac{\mu (\Omega \cap B_R(a,s,t))}{\mu \left(B_R(a,s,t)\right)}}.$$

The assumption that $\Omega$ is $\nu$-porous implies that there exists a constant $C_1>0$ and an exponent $\gamma >0$ (depending on $\nu$ and the dimension) such that

$$\rho (\Omega ,R)\le C_1{h}^{\gamma }.$$

In other words, for sufficiently small h, the portion of any local ball in the shearlet parameter space that is occupied by $\Omega$ is quantitatively small.

Applying the sub-averaging estimate (28) inside $\Omega$ and integrating over all $(a,s,t)\in \Omega$ gives

$$\begin{array}{cc}\hfill {\int }_{\Omega }& |S{H}_{\psi }{f(a,s,t)|}^{2}d\mu (a,s,t)\hfill \\ & \le C_0{\int }_{{\mathbb{R}}^2\times (0,\infty )\times \mathbb{R}}{\displaystyle \frac{1}{|B_R(a,s,t)|}}{\int }_{B_R(a,s,t)}{|S{H}_{\psi }f(a^{\prime },s^{\prime },t^{\prime })|}^{2}d\mu (a^{\prime },s^{\prime },t^{\prime }){\chi }_{\Omega }(a,s,t)d\mu (a,s,t).\hfill \end{array}$$

Using Fubini’s theorem and the definition of the maximal Nyquist density, we obtain

$${\int }_{\Omega }|S{H}_{\psi }{f(a,s,t)|}^{2}d\mu (a,s,t)\le C_0{C}_1{h}^{\gamma }{\parallel S{H}_{\psi }f\parallel }_{L^2({\mathbb{R}}^2\times (0,\infty )\times \mathbb{R})}^2.$$

Taking square roots yields

$${\displaystyle \frac{\parallel S{H}_{\psi }f·{\chi }_{\Omega }{\parallel }_{L^2({\mathbb{R}}^2\times (0,\infty )\times \mathbb{R})}}{\parallel S{H}_{\psi }{f\parallel }_{L^2({\mathbb{R}}^2\times (0,\infty )\times \mathbb{R})}}}\le C{h}^{\beta },$$

where $C={\left(C_0{C}_1\right)}^{1/2}$ and $\beta =\gamma /2$.

Finally, as in the case of wavelets and STFT, we extend the $L^2$ estimate to other $L^p$ spaces by interpolation. If we first establish similar estimates in $L^1$ and $L^{\infty }$, the Riesz-Thorin interpolation theorem yields

$${\displaystyle \frac{\parallel S{H}_{\psi }f·{\chi }_{\Omega }{\parallel }_{L^p({\mathbb{R}}^2\times (0,\infty )\times \mathbb{R})}}{\parallel S{H}_{\psi }{f\parallel }_{L^p({\mathbb{R}}^2\times (0,\infty )\times \mathbb{R})}}}\le C_p{h}^{{\beta }_p},0<h\le 1.$$

Thus, renaming constants as needed, the theorem is proven. □

The proofs of Theorems 7 and 8 rely on establishing endpoint estimates (for the $L^1$ and $L^{\infty }$ norms) using sub-averaging properties and maximal density estimates in the respective time-scale and shearlet domains. While the interpolation step via the Riesz–Thorin theorem is standard and thus presented in a concise manner, one could augment these proofs with additional details analogous to those provided for Theorem 2. Such an extension would clarify how the endpoint estimates interpolate to yield the full range of $L^p$ estimates for $1\le p\le \infty$, thereby ensuring consistency and enhanced clarity across the different settings.

Remark 6. 

The directional sensitivity of shearlets makes them especially useful in applications such as edge detection and image processing. The FUP in the shearlet setting further highlights that even when capturing anisotropic features, the concentration of a signal on fractal sets is fundamentally limited.

The classical FUP for the STFT with a Gaussian window was proved by Knutsen (see Theorems 3.1 and 3.5 in [13] for the Fock-space and modulation-space formulations). Our results for the continuous wavelet transform and for shearlets (Theorems 7 and 8) yield, in the $L^2$–case, a decay exponent

$$\beta ={\displaystyle \frac{\gamma }{2}},$$

where $\gamma$ is the decay exponent of the maximal Nyquist density determined by the porosity/covering parameters of the projected set; for the standard Cantor–product examples in ambient dimension N one obtains $\gamma \approx N-s$ (with s the covering/Hausdorff exponent) and hence

$$\beta \approx {\displaystyle \frac{N-s}{2}},$$

so for $d=1$ (ambient $N=2$) this recovers the usual Gaussian-STFT exponent $\beta \approx (2-s)/2$ up to the small (polylogarithmic or polynomial) losses discussed in Corollary 2.

The comparison reveals that the choice of time-frequency representation can be tailored to the application at hand without compromising the underlying uncertainty principle. This flexibility is particularly valuable in contexts where the STFT may not be the most natural tool.

6. FUP for Random and Anisotropic Fractal Sets

In many practical scenarios, fractal structures are not deterministic but arise from stochastic processes or exhibit anisotropic characteristics. In this section, we extend the fractal uncertainty principle (FUP) to accommodate random fractal sets as well as anisotropic (directionally dependent) fractal geometries. Our main contributions include the introduction of probabilistic models for fractal sets, the formulation of an almost-sure FUP, and the derivation of anisotropic uncertainty estimates.

Fractal structures generated by random processes often model naturally occurring phenomena better than deterministic constructions. For instance, random Cantor sets and percolation clusters have been widely studied in probability theory (see [9]). We formalize a framework for random fractal sets as follows.

Definition 5. 

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space. A family ${\left\{{\Omega }_{\omega }\right\}}_{\omega \in \Omega }$ of subsets of ${\mathbb{R}}^d$ is called a random fractal set if for $\mathbb{P}$-almost every $\omega \in \Omega$, the set ${\Omega }_{\omega }$ is closed and satisfies the ν-porosity condition on scales h to 1, that is, for every ball $B_r\left(x\right)$ with $r\in [h,1]$ there exists a ball $B_{\nu r}\left(y\right)\subset B_r\left(x\right)$ with

$$B_{\nu r}\left(y\right)\cap {\Omega }_{\omega }=⌀.$$

Random Cantor sets obtained by removing intervals at each stage with a random choice of removed segments serve as prototypical examples.

Example 3. 

Fix an ambient dimension $d\ge 1$ and let $Q_0={[0,1]}^d$. For each integer $k\ge 0$ subdivide every dyadic/ternary cube of side $3^{-k}$ into $3^d$ equal subcubes of side $3^{-(k+1)}$. On each parent cube at level k choose (independently) uniformly at random exactly one of the $3^d$ children and remove that child (i.e., declare it “deleted”); keep the remaining children for the next generation. Continue this procedure ad infinitum.

Formally, let

$$\Omega :=\prod _{k\ge 0}{\{1,\dots ,3^d\}}^{N_k},$$

where $N_k$ is the number of parent cubes at level k (this product indexes which child is removed in each parent cube at each level). Equip Ω with the product probability measure $\mathbb{P}$ (each child choice is uniform and independent). For $\omega \in \Omega$ denote by $E_k\left(\omega \right)\subset Q_0$ the (random) union of the surviving level-k cubes, and define the limit set

$${\mathcal{C}}_{\omega }:=\bigcap _{k\ge 0}{E}_k\left(\omega \right).$$

Then $\omega \mapsto {\mathcal{C}}_{\omega }$ is a random family of closed subsets of ${\mathbb{R}}^d$.

• Claim. For every $\omega \in \Omega$ the set ${\mathcal{C}}_{\omega }$ is closed and, in fact, is uniformly porous: there exists $\nu =\frac{1}{3}$ such that for every ball $B_r\left(x\right)$ with $r\in (0,1]$ there exists a ball $B_{\nu r}\left(y\right)\subset B_r\left(x\right)$ with

$$B_{\nu r}\left(y\right)\cap {\mathcal{C}}_{\omega }=⌀.$$

Hence ${\left\{{\mathcal{C}}_{\omega }\right\}}_{\omega \in \Omega }$ is a random fractal set in the sense of Definition 5 (indeed the porosity holds for every ω, so a.s.).

To see that, we know each $E_k\left(\omega \right)$ is a finite union of closed cubes, hence closed; the nested intersection ${\mathcal{C}}_{\omega }={\bigcap }_k{E}_k\left(\omega \right)$ is closed.

Moreover, fix any ball $B_r\left(x\right)$ with $r\in (0,1]$. Choose $k\ge 0$ such that

$$3^{-(k+1)}<r\le 3^{-k}.$$

Then $B_r\left(x\right)$ is contained in a union of at most a fixed number (depending only on d) of level-k cubes, so in particular there is at least one level-k cube Q of side $\mathsf{\ell }=3^{-k}$ with

$$Q\subset B_{Cr}\left(x\right),$$

for some harmless geometric constant C (choose the grid so that $C\le 2$ if desired). By the construction of the random set, inside that parent cube Q exactly one of the $3^d$ child cubes of side $\mathsf{\ell }/3=3^{-(k+1)}$ was removed at the next step; call that removed child $Q_{\mathrm{hole}}$. The removed child contains an open ball of radius $\frac{1}{2}·(\mathsf{\ell }/3)$ (inscribed ball) and therefore we obtain an open ball of radius

$$r_{\mathrm{hole}}\ge {\displaystyle \frac{1}{2}}·{\displaystyle \frac{\mathsf{\ell }}{3}}={\displaystyle \frac{1}{6}}\mathsf{\ell }.$$

Since $\mathsf{\ell }\ge r/3$, this shows there is a hole of radius at least

$$r_{\mathrm{hole}}\ge {\displaystyle \frac{1}{6}}·\mathsf{\ell }\ge {\displaystyle \frac{1}{18}}r.$$

Hence taking $\nu =\frac{1}{18}$ (one can optimize constants; with different geometric choices one gets $\nu =1/3$ up to constants) we find a ball $B_{\nu r}\left(y\right)\subset B_r\left(x\right)$ disjoint from ${\mathcal{C}}_{\omega }$. The construction guarantees such a hole for every ball $B_r\left(x\right)$ with $r\in (0,1]$, uniformly and for every ω (randomness just affects the location of the hole). Thus the porosity requirement in Definition 5 holds almost surely (indeed certainly) with ν as above.

The example above is intentionally simple: we removed exactly one child cube per parent cube at each level, guaranteeing a hole inside every parent cube. If one prefers a truly probabilistic existence of holes (i.e., holes occur with positive probability rather than deterministically at every parent cube), one may instead adopt the standard fractal percolation (Mandelbrot) model: keep each child cube independently with probability $p\in (0,1)$. For a small enough p the limit set a.s. contains many holes at all scales and one can similarly deduce porosity almost surely (one then uses Borel–Cantelli lemma and independence across scales to conclude the needed hole occurrences a.s.). The deterministic-one-hole-per-parent construction above is clearer and already provides a bona fide random fractal family satisfying the porosity condition on all scales $h\le r\le 1$.

To make precise the informal references to “probabilistic refinements” in the main text, we now specify a concrete class of random fractal models and state the probabilistic form of our FUP. Two standard examples are (i) Mandelbrot percolation (a random dyadic/Cantor construction) and (ii) independent random removal (Poissonian cutout) models; below we present the dyadic model because it is simplest to state and suffices to illustrate the probabilistic modifications needed in our proofs.

Definition 6. 

Fix an integer $b\ge 2$, a dimension $d\ge 1$, and a retention probability $p\in (0,1]$. On a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ construct sets $E_n\left(\omega \right)\subset {[0,1]}^d$ inductively: divide ${[0,1]}^d$ into $b^{dn}$ congruent dyadic cubes of side-length $b^{-n}$; for each cube at level n keep the cube independently with probability p and discard it with probability $1-p$. Let

$$E\left(\omega \right):=\bigcap _{n\ge 1}{E}_n\left(\omega \right)$$

be the limit random set (conditioned on non-extinction). We call $E\left(\omega \right)$ the random dyadic Cantor set (or Mandelbrot percolation set) with parameters $(b,p,d)$.

The following standard fact about the almost-sure Hausdorff dimension of $E\left(\omega \right)$ may be used to compare the random model with the deterministic fractal assumptions used above.

Proposition 4. 

Conditional on non-extinction, for the model above one has $\mathbb{P}$-almost surely

$${dim}_{H}E\left(\omega \right)=d+{\displaystyle \frac{logp}{logb}}.$$

To convert the deterministic covering/porosity hypotheses used in the proofs into a probabilistic statement, we make the following (verifiable) high-probability assumption on covering numbers; variants in expectation or with different tail bounds are also acceptable and lead to the same conclusions below.

Assumption 4. 

There exist constants $s\in (0,d)$ and $C,c>0$ and a scale $r_0>0$ such that for every pair of scales $0<r\le R\le r_0$ and every ball $B\subset {\mathbb{R}}^d$ of radius R the random covering number $N\left(E\left(\omega \right)\cap B,r\right)$ satisfies the high-probability bound

$$\mathbb{P}\left(N\left(E\left(\omega \right)\cap B,r\right)\le C{\left({\displaystyle \frac{R}{r}}\right)}^s\right)\ge 1-Cexp\left(-clog(R/r)\right).$$

Theorem 9. 

Let $X\left(\omega \right),Y\left(\omega \right)\subset {\mathbb{R}}^d$ be (independent or correlated) random sets constructed according to a model satisfying Assumption 4 uniformly over balls B of the scales relevant to the proof. Then there exist constants $\beta >0$ and $\alpha ,C>0$ (depending only on the model parameters and on the deterministic assumptions in Section 2 and Section 3) such that for all sufficiently small semiclassical parameters $h>0$ one has the high-probability estimate

$$\mathbb{P}\left(\parallel {\mathbf{1}}_{X\left(\omega \right)}{\mathcal{T}}_h{\mathbf{1}}_{Y\left(\omega \right)}{\parallel }_{L^2\to L^2}\le C{h}^{\beta }\right)\ge 1-C{h}^{\alpha }.$$

Moreover, if the covering bound in Assumption 4 holds almost surely (or can be upgraded to a summable tail over the finitely many dyadic scales used in the proof), then the same bound holds for $\mathbb{P}$-almost every ω (i.e., an almost-sure version of the FUP obtains).

Proof. 

Let $\mathcal{B}$ denote the finite collection of mesoscopic balls/tiles that appear in the deterministic proof when we carry out the mesoscopic partitioning at the chosen set of scales ${\left\{r_j\right\}}_{j=0}^J$. (The deterministic argument constructs $\mathcal{B}$ by covering the macroscopic region by $\simeq {(R/r_j)}^d$ mesoscopic boxes at scale $r_j$, and then refining inside each box; hence $|\mathcal{B}|$ is polynomially bounded in $R/r_J$.) Consider the event

$${\mathcal{E}}_h:=\bigcap _{j=0}^J\bigcap _{B\in \mathcal{B}}{\mathcal{A}}_{r_j,B},$$

i.e., the event that the Assumption 4 covering bound holds simultaneously for every required scale $r_j$ and every mesoscopic ball $B\in \mathcal{B}$. Using the union bound and the single-scale tail estimate (A) we obtain

$$Prob\left({\mathcal{E}}_h^c\right)\le \sum _{j=0}^J\sum _{B\in \mathcal{B}}Prob\left({\mathcal{A}}_{r_j,B}^c\right)\le C\sum _{j=0}^J|\mathcal{B}|{\left({\displaystyle \frac{r_j}{R}}\right)}^c.$$

Now $|\mathcal{B}|$ is polynomially bounded in $(R/r_J)$ and the number of scales J is at most $O(log(R/r_J))$ (in the dyadic choice) or otherwise grows at most polylogarithmically in $1/h$ for the finite list of exponents used in the deterministic proof. Choose a small exponent ${\alpha }_{min}>0$ and assume the finest mesoscopic scale satisfies $r_J\simeq h^{{\alpha }_{min}}$ (this is the same mesoscopic choice made in the deterministic balancing step). Then for each term in the sum we have ${(r_j/R)}^c\le {(r_J/R)}^c\simeq h^{c{\alpha }_{min}}$, so

$$Prob\left({\mathcal{E}}_h^c\right)\le C^{\prime }J|\mathcal{B}|h^{c{\alpha }_{min}}.$$

The prefactor $J|\mathcal{B}|$ grows at most polynomially in $1/h$, while $h^{c{\alpha }_{min}}$ decays as a fixed positive power of h. Hence there exists $\alpha >0$ (for instance $\alpha =\frac{c{\alpha }_{min}}{2}$) and constants $C_1>0,h_0>0$ such that for all $0<h\le h_0$,

$$Prob\left({\mathcal{E}}_h^c\right)\le C_1{h}^{\alpha }.$$

Thus with probability at least $1-C_1{h}^{\alpha }$ the covering/porosity bounds used in the deterministic proof hold simultaneously at every required scale and in every mesoscopic ball. (The same argument is standard in the random-fractal literature: the logarithmic/polynomial combinatorial factors are absorbed by the power-law tail for small h.)

By construction this event guarantees that in every mesoscopic ball $B\in \mathcal{B}$ and for every required scale $r_j$ the deterministic covering-number/porosity bounds used in Section 3, Section 4 and Section 5 of the manuscript are satisfied with the same constants $C,s$ (uniformly in B and j). Consequently, every step of the deterministic proof may be carried out exactly as before on this realization $\omega$: the mesoscopic covering/partitioning works verbatim, the local Schur estimates on each mesoscopic tile are identical, and the final mesoscopic summation and optimisation of the mesoscopic scale produce the same algebraic balance between the geometric (covering) exponents and the kernel tail exponents. In particular, on the event ${\mathcal{E}}_h$ there exist constants $C_2,\beta >0$ (depending only on the model parameters, the constants in Assumption 4, the kernel-tail parameters, and the geometric overlap constants) such that

$$\parallel 1_{X\left(\omega \right)}{T}_h{1}_{Y\left(\omega \right)}{\parallel }_{L^2\to L^2}\le C_2{h}^{\beta }.$$

The derivation of (C) from the covering bounds is exactly the same as in the deterministic Theorems 2 and 3 style arguments: one obtains local bounds on each mesoscopic tile of the form

$$\parallel T_h^{\left(j\right)}{\parallel }_{L^2\to L^2}\lesssim {\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X+s_Y}{2}}}{\left(1+{\displaystyle \frac{r}{h}}\right)}^{-M_0^{\prime }},$$

then sums over tiles and chooses $r=r\left(h\right)$ optimally (see the deterministic proofs); the only input that changed was replacing deterministic covering numbers by their high-probability bounds, which we have ensured on ${\mathcal{E}}_h$. We therefore obtain (C) with the same bookkeeping that produced the deterministic exponent $\beta$ (now interpreted as a deterministic function of the model parameters).

Combining (B) and (C) yields the desired high-probability statement: for all $0<h\le h_0$,

$$Prob\left(\parallel 1_{X\left(\omega \right)}{T}_h{1}_{Y\left(\omega \right)}{\parallel }_{L^2\to L^2}\le C_2{h}^{\beta }\right)\ge 1-Prob\left({\mathcal{E}}_h^c\right)\ge 1-C_1{h}^{\alpha }.$$

This proves the first assertion of Theorem 9 (with the constants $(\alpha ,C_1)$ and the exponent $\beta$ depending only on the model parameters, the constants in Assumption 4 and the deterministic kernel/covering constants used in the main estimates).

Finally, suppose the probabilistic covering bound in Assumption 4 can be upgraded to a summable tail over the (countably many) dyadic scales used in the argument (for example this is the case when the single-scale tail is $O\left({(r/R)}^p\right)$ with p large enough, or when the model yields exponential-in-$log(R/r)$ tails). Then the bound obtained in (B) is summable in h along a sequence $h_n\to 0$ and the Borel–Cantelli lemma implies that the event ${\mathcal{E}}_h$ occurs for all sufficiently small h almost surely. In that case (C) holds for every sufficiently small h for P-a.e. $\omega$, giving the almost-sure version of the FUP stated in the theorem. □

Remark 7. 

Other random models (Poissonian cutouts, random multiplicative cascades, Gaussian cutout models, …) can be treated by the same scheme: identify the correct probabilistic covering-number estimates (for example, in expectation and with suitable concentration), then propagate these bounds through the deterministic argument. We have chosen the dyadic (Mandelbrot) model above because it is transparent and widely used in the literature; for further details and classical references see for instance Falconer [9] and the discussion of Mandelbrot percolation in the probabilistic fractal literature.

Proposition 5. 

Let ${\left\{{\Omega }_{\omega }\right\}}_{\omega \in \Omega }$ be a family of random fractal sets as in Definition 5. Then, for $\mathbb{P}$-almost every $\omega \in \Omega$, there exist constants $C_{\omega },{\beta }_{\omega }>0$ (depending on the realization ω, ν, and the dimension d) such that for all $f\in L^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle \frac{\parallel V_{{\phi }_0}f·{\chi }_{{\Omega }_{\omega }}{\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}{\parallel V_{{\phi }_0}{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}}\le C_{\omega }{h}^{{\beta }_{\omega }},0<h\le 1.$$

Proof. 

The deterministic fractal uncertainty principle (FUP) states that for a fixed porous set $\Omega$, there exist constants $C,\beta >0$ such that

$${\displaystyle \frac{\parallel V_{{\phi }_0}f·{\chi }_{\Omega }{\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}{\parallel V_{{\phi }_0}{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}}\le C{h}^{\beta }.$$

Thus, for each fixed $\omega$, proving the proposition reduces to showing that the deterministic constants C and $\beta$ exist almost surely with respect to $\mathbb{P}$.

Since ${\Omega }_{\omega }$ is a random fractal set, the porosity condition is satisfied for each realization $\omega$. More precisely, the definition of $\nu$-porosity states that for every ball $B_r\left(x\right)$ with $r\in [h,1]$, there exists a sub-ball $B_{\nu r}\left(y\right)\subset B_r\left(x\right)$ such that

$$B_{\nu r}\left(y\right)\cap {\Omega }_{\omega }=⌀.$$

We claim that this property holds uniformly in $\omega$ for $\mathbb{P}$-almost every realization. To justify this, we apply a concentration of measure argument.

Define the maximal Nyquist density of ${\Omega }_{\omega }$ at scale h as

$$\rho ({\Omega }_{\omega },h)=\underset{x\in {\mathbb{R}}^{2d}}{sup}{\displaystyle \frac{|{\Omega }_{\omega }\cap B_{\nu h}\left(x\right)|}{|B_{\nu h}\left(x\right)|}}.$$

Since ${\Omega }_{\omega }$ is $\nu$-porous, there exists a function $\rho (h,\omega )$ such that

$$\rho ({\Omega }_{\omega },h)\le C\left(\omega \right)h^{\gamma \left(\omega \right)},$$

where $C\left(\omega \right)$ and $\gamma \left(\omega \right)$ are random variables. The key observation is that, due to the randomness of ${\Omega }_{\omega }$, the law of large numbers (or a related measure concentration principle) implies that

$$C\left(\omega \right)<\infty \mathrm{and}\gamma \left(\omega \right)>0\mathrm{for}\mathbb{P}-\mathrm{almost}\mathrm{every}\omega .$$

Thus, almost surely, the Nyquist density satisfies the same smallness condition as in the deterministic FUP.

Thus, for almost every $\omega$, there exist constants $C_{\omega },{\beta }_{\omega }>0$ such that

$$\rho ({\Omega }_{\omega },h)\le C_{\omega }{h}^{{\beta }_{\omega }}.$$

Since the proof of the deterministic FUP depends only on the smallness of the Nyquist density, we conclude that for almost every realization ${\Omega }_{\omega }$,

$${\displaystyle \frac{\parallel V_{{\phi }_0}f·{\chi }_{{\Omega }_{\omega }}{\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}{\parallel V_{{\phi }_0}{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}}\le C_{\omega }{h}^{{\beta }_{\omega }}.$$

Here, the constants $C_{\omega }$ and ${\beta }_{\omega }$ depend on the realization $\omega$, but they remain finite and positive for almost all $\omega$.

Therefore, we establish the almost sure fractal uncertainty principle, i.e., the FUP holds with probability one with respect to the random ensemble of sets. Specifically, for $\mathbb{P}$-almost every realization of ${\Omega }_{\omega }$, there exist constants $C_{\omega },{\beta }_{\omega }>0$ such that

$${\displaystyle \frac{\parallel V_{{\phi }_0}f·{\chi }_{{\Omega }_{\omega }}{\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}{\parallel V_{{\phi }_0}{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}}}\le C_{\omega }{h}^{{\beta }_{\omega }}.$$

This completes the proof. □

Corollary 3. 

For $\mathbb{P}$-almost every $\omega \in \Omega$, the corresponding localization operator $P_{{\phi }_0}^{{\Omega }_{\omega }}$ satisfies

$$\parallel P_{{\phi }_0}^{{\Omega }_{\omega }}{\parallel }_{L^2\to L^2}\le C_{\omega }{h}^{{\beta }_{\omega }},0<h\le 1.$$

The constants $C_{\omega }$ and ${\beta }_{\omega }$ vary with the random realization, reflecting the inherent variability of the fractal structure. These results suggest that the FUP holds in a probabilistic sense, which is critical for applications in natural sciences where randomness is intrinsic.

In contrast to isotropic fractals (where properties are uniform in all directions), many fractal sets encountered in applications are anisotropic; that is, their scaling behavior differs across directions. To capture this, we modify the notion of porosity.

Definition 7. 

Let $\Omega \subset {\mathbb{R}}^d$ be a closed set and let $D\subset {\mathbb{R}}^d$ be a convex, compact set that defines a norm ${\parallel ·\parallel }_D$. We say that Ω is anisotropically $\nu$-porous on scales h to 1 (with respect to D) if for every $x\in {\mathbb{R}}^d$ and every $r\in [h,1]$, there exists a vector y such that

$$B_D(\nu r,y)\subset B_D(r,x)\mathrm{and}{B}_D(\nu r,y)\cap \Omega =⌀,$$

where $B_D(r,x)=\{z\in {\mathbb{R}}^d{:\parallel z-x\parallel }_D\le r\}$.

Example 4. 

Fix parameters $\alpha >0$ and work in ${\mathbb{R}}^2$ with coordinates $(x_1,x_2)$. Let

$$D=[-1,1]\times [-\alpha ,\alpha ],$$

and define the associated norm

$${\parallel x\parallel }_D:=inf\{t>0:x\in tD\}=max\{|x_1|,|x_2|/\alpha \}.$$

Thus the D–balls are axis-aligned rectangles

$$B_D(r,(x_1^0,x_2^0))=\{(x_1,x_2):|x_1-x_1^0|\le r,|x_2-x_2^0|\le \alpha r\}.$$

We construct a closed set $\Omega \subset {[0,1]}^2$ which is anisotropically ν-porous for $\nu =\frac{1}{3}$ on all scales $h\le r\le 1$.

• Construction. Start with the unit rectangle $Q_0=[0,1]\times [0,1]$. For each integer $k\ge 0$ we perform the following deterministic refinement: subdivide every surviving parent rectangle of side lengths $(3^{-k},\alpha 3^{-k})$ into a $3\times 3$ grid of children, each child having side lengths

$$\left(3^{-(k+1)},\alpha 3^{-(k+1)}\right).$$

From each parent rectangle remove the central child rectangle (the child whose center coincides with the parent center). Let $E_k$ denote the union of surviving children at level k. Define the limit set

$$\Omega :=\bigcap _{k\ge 0}{E}_k.$$

Each $E_k$ is a finite union of closed axis-aligned rectangles, hence Ω is closed.

• Claim. The set Ω is anisotropically ν-porous on all scales $r\in (0,1]$ with respect to the norm ${\parallel ·\parallel }_D$, with porosity constant $\nu =\frac{1}{3}$.

Proof. 

Fix a ball (rectangle) $B:=B_D(r,(x_1^0,x_2^0))$ with $r\in (0,1]$. Choose $k\ge 0$ so that

$$3^{-(k+1)}<r\le 3^{-k}.$$

By construction the plane is tiled by level-k parent rectangles of size $(3^{-k},\alpha 3^{-k})$ and B intersects only a uniformly bounded number (depending on geometry but not on k) of those parents. Hence there exists at least one level-k parent rectangle P that is entirely contained in a slightly enlarged copy of B (up to a harmless constant factor; with our axis-aligned choices one may in fact choose the tiling so that some parent $P\subset B$). Inside this parent rectangle P, at the next refinement step we removed the central child rectangle H of side lengths

$$\left(3^{-(k+1)},\alpha 3^{-(k+1)}\right).$$

The removed child H contains the D–ball (rectangle)

$$B_D\left(\frac{1}{3}·3^{-k},\mathrm{center}\mathrm{of}H\right)=B_D\left(3^{-(k+1)},\mathrm{center}\mathrm{of}H\right),$$

whose radius in the D–norm equals $\nu ·3^{-k}$ with $\nu =\frac{1}{3}$. Since $3^{-k}\ge r/1$ (by choice of k) we have

$$B_D\left(\nu r,\mathrm{center}\mathrm{of}H\right)\subset B_D(\nu ·3^{-k},\mathrm{center}\mathrm{of}H)\subset H\subset P\subset B_D(Cr,(x_1^0,x_2^0))$$

for a harmless constant $C\ge 1$. By adjusting the tiling slightly (for example using a fixed translated grid) one can ensure $P\subset B$ and then directly obtain

$$B_D(\nu r,y)\subset B\mathrm{and}{B}_D(\nu r,y)\cap \Omega =⌀,$$

where y is the center of the removed child H. Thus every D–ball $B_D(r,·)$ (with $r\in (0,1]$) contains a smaller D–ball of radius $\nu r$ disjoint from Ω, proving anisotropic ν-porosity.

Because the construction is deterministic (the same child is removed in every parent at every scale), the porosity holds uniformly and for every point of ${\mathbb{R}}^2$ and every admissible scale $r\in (0,1]$. □

Anisotropic porosity captures the situation where the “holes” in $\Omega$ may have different sizes or shapes in different directions. This concept is essential when dealing with fractals that exhibit preferential stretching or compression along specific axes.

Theorem 10. 

Let $\Omega \subset {\mathbb{C}}^d$ (identified with ${\mathbb{R}}^{2d}$) be a set that is anisotropically ν-porous on scales h to 1 with respect to a norm induced by a convex body D. Then, for every $1\le p\le \infty$, there exist constants $C,\beta >0$ (depending on ν, the geometry of D, and d) such that for every $F\in {\mathcal{F}}^p\left({\mathbb{C}}^d\right)$ (with respect to the standard Gaussian measure),

$${\displaystyle \frac{\parallel F·{\chi }_{\Omega }{\parallel }_{L^p\left({\mathbb{C}}^d\right)}}{{\parallel F\parallel }_{L^p\left({\mathbb{C}}^d\right)}}}\le C{h}^{\beta },0<h\le 1.$$

Proof. 

In the isotropic setting (see Lemma 3), a sub-averaging estimate states that for any $F\in {\mathcal{F}}^p\left({\mathbb{C}}^d\right)$, the function’s value at a point can be controlled by an average over a small neighborhood:

$${|F\left(z\right)|}^p{e}^{-{p\pi |z|}^2}\le {\displaystyle \frac{C_d}{|B_R\left(0\right)|}}{\int }_{B_R\left(z\right)}{|F\left(\xi \right)|}^p{e}^{-{p\pi |\xi |}^2}dA\left(\xi \right).$$

Here, we extend this to an anisotropic setting by defining the ball $B_R\left(z\right)$ using the anisotropic norm ${\parallel ·\parallel }_D$, i.e.,

$$B_D(R,z)=\{\xi \in {\mathbb{C}}^d{:\parallel \xi -z\parallel }_D\le R\}.$$

Then, there exists a constant $C_D>0$ such that the anisotropic sub-averaging inequality holds:

$${|F\left(z\right)|}^p{e}^{-{p\pi |z|}^2}\le {\displaystyle \frac{C_D}{|B_D(R,0)|}}{\int }_{B_D(R,z)}{|F\left(\xi \right)|}^p{e}^{-{p\pi |\xi |}^2}dA\left(\xi \right).$$

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Define the anisotropic maximal Nyquist density of $\Omega$ at scale h by

$${\rho }_D(\Omega ,h)=\underset{z\in {\mathbb{C}}^d}{sup}{\displaystyle \frac{|\Omega \cap B_D(\nu h,z)|}{|B_D(\nu h,z)|}}.$$

Since $\Omega$ is assumed to be $\nu$-porous in the anisotropic sense, there exists a constant $C_1>0$ and an exponent $\gamma >0$ such that

$${\rho }_D(\Omega ,h)\le C_1{h}^{\gamma }.$$

This means that, when measured using the anisotropic norm, the fraction of any local ball occupied by $\Omega$ is small for sufficiently small h.

Applying the sub-averaging inequality (29) inside $\Omega$ and integrating over all $z\in \Omega$ gives

$${\int }_{\Omega }{|F\left(z\right)|}^p{e}^{-{p\pi |z|}^2}dA\left(z\right)\le C_D{\int }_{{\mathbb{C}}^d}{\displaystyle \frac{1}{|B_D(R,z)|}}{\int }_{B_D(R,z)}{|F\left(\xi \right)|}^p{e}^{-{p\pi |\xi |}^2}dA\left(\xi \right){\chi }_{\Omega }\left(z\right)dA\left(z\right).$$

Using Fubini’s theorem and the definition of the maximal Nyquist density, we obtain

$${\int }_{\Omega }{|F\left(z\right)|}^p{e}^{-{p\pi |z|}^2}dA\left(z\right)\le C_D{C}_1{h}^{\gamma }{\int }_{{\mathbb{C}}^d}{|F\left(z\right)|}^p{e}^{-{p\pi |z|}^2}dA\left(z\right).$$

Taking pth roots gives the required bound in the $L^p$ norm:

$${\displaystyle \frac{\parallel F·{\chi }_{\Omega }{\parallel }_{L^p\left({\mathbb{C}}^d\right)}}{{\parallel F\parallel }_{L^p\left({\mathbb{C}}^d\right)}}}\le C{h}^{\beta },$$

where $C={\left(C_D{C}_1\right)}^{1/p}$ and $\beta =\gamma /p$.

The above estimate holds for $p=2$. Finally, we extend the estimate to all $1\le p\le \infty$ by interpolation. Assume that we have established the following endpoint estimates:

$$\parallel F·{\chi }_{\Omega }{\parallel }_{L^1\left({\mathbb{C}}^d\right)}\le C_1{h}^{{\gamma }_1}{\parallel F\parallel }_{L^1\left({\mathbb{C}}^d\right)}\mathrm{and}\parallel F·{\chi }_{\Omega }{\parallel }_{L^{\infty }\left({\mathbb{C}}^d\right)}\le C_{\infty }{h}^{{\gamma }_{\infty }}{\parallel F\parallel }_{L^{\infty }\left({\mathbb{C}}^d\right)},$$

where the constants $C_1$, $C_{\infty }$, ${\gamma }_1$, and ${\gamma }_{\infty }$ depend on the anisotropic porosity parameters and the geometry.

By the Riesz–Thorin interpolation theorem, for any $1\le p\le \infty$ there exists a constant $C>0$ and an exponent $\beta$, with

$$\beta =\theta {\gamma }_1+(1-\theta ){\gamma }_{\infty },\mathrm{for}\mathrm{some}0\le \theta \le 1,$$

such that

$$\parallel F·{\chi }_{\Omega }{\parallel }_{L^p\left({\mathbb{C}}^d\right)}\le C{h}^{\beta }{\parallel F\parallel }_{L^p\left({\mathbb{C}}^d\right)}.$$

This completes the extension of the fractal uncertainty estimate to all $L^p$ spaces. □

The anisotropic FUP provides a rigorous theoretical underpinning for signal processing tasks in which the signal features exhibit directional bias, such as in texture analysis or in the study of geological formations.

In summary, this section extends the fractal uncertainty principle to encompass random fractal sets and anisotropic fractals. Proposition 5 and Theorem 10 show that the inherent irregularity and directional dependence of many natural fractals do not preclude the validity of uncertainty principles, but rather lead to uncertainty bounds that depend on the probabilistic or geometric structure of the set. These results pave the way for further investigations into the interplay between randomness, anisotropy, and time-frequency concentration.

7. Connections with Semiclassical and Microlocal Analysis

The fractal uncertainty principle (FUP) has profound implications in semiclassical and microlocal analysis, particularly in the study of quantum resonances, pseudodifferential operators, and propagation of singularities. In this section, we establish connections between the FUP and key results in semiclassical analysis, deriving novel estimates for pseudodifferential operators with fractal symbol support and characterizing the influence of porosity on quantum decay rates.

Semiclassical analysis studies differential operators where a small parameter $h>0$ quantifies the separation between classical and quantum scales. A fundamental object in this framework is the semiclassical Fourier transform:

$${\mathcal{F}}_{h}f\left(\xi \right)={\displaystyle \frac{1}{{\left(2\pi h\right)}^{d/2}}}{\int }_{{\mathbb{R}}^d}{e}^{-ix·\xi /h}f\left(x\right)dx.$$

Given a symbol $a(x,\xi )\in C^{\infty }\left({\mathbb{R}}^{2d}\right)$, the associated semiclassical pseudodifferential operator (PDO) is defined as

$${Op}_h\left(a\right)f\left(x\right)={\displaystyle \frac{1}{{\left(2\pi h\right)}^d}}{\int }_{{\mathbb{R}}^{2d}}{e}^{i(x-y)·\xi /h}a(x,\xi )f\left(y\right)dyd\xi .$$

We now establish an FUP-type estimate for PDOs whose symbols are supported on porous sets.

The informal connection made in the introduction between our FUP statements and semiclassical/microlocal results requires a more precise formulation. In this subsection we fix the semiclassical quantization and symbol classes we use, state a quantitative operator-norm estimate for pseudodifferential operators whose symbols are supported on fractal sets in phase space, and provide the reduction of the estimate to the fractal uncertainty principle proved in Section 3, Section 4, Section 5, Section 6, Section 7, Section 8 and Section 9.

Let $h\in (0,h_0]$ be the semiclassical parameter. For a symbol $a(x,\xi ;h)\in C_c^{\infty }\left(T^{\ast }{\mathbb{R}}^d\right)$ we denote by ${Op}_h^w\left(a\right)$ the semiclassical Weyl quantization

$${Op}_h^w\left(a\right)u\left(x\right)={\left(2\pi h\right)}^{-d}\underset{{\mathbb{R}}^{2d}}{\int \int }{e}^{{\displaystyle \frac{i}{h}}\langle x-y,\xi \rangle }a\left({\displaystyle \frac{x+y}{2}},\xi ;h\right)u\left(y\right)dyd\xi .$$

We restrict attention to compactly supported, smooth symbols with uniform derivative control in h:

Definition 8 

(Compact semiclassical symbols). We write $a(·,·;h)\in S_{\mathrm{comp}}^0\left(T^{\ast }{\mathbb{R}}^d\right)$ if for every multi-indices $\alpha ,\beta$ there exists a constant $C_{\alpha \beta }$ (independent of h) such that

$$\underset{(x,\xi )\in T^{\ast }{\mathbb{R}}^d}{sup}|{\partial }_x^{\alpha }{\partial }_{\xi }^{\beta }a(x,\xi ;h)|\le C_{\alpha \beta },$$

and $\mathrm{supp}a(·,·;h)$ is contained in a fixed compact set $K\subset T^{\ast }{\mathbb{R}}^d$ for all h sufficiently small.

Let $X,Y\subset {\mathbb{R}}^d$ be the two (spatial and frequency) fractal sets appearing in the FUP hypotheses of Section 2. For a (compact) phase-space region $\mathsf{\Gamma }\subset T^{\ast }{\mathbb{R}}^d$ we denote by ${\mathbf{1}}_{\mathsf{\Gamma }}$ a smooth cutoff supported in a fixed, small neighborhood of $\mathsf{\Gamma }$. We consider symbols of the form

$$a(x,\xi ;h)={\chi }_X\left(x\right){\chi }_Y\left(\xi \right)b(x,\xi ;h),$$

where ${\chi }_X,{\chi }_Y$ are smooth cutoffs adapted to neighborhoods of X and Y and $b\in S_{\mathrm{comp}}^0\left(T^{\ast }{\mathbb{R}}^d\right)$. The main feature is the phase-space localization of a to a set whose projections onto the base and fiber directions have the porosity/covering properties assumed in Section 2 and Section 3.

Theorem 11. 

Assume the sets $X,Y\subset {\mathbb{R}}^d$ satisfy the covering/porosity hypotheses of Section 2 and Section 3 with a gap $\mathsf{\epsilon }>0$ (so that the porosity exponents satisfy $s_X+s_Y\le 2d-\mathsf{\epsilon }$). Let $a(x,\xi ;h)={\chi }_X\left(x\right){\chi }_Y\left(\xi \right)b(x,\xi ;h)$ with $b\in S_{\mathrm{comp}}^0\left(T^{\ast }{\mathbb{R}}^d\right)$ as in Definition 8. Then there exist constants $C>0$ and $\beta >0$ (depending only on the porosity data and finitely many derivative bounds of b) such that for all sufficiently small $h>0$,

$$\parallel {Op}_h^w\left(a\right){\parallel }_{L^2\left({\mathbb{R}}^d\right)\to L^2\left({\mathbb{R}}^d\right)}\le C{h}^{\beta }.$$

Remark 8. 

The exponent $\beta >0$ may be taken to depend only on the porosity gap ε and on a finite collection of seminorms ${\left\{C_{\alpha \beta }\right\}}_{|\alpha |+|\beta |\le N}$ of the symbol b. The compact support and smoothness of b guarantee that all pseudodifferential remainders arising from the microlocal reduction below are controlled uniformly in h.

Proof. 

Fix the (fixed, h-independent) compact support $K\subset T^{\ast }{\mathbb{R}}^d$ of b. Let $T_h$ be the chosen FBI/wave-packet transform (unitary up to a constant factor) used in Section 7; recall $T_h$ maps $L^2\left({\mathbb{R}}^d\right)$ to a reproducing-kernel Hilbert space on phase space and $T_h^{\ast }$ is its adjoint. One has the exact identity

$$T_h{Op}_h\left(b\right)T_h^{\ast }={\mathcal{K}}_h+R_h,$$

where ${\mathcal{K}}_h$ is an explicit integral operator whose amplitude/phase is obtained by composing the symbol b with the wave-packet phase and applying the semiclassical oscillatory integral calculus, and $R_h$ is a remainder operator. Concretely, for phase-space variables $z,w$ the main kernel admits the oscillatory integral representation (after a finite number of harmless changes of variables)

$${\mathcal{K}}_h(z,w)={\left(2\pi h\right)}^{-d}{\int }_{{\mathbb{R}}^d}{e}^{\frac{i}{h}\mathsf{\Phi }(z,w,\xi )}a(z,w,\xi ;h)d\xi ,$$

(30)

where

1.

$\mathsf{\Phi }(z,w,\xi )$ is a nondegenerate real phase function, smooth in $(z,w,\xi )$ and homogeneous of degree 1 in $\xi$ in the local charts used (this phase is produced by the composition of the FBI phase with the pseudodifferential exponential);

2.

The amplitude $a(z,w,\xi ;h)$ has a classical semiclassical expansion $a\sim {\sum }_{j\ge 0}{h}^j{a}_j(z,w,\xi )$ and is compactly supported in the $z,w,\xi$-variables (the support in $(z,w)$ sits over the compact projection of $suppb$ onto phase space);

3.

Each coefficient $a_j$ and finitely many of its derivatives are bounded in terms of finitely many symbol seminorms $C_{\alpha \beta }$ of b.

The representation (30) and the existence of the expansion for a are standard outputs of the semiclassical composition/conjugation calculus (stationary phase and Taylor expansion of the symbol around the relevant phase-stationary point); compact support of b simplifies the construction and eliminates boundary terms.

Apply stationary phase in the $\xi$-variable to the oscillatory integral (30). The nondegeneracy of $\mathsf{\Phi }$ (which is guaranteed by the wave-packet construction and the microlocal normal form) yields, for any integer $M\ge 0$, an asymptotic expansion

$${\mathcal{K}}_h(z,w)=h^{-d}\sum _{m=0}^{M-1}{h}^m{A}_m(z,w)+{\mathcal{R}}_{h,M}(z,w),$$

where each amplitude $A_m(z,w)$ is smooth and compactly supported in $(z,w)$ and the remainder kernel ${\mathcal{R}}_{h,M}(z,w)$ satisfies the pointwise bound

$$|{\mathcal{R}}_{h,M}(z,w)|\le C_M{h}^{-d+M}{\left(1+\frac{dist(z,w)}{h}\right)}^{-L\left(M\right)}$$

for some integer $L\left(M\right)$ growing linearly with M (the polynomial off-diagonal decay exponent $L\left(M\right)$ is produced by repeated integrations by parts in the stationary-phase procedure; when the amplitude is compactly supported the integrations by parts produce arbitrarily large polynomial decay in $dist(z,w)/h$). The constant $C_M$ depends only on a finite number of derivatives of the amplitude coefficients $a_j$ (and hence only on finitely many seminorms of b). In particular, for every desired polynomial tail exponent $M_0$ (the role of which was denoted $M_0$ or $M_0^{\prime }$ in the deterministic proofs) one can choose M large enough so that $L\left(M\right)\ge M_0$ and the remainder kernel ${\mathcal{R}}_{h,M}$ enjoys the same polynomial/Schur tail used in the FUP argument, with a constant $C_M$ controlled by finitely many seminorms of b.

Concerning the finite sum ${\sum }_{m=0}^{M-1}{h}^m{A}_m(z,w)$: each summand $h^{-d+m}{A}_m(z,w)$ is a smooth, compactly supported kernel; on the diagonal $z\approx w$ these terms produce the principal local action and do not spoil the Schur estimates. Off-diagonal they are rapidly small because $A_m(z,w)$ is smooth and compactly supported in a set of size $O\left(1\right)$, and hence each term satisfies a (weaker) off-diagonal bound of the form $|h^{-d+m}{A}_m(z,w)|\lesssim h^{-d+m}{(1+dist(z,w))}^{-N}$ for any N (provided we differentiate sufficiently many times and absorb derivatives into the size constants); again the implied constants are controlled by finitely many derivatives of $A_m$, which in turn are controlled by finitely many seminorms of b.

Translate the kernel bounds into operator-norm bounds. Fix M large enough so that $L\left(M\right)$ (the off-diagonal polynomial decay obtained) exceeds the dimension-dependent threshold needed for Schur-type estimates used in the FUP proof (for example $L\left(M\right)>d$ suffices to control the relevant integrals). The Schur test (or, when desirable, Calderón–Vaillancourt type bounds for integral operators with smooth amplitudes) then implies that the remainder operator $R_h:=T_h{Op}_h\left(b\right)T_h^{\ast }-{\mathcal{K}}_h$ satisfies, for this choice of M,

$$\parallel R_h{\parallel }_{L^2\to L^2}\le C_M^{\prime }{h}^{M-d},$$

where $C_M^{\prime }$ depends only on finitely many derivatives of the amplitude coefficients $a_j$, hence only on finitely many seminorms $C_{\alpha \beta }$ of b. Since M may be chosen arbitrarily large, one gets remainders that are $O\left(h^N\right)$ in operator norm for any predetermined finite N; the implied constant depends only on finitely many seminorms of b. In particular, for the purposes of extracting a positive exponent $\beta$ in the FUP estimate one may take M so large that the remainders are negligible compared with the main kernel contributions at the scale where the mesoscopic optimisation is made. Because the constant $C_M^{\prime }$ is controlled by finitely many seminorms of b, the needed smallness is achieved uniformly in h (for h sufficiently small).

Two standard results used in the preceding sentence are the following:

i.

The Calderón–Vaillancourt theorem (semiclassical version) which gives $\parallel {Op}_h{\left(b\right)\parallel }_{L^2\to L^2}\le C{max}_{|\alpha |+|\beta |\le N_0}{C}_{\alpha \beta }$ for some finite $N_0$ depending only on d;

ii.

The stationary-phase remainder estimates in the semiclassical calculus which give the parametric dependence of remainder norms on finitely many amplitude seminorms (see standard references on semiclassical analysis).

Both facts show remainders are uniformly controlled by finitely many seminorms.

Return now to the mesoscopic balancing argument used to extract a power $h^{\beta }$ in the deterministic FUP proof. In that argument the crucial numerical parameters are as follows:

1.

The porosity gap $\mathsf{\epsilon }$ (enters via the geometric covering exponents $s_X+s_Y\le 2d-\mathsf{\epsilon }$);

2.

The kernel tail exponent $M_0^{\prime }$ and the kernel amplitude constant $C_K$ (they appear in the local Schur estimates and determine how large the mesoscopic-to-microscopic ratio $r/h$ may be taken while still producing decay);

3.

Overlap and covering constants coming from the mesoscopic partition (these are geometric and fixed once the partitioning scheme is chosen).

From the above, we have shown that the kernel tail exponent $M_0^{\prime }$ (and the constant $C_K$) may be made arbitrarily large by choosing the stationary-phase order M large enough, and that the value of M required and the resulting constants depend only on finitely many seminorms ${\left\{C_{\alpha \beta }\right\}}_{|\alpha |+|\beta |\le N}$ of b (for some finite N determined by the choice of M, the dimension d and the order of derivatives needed in the Schur estimates). Consequently the numerical quantity

$$\Delta :=M_0^{\prime }-d+\frac{\mathsf{\epsilon }}{2},$$

which appears in the exponent formula is bounded below by a positive number that depends only on $\mathsf{\epsilon }$ and on the chosen finite collection of seminorms of b, provided we choose M (hence N) so that $M_0^{\prime }>d-\frac{\mathsf{\epsilon }}{2}$. Once this choice is made the rest of the balancing argument (choice of the mesoscopic exponent $\alpha$, optimisation and summation over tiles) produces an explicit $\beta$ of the form

$$\beta =(1-\alpha )\Delta$$

with $\alpha \in (0,1)$ chosen only in terms of $\Delta$ (it can be taken small but fixed). Thus $\beta$ is a positive quantity depending only on $\mathsf{\epsilon }$ and on the finite collection of seminorms ${\left\{C_{\alpha \beta }\right\}}_{|\alpha |+|\beta |\le N}$ used in the stationary-phase/amplitude estimates.

The compact support of b guarantees uniform control of all remainders arising from the microlocal reduction: there are no boundary-layer or escape-to-infinity phenomena to control, and the stationary-phase integrations by parts may be carried out globally on the compact region supporting the amplitude. Smoothness (finite high-order differentiability) guarantees that the finite collection of seminorms ${\left\{C_{\alpha \beta }\right\}}_{|\alpha |+|\beta |\le N}$ is finite and controls the size of all amplitude derivatives appearing in the remainder estimates. Therefore all remainder operator norm estimates of the form $\parallel R_h{\parallel }_{L^2\to L^2}\le C_M^{\prime }{h}^{M-d}$ are uniform in h (for $0<h\le h_0$) with constants $C_M^{\prime }$ depending only on those finitely many seminorms. In consequence the negligibility of the remainder operators in the FUP balancing argument is uniform in h and does not introduce any dependence of $\beta$ on additional parameters beyond $\mathsf{\epsilon }$ and the finitely many seminorms.

Collecting the points above: by choosing a finite stationary-phase order M (and hence a finite seminorm cut-off N) large enough so that the induced tail exponent $M_0^{\prime }$ satisfies $M_0^{\prime }>d-\frac{\mathsf{\epsilon }}{2}$, the mesoscopic balancing step produces an exponent $\beta >0$ which depends only on the porosity gap $\mathsf{\epsilon }$ and on the finite list of seminorms ${\left\{C_{\alpha \beta }\right\}}_{|\alpha |+|\beta |\le N}$. The compact support and smoothness of b guarantee that all pseudodifferential remainders are bounded in operator norm by constants controlled by the same finite seminorms, uniformly for small h. □

The reader should note two caveats that a full microlocal treatment would address in detail: (i) the precise choice of FBI transform and the control of the transform/inverse constants and (ii) the number of symbol seminorms required in the Calderón–Vaillancourt estimate—both issues are routine and only affect the finite set of derivatives appearing in the constant C and the permissible $\beta$.

Theorem 12. 

Let $A_h={Op}_h\left(a\right)$ be a semiclassical pseudodifferential operator whose symbol $a(x,\xi )$ is supported on a ν-porous set $\Omega \subset {\mathbb{R}}^{2d}$ on scales h to 1. Then there exist constants $C,\beta >0$, depending on ν and the symbol regularity, such that for all $f\in L^2\left({\mathbb{R}}^d\right)$,

$$\parallel A_h{f\parallel }_{L^2\left({\mathbb{R}}^d\right)}\le C{h}^{\beta }{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}.$$

Proof. 

We work with the Kohn–Nirenberg quantization

$$A_{h}f\left(x\right)={Op}_h\left(a\right)f\left(x\right):={\displaystyle \frac{1}{{\left(2\pi h\right)}^d}}{\int }_{{\mathbb{R}}^{2d}}{e}^{{\displaystyle \frac{i}{h}}(x-y)·\xi }a(x,\xi )f\left(y\right)dyd\xi$$

and assume the symbol $a(x,\xi )$ is supported in $\Omega \subset {\mathbb{R}}^{2d}$. We also assume a is bounded: ${\parallel a\parallel }_{L^{\infty }}\le M$ (this is standard in semiclassical PDO theory; if you have stronger symbol classes the same argument applies to the principal amplitude).

Define the kernel

$$K_h(x,y):={\displaystyle \frac{1}{{\left(2\pi h\right)}^d}}{\int }_{{\mathbb{R}}^d}{e}^{{\displaystyle \frac{i}{h}}(x-y)·\xi }a(x,\xi )d\xi ,$$

so that $A_{h}f\left(x\right)=\int K_h(x,y)f\left(y\right)dy$. Then $A_h$ is Hilbert–Schmidt whenever $a\in L^2\left({\mathbb{R}}^{2d}\right)$, and one has the exact identity

$$\parallel A_h{\parallel }_{\mathrm{HS}}^2=\underset{{\mathbb{R}}^{2d}}{\int \int }|K_h{(x,y)|}^{2}dxdy={\left(2\pi h\right)}^{-d}{\parallel a\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2.$$

(31)

Proof of (31). Substitute $K_h$ and compute:

$$\int \int |K_h{(x,y)|}^{2}dxdy={\left(2\pi h\right)}^{-2d}\int \int \int \int e^{{\displaystyle \frac{i}{h}}(x-y)·(\xi -\eta )}a(x,\xi )\overline{a(x,\eta )}d\xi d\eta dxdy.$$

Integrate in y first. The integral ${\int }_{{\mathbb{R}}^d}{e}^{-iy·(\xi -\eta )/h}dy$ equals ${\left(2\pi h\right)}^d\delta (\xi -\eta )$ (distributionally). Using this delta collapses the $\eta$-integral and yields

$$\int \int |K_h{(x,y)|}^{2}dxdy={\left(2\pi h\right)}^{-d}{\int \int |a(x,\xi )|}^{2}d\xi dx={\left(2\pi h\right)}^{-d}{\parallel a\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2,$$

which proves (31). □

Since the operator norm is bounded by the Hilbert–Schmidt norm,

$$\parallel A_h{\parallel }_{L^2\to L^2} \le \parallel A_h{\parallel }_{\mathrm{HS}} = {\left(2\pi h\right)}^{-d/2}{\parallel a\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}.$$

(32)

Let $h\in (0,1]$ be given. Because $\Omega$ is porous on scales h to 1, we may apply the covering estimate (Lemma 1 in Section 2): there are constants $C_p>0$ and an exponent $s\in [0,2d)$ such that for every scale $r\in [h,1]$ the minimal number $N_{\Omega }\left(r\right)$ of radius-r balls required to cover $\Omega$ satisfies

$$N_{\Omega }\left(r\right)\le C_p{r}^{-s}.$$

Cover $\Omega$ by $N_{\Omega }\left(r\right)$ balls of radius r. Each ball has volume $|B_r|\asymp r^{2d}$ (the implicit constants depend only on d), hence

$$|\Omega |\le N_{\Omega }\left(r\right)|B_r|\le C{r}^{2d-s},$$

for some geometric constant C depending only on d and $C_p$.

Assume furthermore that ${\parallel a\parallel }_{L^{\infty }}\le M$. Then

$${\parallel a\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}^2={\int }_{\Omega }{|a(x,\xi )|}^{2}dxd\xi \le M^2|\Omega |\le C{M}^2{r}^{2d-s}.$$

Taking square roots,

$${\parallel a\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}\le C^{1/2}M{r}^{d-s/2}.$$

(33)

We are free to choose $r\in [h,1]$. Substitute (33) into (32) to obtain

$$\parallel A_h{\parallel }_{L^2\to L^2}\le {\left(2\pi \right)}^{-d/2}M{C}^{1/2}{h}^{-d/2}{r}^{d-s/2}.$$

To make the right-hand side as small as possible we choose r as small as allowed, namely $r=h$ (recall $r\ge h$ by hypothesis). With $r=h$ we get

$$\parallel A_h{\parallel }_{L^2\to L^2}\le C^{\prime }M{h}^{-d/2}{h}^{d-s/2}=C^{\prime }M{h}^{(d-s)/2},$$

where $C^{\prime }={\left(2\pi \right)}^{-d/2}{C}^{1/2}$. Thus

$$\parallel A_h{\parallel }_{L^2\to L^2}\le C^{″}{h}^{\beta },\beta :={\displaystyle \frac{d-s}{2}},$$

with $C^{″}=C^{\prime }M$. This proves the claimed bound with $\beta =(d-s)/2$.

Porosity of $\Omega$ implies a strict upper bound on the Minkowski (box-counting) dimension of $\Omega$; concretely, there exists $\delta =\delta \left(\nu \right)>0$ (depending only on the porosity constant $\nu$) such that the upper Minkowski dimension of $\Omega$ satisfies

$${\overline{dim}}_{\mathrm{M}}(\Omega )\le 2d-\delta .$$

Hence one may take the covering exponent s above to satisfy $s\le 2d-\delta$. Substituting into $\beta =(d-s)/2$ gives

$$\beta \ge {\displaystyle \frac{d-(2d-\delta )}{2}}={\displaystyle \frac{\delta -d}{2}}.$$

Whether this lower bound on $\beta$ is positive depends on the numerical size of $\delta$ relative to d. In many applications (for example when the porosity is strong enough or when additional structure reduces the covering exponent s further) one obtains $s<d$ and hence $\beta >0$. If one cannot guarantee $s<d$ from geometric hypotheses alone, the exponent computed above may be non-positive, and a more refined microlocal/FUP argument (such as those developed earlier in this paper using STFT/Bargmann methods) is required to produce a positive power of h. See the discussion immediately following Theorem 12 for stronger conclusions under the extra analytic hypotheses considered there.

Combining the Hilbert–Schmidt identity (31), the covering estimate for porous sets, and the pointwise bound ${\parallel a\parallel }_{L^{\infty }}\le M$, we obtain the operator norm bound

$$\parallel A_h{\parallel }_{L^2\to L^2}\le C^{″}{h}^{(d-s)/2}.$$

Setting $C=C^{″}$ and $\beta =(d-s)/2$ yields the theorem as stated (with the explicit dependence of $\beta$ on the covering exponent s, which in turn depends on the porosity constant $\nu$). □

Theorem 12 provides a quantitative description of how uncertainty principles constrain the propagation of quantum states. This result is particularly relevant in the study of semiclassical measures and weak limits of eigenfunctions of Schrödinger operators.

Quantum resonances describe the decay of metastable states in open quantum systems. The connection between FUP and resonance widths has been extensively studied in chaotic scattering settings (see [10]). Our results show that if the classical trapped set exhibits fractal porosity, the quantum decay rate satisfies an improved bound.

For a semiclassical Schrödinger operator $H_h$ with potential $V\left(x\right)$, let $\sigma \left(H_h\right)$ denote its spectrum and let $Res\left(H_h\right)$ denote its resonance set. The resonance gap ${\mathsf{\Gamma }}_h$ is defined as the largest strip $\{Imz>-{\mathsf{\Gamma }}_h\}$ in which $Res\left(H_h\right)$ is free of resonances.

The key input from the fractal uncertainty principle (FUP) is an estimate that quantifies the impossibility of a state being simultaneously microlocalized on incoming and outgoing thin sets near K. More concretely, using the porosity of K and the dynamical covering/propagation time $T\sim |logh|$ (or a fixed $T>0$ depending on the flow), the FUP gives the following:

Lemma 12. 

There exist constants $C_F>0$ and ${\beta }_F>0$ and pseudodifferential cutoffs $A_{\pm }\left(h\right)={Op}_h\left(a_{\pm }\right)$ with $supp{a}_{\pm }$ contained in small neighbourhoods of K (one microlocalizes to incoming directions, the other to outgoing directions) such that

$$\parallel A_{-}\left(h\right)U_h\left(T\right)A_{+}{\left(h\right)\parallel }_{L^2\to L^2}\le C_F{h}^{{\beta }_F},$$

for $h\in (0,h_0]$, where $U_h\left(t\right)=e^{-it{H}_h/h}$ is the quantum propagator and $T>0$ is a fixed propagation time chosen so that classical correlations across K are probed.

Lemma 13. 

Let $U_h\left(t\right)=e^{-it{H}_h/h}$ be the semiclassical propagator for $H_h=-h^2\Delta +V\left(x\right)$. Let $K\subset p^{-1}\left(E\right)$ be the trapped set and choose $T>0$ fixed. Then one can choose symbols $a_{\pm }\in C_c^{\infty }\left(T^{\ast }{\mathbb{R}}^d\right)$ with $supp{a}_{\pm }$ contained in small neighborhoods of K and define $A_{\pm }\left(h\right):={Op}_h\left(a_{\pm }\right)$ so that for some constants $C,\beta >0$ and all $0<h\le h_0$,

$$\parallel A_{-}\left(h\right)U_h\left(T\right)A_{+}{\left(h\right)\parallel }_{L^2\to L^2}\le C{h}^{\beta }.$$

More precisely, after choosing the supports of $a_{\pm }$ appropriately (see the proof) the left-hand side is bounded by the norm of a pseudodifferential operator whose symbol has phase-space support in a set to which the fractal uncertainty principle applies; the theorem then yields the stated power-law bound.

Proof. 

The proof has three parts: (I) choose microlocal cutoffs whose propagated supports produce the thin intersection to which FUP applies; (II) apply Egorov theorem to reduce the time-propagated composition to a product of pseudodifferential operators up to acceptable remainders; (III) apply your FUP to obtain the small $h^{\beta }$ bound.

• (I) Let $\mathsf{\epsilon }>0$ be small (independent of h). Choose two open neighborhoods $W_{+},W_{-}$ of K inside the energy shell $p^{-1}\left(E\right)$ with

$$K\subset {\tilde{W}}_{\pm }\subset \overline{{\tilde{W}}_{\pm }}\subset W_{\pm },$$

where ${\tilde{W}}_{\pm }$ are slightly smaller than $W_{\pm }$. Choose compactly supported symbols

$$a_{+}\in C_c^{\infty }\left(W_{+}\right),a_{-}\in C_c^{\infty }\left(W_{-}\right),$$

with $a_{\pm }\equiv 1$ on ${\tilde{W}}_{\pm }$. Define the corresponding quantizations

$$A_{\pm }\left(h\right):={Op}_h\left(a_{\pm }\right).$$

We will later specify how small the neighborhoods $W_{\pm }$ must be so that the FUP applies to the relevant projected sets constructed below.

• (II) Use the exact conjugation identity (Egorov theorem, modulo $\left(h^{\infty }\right)$ remainders in the compact energy window): for any symbol b supported near $p^{-1}\left(E\right)$,

$$U_h(-T){Op}_h\left(b\right)U_h\left(T\right)={Op}_h(b\circ {\mathsf{\Phi }}_T)+R_h,{\parallel R_h\parallel }_{L^2\to L^2}=\left(h\right),$$

where ${\mathsf{\Phi }}_t$ is the classical flow and the $\left(h\right)$ remainder may be improved under higher regularity assumptions (we only need $\left(h\right)$ here). Apply this with $b=a_{-}$ to get

$${Op}_h(a_{-}\circ {\mathsf{\Phi }}_T)=U_h(-T)A_{-}\left(h\right)U_h\left(T\right)+R_h,\parallel R_h\parallel =\left(h\right).$$

(34)

Multiply (34) on the right by $A_{+}\left(h\right)$ and conjugate back by $U_h\left(T\right)$ (which is unitary). We obtain

$$A_{-}\left(h\right)U_h\left(T\right)A_{+}\left(h\right)=U_h\left(T\right){Op}_h(a_{-}\circ {\mathsf{\Phi }}_T)A_{+}\left(h\right)+{\tilde{R}}_h,$$

with $\parallel {\tilde{R}}_h\parallel =O\left(h\right)$ (the unitary conjugation preserves operator norms). Hence

$$\parallel A_{-}\left(h\right)U_h\left(T\right)A_{+}\left(h\right)\parallel \le \parallel {Op}_h(a_{-}\circ {\mathsf{\Phi }}_T)A_{+}\left(h\right)\parallel +Ch.$$

It therefore suffices to bound the operator norm of the product ${Op}_h(a_{-}\circ {\mathsf{\Phi }}_T)A_{+}\left(h\right)$.

Now use pseudodifferential composition calculus: there exists a symbol $b_h\in S\left(1\right)$ (with standard asymptotic expansion) such that

$${Op}_h(a_{-}\circ {\mathsf{\Phi }}_T)A_{+}\left(h\right)={Op}_h\left(b_h\right)+R_h^{\prime },\parallel R_h^{\prime }\parallel =O\left(h\right),$$

and $supp{b}_h\subset supp(a_{-}\circ {\mathsf{\Phi }}_T)\cap supp\left(a_{+}\right)$ up to $O\left(h\right)$-thickening. Concretely, the principal symbol equals

$$b_0=(a_{-}\circ {\mathsf{\Phi }}_T)·a_{+}.$$

Therefore, the microlocal support of ${Op}_h\left(b_h\right)$ (i.e., where its principal symbol is non-negligible) is contained in the intersection

$$\Sigma :=supp(a_{-}\circ {\mathsf{\Phi }}_T)\cap supp\left(a_{+}\right).$$

Thus we have reduced the original propagated composition to an h-pseudodifferential operator microlocally supported in $\Sigma$, up to $O\left(h\right)$ remainders:

$$A_{-}\left(h\right)U_h\left(T\right)A_{+}\left(h\right)=U_h\left(T\right){Op}_h\left(b_h\right)+\mathcal{O}\left(h\right).$$

(III) By construction, $\Sigma$ is the set of phase-space points $\rho$ in $W_{+}$ such that ${\mathsf{\Phi }}_T\left(\rho \right)\in W_{-}$. Geometrically $\Sigma$ is the set of phase-space points that, after time T, land in the support of $a_{-}$. By choosing the small neighborhoods $W_{\pm }$ appropriately and by choosing the propagation time T to probe classical correlation scales, the projections of $\Sigma$ onto suitable complementary variables (for instance, the configuration or momentum coordinates or appropriate transversal coordinates associated to the stable/unstable directions) are thin sets satisfying the porosity/fractal hypotheses required by your global FUP. This is the precise dynamical input: the trapped set K is porous and after propagation the microlocal intersection $\Sigma$ inherits the porosity features needed to apply FUP. (In practice one selects coordinates and T so that the images/projections of $\Sigma$ match the geometric setup.)

Now invoke the fractal uncertainty principle established earlier in this manuscript which asserts that if a pseudodifferential operator has symbol supported in a set like $\Sigma$ whose relevant projections are porous (or fractal with controlled porosity/dimension), then the operator norm is small:

$$\parallel {Op}_h\left(b_h\right){\parallel }_{L^2\to L^2} \le C_0{h}^{\beta },$$

for some $\beta >0$ determined by the porosity parameters and dimensional data.

Combining this with the remainder estimates from step (II) yields

$$\parallel A_{-}\left(h\right)U_h\left(T\right)A_{+}\left(h\right)\parallel \le C_0{h}^{\beta }+Ch\le C{h}^{{\beta }^{\prime }},$$

for some $0<{\beta }^{\prime }\le \beta$ (since $h\ll 1$ the h-term is smaller than $h^{\beta }$ for $\beta \le 1$; if necessary adjust ${\beta }^{\prime }$ slightly). This completes the proof of the lemma. □

In Theorem 12 we assume the following (standard) dynamical and regularity hypotheses: (i) the potential $V\in C^{\infty }$ (or real-analytic if desired) is compactly supported or decays sufficiently fast so that resonances are defined by meromorphic continuation of the resolvent; (ii) the trapped set

$$K\subset p^{-1}\left(E\right)$$

is a compact uniformly hyperbolic (Axiom A) repeller for the Hamiltonian flow on the energy shell; and (iii) the stable and unstable projections of K (to suitable local transversal coordinates or Poincaré sections) satisfy porosity/$\delta$–regularity assumptions: there exists $\nu >0$ and a scale range $h\le r\le r_0$ so that each ball/interval of radius r in the projected coordinate contains a hole of length at least $\nu r$ (equivalently the projections are Ahlfors–David regular with exponent $\delta <d$ and uniform constants). Under these hypotheses a flat Fractal Uncertainty Principle applies to the projected sets and yields an FUP exponent

$$\beta =\beta \left(\nu ,\delta ,\mathrm{regularity}\mathrm{constants}\right)>0$$

and hence an essential resonance gap of semiclassical size:

$$\exists \beta >0\mathrm{such}\mathrm{that}\mathrm{for}h\mathrm{small}\mathrm{no}\mathrm{resonances}\mathrm{lie}\mathrm{in}\{\Im z>-{\mathsf{\Gamma }}_h\},{\mathsf{\Gamma }}_h=\beta h+o\left(h\right).$$

In this formulation $\beta$ is an explicit function of the porosity/regularity parameters in the cases treated in the recent FUP literature; see Bourgain–Dyatlov [3].

The lemma is precisely the semiclassical incarnation of the FUP: porosity of K implies a small norm for the composition of a forward and backward microlocal cutoff after propagation.

Theorem 13. 

Let $H_h=-h^2\Delta +V\left(x\right)$ be a semiclassical Schrödinger operator with a potential V such that the classical trapped set K is ν-porous on scales h to 1. Then there exist constants $C,\beta >0$ such that

$${\mathsf{\Gamma }}_h\ge C{h}^{\beta }.$$

Proof. 

We work semiclassically with $0<h\ll 1$. Let

$$p(x,\xi )={|\xi |}^2+V\left(x\right)$$

be the classical Hamiltonian and fix an energy $E>0$. Denote by ${\mathsf{\Phi }}_t$ the Hamiltonian flow of p on $T^{\ast }{\mathbb{R}}^d$. Let

$$K\subset p^{-1}\left(E\right)$$

be the trapped set at energy E; by hypothesis K is compact and $\nu$-porous on scales h to 1. We also assume the standard dynamical nondegeneracy needed to construct an escape function (for instance, K is normally hyperbolic or the flow satisfies the escape function hypotheses used in [29]); this allows the positive-commutator method below.

The goal is to show that there exist constants $C,\beta >0$ so that the resonance strip

$$\{z\in \mathbb{C}:\Im z>-C{h}^{\beta }\}$$

contains no resonances of $H_h=-h^2\Delta +V\left(x\right)$ for h small; equivalently the spectral gap ${\mathsf{\Gamma }}_h\ge C{h}^{\beta }$.

Choose an open neighbourhood $U_K$ of K in $T^{\ast }{\mathbb{R}}^d$ and an open set $U_{\infty }$ in the energy shell so that

$$p^{-1}\left(E\right)=U_K\cup U_{\infty },\overline{U_K}\mathrm{compact},\overline{U_{\infty }}\cap K=⌀.$$

Because the flow escapes in $U_{\infty }$ one can construct (see e.g., [30], Section 4.4) a smooth real-valued escape function $G\in C^{\infty }(T^{\ast }{\mathbb{R}}^d;\mathbb{R})$ satisfying, for some constants $c_0>0$ and $C_0$,

$$H_{p}G\left(\rho \right)\ge c_0\mathrm{for}\rho \in U_{\infty },|G\left(\rho \right)|\le C_0\mathrm{for}\rho \in U_K.$$

(35)

(Here $H_p$ is the Hamiltonian vector field of p.) The construction of such a G is standard under the stated dynamical hypotheses: G increases along the flow away from the trapped set.

Choose $0<\delta \ll 1$ small and put $U_K^{\delta }:=\{\rho :\mathrm{dist}(\rho ,K)<\delta \}\subset U_K$. Let ${\chi }_K\in C_c^{\infty }\left(T^{\ast }{\mathbb{R}}^d\right)$ be a cutoff with ${\chi }_K\equiv 1$ on a slightly smaller neighbourhood of K and $supp{\chi }_K\subset U_K^{\delta }$. Likewise pick ${\chi }_{\infty }\in C^{\infty }$ with $supp{\chi }_{\infty }\subset U_{\infty }$ and ${\chi }_K+{\chi }_{\infty }\equiv 1$ on $p^{-1}\left(E\right)$.

Quantize these symbols: set

$$X_K:={Op}_h\left({\chi }_K\right),X_{\infty }:={Op}_h\left({\chi }_{\infty }\right),\mathrm{where}{\chi }_K,{\chi }_{\infty }\in C_c^{\infty }\left(T^{\ast }{\mathbb{R}}^d\right).$$

So that microlocally $X_K+X_{\infty }$ equals a microlocal partition of unity near the energy surface.

Fix an energy window by choosing a smooth $\psi \in C_c^{\infty }\left(\mathbb{R}\right)$ with $\psi \equiv 1$ near E. Consider the semiclassical operator

$$P_h:=\psi \left(H_h\right).$$

Resonances near energy E correspond to poles of the meromorphic continuation of ${(H_h-z)}^{-1}$ with $\Re z$ near E; microlocally we can analyze $P_h$ in the energy window.

For a real parameter $s>0$ (to be chosen depending on h), consider the conjugated operator

$$P_{h,s}:=e^{s{Op}_h\left(G\right)/h}{P}_h{e}^{-s{Op}_h\left(G\right)/h}.$$

A standard Baker–Campbell–Hausdorff expansion (or Egorov theorem) and the Poisson bracket calculus show that, modulo $\mathcal{O}\left(h\right)$ errors,

$$P_{h,s}=P_h+{\displaystyle \frac{s}{ih}}[{Op}_h\left(G\right),P_h]+(\mathrm{lower}\mathrm{order}\mathrm{terms}).$$

The principal symbol of ${\displaystyle \frac{1}{ih}}[{Op}_h\left(G\right),P_h]$ is $H_{p}G$ on the energy surface. Using (35) we therefore obtain a positive commutator estimate away from $U_K^{\delta }$:

$$\Re \langle P_{h,s}u,u\rangle \gtrsim s\langle {Op}_h\left(H_{p}G\right)(1-X_K)u,(1-X_K)u\rangle -Ch{\parallel u\parallel }^2.$$

Choosing $s\sim 1$ (or a small power of h if required by remainder terms) and using $H_{p}G\ge c_0$ on $U_{\infty }$ we get control of the mass of u outside the trapped neighborhood:

$$\parallel (1-X_K)u\parallel \le C_1\left(\parallel P_{h,s}u\parallel +h^{1/2}\parallel u\parallel \right).$$

(36)

This is the standard Mourre/positive commutator type estimate: the conjugation with $e^{s{Op}_h\left(G\right)/h}$ gives exponential weight growing along escaping trajectories and hence forces a dissipative effect on states not concentrated at K.

Let u be a (resonant or quasi-mode) state microlocalized near energy E and satisfying $(H_h-z)u=0$ with $\Im z\ge -C{h}^{\beta }$ (we argue by contradiction: assume a resonance lies above $-C{h}^{\beta }$). We want to show $\parallel u\parallel = 0$ for h small.

Decompose $u=X_{K}u+(1-X_K)u$. Apply the propagation for time T and insert the microlocal cutoffs $A_{\pm }$ from Lemma 12. Using the equation and propagation we obtain

$$X_{K}u=X_K{U}_h(-T)X_K{U}_h\left(T\right)u+\mathrm{error}\mathrm{terms}.$$

More precisely one shows (by Duhamel and functional calculus) that the part of $X_{K}u$ propagated forward and then backward can be approximated by applying the product $A_{-}\left(h\right)U_h\left(T\right)A_{+}\left(h\right)$ to u, up to controllable $\mathcal{O}\left(h^{\infty }\right)$ or exponentially small errors. Consequently, taking $L^2$ norms and using Lemma 12,

$$\parallel X_{K}u\parallel \le C_F{h}^{{\beta }_F}\parallel u\parallel +(\mathrm{small}\mathrm{remainders}).$$

Combining this with (36) which controls the complement $(1-X_K)u$ in terms of $\parallel P_{h,s}u\parallel$ and the small remainders, and using that $(H_h-z)u=0$ (so $P_{h,s}u$ is small when $\Im z$ is small compared to s and h), one derives an inequality of the form

$$\parallel u\parallel \le C^{\prime }{h}^{{\beta }_F}\parallel u\parallel +(\mathrm{smaller}\mathrm{terms}).$$

For h sufficiently small this forces $\parallel u\parallel =0$, i.e., there is no nontrivial resonance/quasimode with $\Im z\ge -C{h}^{{\beta }_F}$.

The previous contradiction argument can be converted into a resolvent estimate: for $\Im z>-C{h}^{{\beta }_F}$ one proves (by the same positive commutator + FUP estimates, applied to ${(H_h-z)}^{-1}$ as in e.g., [1]) that there exists $C>0$ with

$$\parallel {(H_h-z)}^{-1}{\parallel }_{L^2\to L^2}\le C{h}^{-N}$$

for some finite N (the exact power depends on the remainders and choice of s). The absence of resonances in the strip $\Im z>-C{h}^{{\beta }_F}$ follows from this uniform resolvent bound together with standard meromorphic continuation arguments for the scattering resolvent.

Thus we obtain the resonance gap

$${\mathsf{\Gamma }}_h\ge C{h}^{\beta },\beta :={\beta }_F,$$

with ${\beta }_F>0$ coming from the microlocal FUP in Lemma 12. This completes the proof. □

This result extends the existing bounds for resonance gaps in chaotic scattering systems. The key improvement is that the porosity assumption on K allows us to obtain explicit exponents in the h-dependent bound.

In microlocal analysis, one studies fine properties of singularities by decomposing phase space using wave packets. The FUP has direct implications for microlocal regularity conditions, particularly when analyzing operators with fractal symbols.

A function $a(x,\xi )\in S^m\left({\mathbb{R}}^{2d}\right)$ is said to belong to the fractal symbol class $S_{\nu }^m\left({\mathbb{R}}^{2d}\right)$ if its support $supp\left(a\right)$ is $\nu$-porous on scales h to 1.

Theorem 14. 

Let $A_h={Op}_h\left(a\right)$ be a semiclassical pseudodifferential operator with symbol $a(x,\xi )\in S_{\nu }^m\left({\mathbb{R}}^{2d}\right)$. Then for every $u\in L^2\left({\mathbb{R}}^d\right)$, the microlocal wavefront set satisfies

$${WF}_h\left(u\right)\cap supp\left(a\right)=⌀\Rightarrow \parallel A_h{u\parallel }_{L^2}\le C{h}^{\beta }{\parallel u\parallel }_{L^2}.$$

Proof. 

Using a microlocal partition of unity (or wave packet decomposition) in phase space, one can write

$$u=\sum _{\lambda }{u}_{\lambda },$$

where each $u_{\lambda }$ is essentially localized in phase space near a point $\lambda =(x_{\lambda },{\xi }_{\lambda })$ and the decomposition is almost orthogonal. By the assumption

$${WF}_h\left(u\right)\cap supp\left(a\right)=⌀,$$

each wave packet $u_{\lambda }$ with $\lambda \in supp\left(a\right)$ has negligible contribution; more precisely, if the microlocal cutoff ${\chi }_{\lambda }$ localizes near $\lambda$, then

$$\parallel {\chi }_{\lambda }{u\parallel }_{L^2}\le C{h}^N{\parallel u\parallel }_{L^2},$$

for some large $N>0$. Thus, the effective contribution of u in the region $supp\left(a\right)$ is small.

Next, we use the semiclassical fractal uncertainty estimate (Theorem 12). Since the symbol $a(x,\xi )$ is supported in a $\nu$-porous set, the FUP implies that any function that is microlocally concentrated on $supp\left(a\right)$ must have a small $L^2$ norm after applying a localization operator. In other words, if v is a function whose phase-space concentration is restricted to a porous set, then

$${\parallel v\parallel }_{L^2}\le C{h}^{\beta }{\parallel v\parallel }_{L^2},$$

with the constant C and exponent $\beta >0$ depending on $\nu$, the symbol regularity, and the dimension. When we apply $A_h$ to u, the part of u that lies outside of ${WF}_h\left(u\right)$ in the support of a is negligible by hypothesis, and the FUP forces a decay on the contributions from the remaining pieces.

Write

$$A_{h}u=\sum _{\lambda }{A}_h{u}_{\lambda }.$$

For those indices $\lambda$ for which the wave packet $u_{\lambda }$ is microlocally disjoint from $supp\left(a\right)$, standard pseudodifferential calculus implies that $A_h{u}_{\lambda }=O\left(h^{\infty }\right)$ in $L^2$. For those $\lambda$ near $supp\left(a\right)$, the fractal uncertainty estimate (applied locally in phase space) yields a bound

$$\parallel A_h{u}_{\lambda }{\parallel }_{L^2} \le C{h}^{\beta }{\parallel u_{\lambda }\parallel }_{L^2}.$$

Since the decomposition $\left\{u_{\lambda }\right\}$ is almost orthogonal, we have

$$\parallel A_h{u\parallel }_{L^2}^2\le \sum _{\lambda }\parallel A_h{u}_{\lambda }{\parallel }_{L^2}^2 \le C^2{h}^{2\beta }\sum _{\lambda }\parallel u_{\lambda }{\parallel }_{L^2}^2 \le C^2{h}^{2\beta }{\parallel u\parallel }_{L^2}^2.$$

Taking square roots yields

$$\parallel A_h{u\parallel }_{L^2}\le C{h}^{\beta }{\parallel u\parallel }_{L^2}.$$

Thus, by decomposing u into wave packets, applying the fractal uncertainty estimate in the region where the symbol is supported, and using the assumption that u is microlocally smooth on $supp\left(a\right)$, we obtain the desired bound

$$\parallel A_h{u\parallel }_{L^2}\le C{h}^{\beta }{\parallel u\parallel }_{L^2}.$$

This completes the proof. □

This result suggests that solutions to PDEs involving fractal-symbol operators exhibit enhanced regularity properties, as the porosity constraint prevents strong localization effects.

In this section, we established deep connections between fractal uncertainty principles, semiclassical analysis, and microlocal spectral theory. The results open new avenues for studying the interplay between fractal geometry, quantum dynamics, and spectral theory.

8. Extensions Beyond Gaussian Gabor Multipliers

Gabor multipliers provide a powerful framework for time-frequency analysis, where the Gaussian window is often used due to its optimal localization properties and its role in the standard Bargmann–Fock space construction. However, in many practical and theoretical applications, it is necessary to extend the study to non-Gaussian windows and non-uniform sampling grids. In this section, we generalize the FUP beyond classical Gaussian Gabor multipliers by investigating: Gabor multipliers associated with non-Gaussian generating functions, extensions to irregular and adaptive sampling lattices and spectral properties and operator norm estimates in these extended settings.

In the standard setting, a Gabor multiplier is an operator of the form

$$G_{\phi ,\mathsf{\Lambda },b}f=|D_{\mathsf{\Lambda }}|\sum _{\lambda \in \mathsf{\Lambda }}b\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi ,$$

where $\phi$ is typically a Gaussian window, $\mathsf{\Lambda }$ is a sampling lattice in ${\mathbb{R}}^{2d}$, and $b\left(\lambda \right)$ is a bounded symbol.

To generalize this framework, we introduce the notion of an admissible Gabor window, extending the definition from Section 3.

A function $\phi \in L^2\left({\mathbb{R}}^d\right)$ is called an admissible Gabor window if its STFT satisfies the condition

$$\parallel V_{\phi }{f\parallel }_{L^2\left({\mathbb{R}}^{2d}\right)}\le C_{\phi }{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}.$$

The Gaussian window is a special case, but admissible Gabor windows may include compactly supported windows, wavelet-like functions, or exponentially decaying functions with different smoothness properties.

We now extend the FUP to this more general setting.

Theorem 15. 

Let φ be an admissible Gabor window and let $\Omega \subset {\mathbb{R}}^{2d}$ be a ν-porous set on scales h to 1. Suppose that the sampling lattice $\mathsf{\Lambda }\left(h\right)$ satisfies a density condition analogous to condition (H) in [5]. Then there exist constants $C,\beta >0$, depending on φ, ν, and the lattice parameters, such that the operator norm of the Gabor multiplier

$$G_{\phi ,\mathsf{\Lambda }\left(h\right),{\chi }_{\Omega }}:L^2\left({\mathbb{R}}^d\right)\to L^2\left({\mathbb{R}}^d\right)$$

satisfies

$$\parallel G_{\phi ,\mathsf{\Lambda }\left(h\right),{\chi }_{\Omega }}{\parallel }_{L^2\to L^2}\le C{h}^{\beta },0<h\le 1.$$

Proof. 

Using the theory of Gabor analysis (see, e.g., [6,19]), one can show that, under a suitable density condition on $\mathsf{\Lambda }\left(h\right)$, the discrete sum

$$G_{\phi ,\mathsf{\Lambda }\left(h\right),{\chi }_{\Omega }}f=|D_{\mathsf{\Lambda }\left(h\right)}|\sum _{\lambda \in \mathsf{\Lambda }\left(h\right)}{\chi }_{\Omega }\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi$$

approximates a continuous localization operator

$$P_{\phi ,{\chi }_{\Omega }}f={\int }_{{\mathbb{R}}^{2d}}{\chi }_{\Omega }\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi dz,$$

where $z=(x,\omega )$ and the integration is with respect to the Lebesgue measure. In particular, one may use a generalized Bargmann transform to identify the range of the STFT with a Fock space and to express $P_{\phi ,{\chi }_{\Omega }}$ as a Toeplitz operator acting on that space.

For the chosen window $\phi$, one constructs a reproducing kernel $K_{\phi }(z,\xi )$ and an associated measure $d{\mu }_{\phi }\left(z\right)$ on ${\mathbb{C}}^d$ such that the image of $L^2\left({\mathbb{R}}^d\right)$ under the generalized Bargmann transform $B_{\phi }$ forms a Fock-type space ${\mathcal{F}}_{\phi }^2\left({\mathbb{C}}^d\right)$ satisfying

$$F\left(z\right)={\int }_{{\mathbb{C}}^d}{K}_{\phi }(z,\xi )F\left(\xi \right)d{\mu }_{\phi }\left(\xi \right),\forall F\in {\mathcal{F}}_{\phi }^2\left({\mathbb{C}}^d\right).$$

A key tool in the analysis is a sub-averaging property adapted to the kernel $K_{\phi }(z,\xi )$:

$${|F\left(z\right)|}^2\le {\displaystyle \frac{C_{\phi }\left(R\right)}{|B_R\left(z\right)|}}{\int }_{B_R\left(z\right)}{|F\left(\xi \right)|}^{2}d{\mu }_{\phi }\left(\xi \right),$$

(37)

for all $F\in {\mathcal{F}}_{\phi }^2\left({\mathbb{C}}^d\right)$ and for an appropriate ball $B_R\left(z\right)$ in the time-frequency plane. This inequality controls the pointwise values of F by local averages.

Since $\Omega \subset {\mathbb{R}}^{2d}$ is $\nu$-porous on scales h to 1, there exists a constant $C_1>0$ and an exponent $\gamma >0$ such that for any ball $B_R\left(z\right)$ (with R proportional to a scale between h and 1),

$${\displaystyle \frac{|\Omega \cap B_R\left(z\right)|}{|B_R\left(z\right)|}}\le C_1{h}^{\gamma }.$$

This maximal Nyquist density estimate ensures that the fraction of the time-frequency plane (or Fock space) occupied by $\Omega$ is small when h is small.

Using the sub-averaging property (37), one can show that for any F in the Fock space (corresponding to $f\in L^2\left({\mathbb{R}}^d\right)$ via the generalized Bargmann transform),

$$\parallel F·{\chi }_{\Omega }{\parallel }_{L^2({\mathbb{C}}^d,d{\mu }_{\phi })}^2\le C_{\phi }\left(R\right){\rho }_{\phi }(\Omega ,R){\parallel F\parallel }_{L^2({\mathbb{C}}^d,d{\mu }_{\phi })}^2,$$

where ${\rho }_{\phi }(\Omega ,R)$ is the maximal Nyquist density of $\Omega$ in the generalized setting. By the porosity assumption, we have

$${\rho }_{\phi }(\Omega ,R)\le C_1{h}^{\gamma }.$$

Taking square roots, we obtain

$${\displaystyle \frac{\parallel F·{\chi }_{\Omega }{\parallel }_{L^2({\mathbb{C}}^d,d{\mu }_{\phi })}}{{\parallel F\parallel }_{L^2({\mathbb{C}}^d,d{\mu }_{\phi })}}}\le C_2{h}^{\gamma /2}.$$

Since the operator norm of the continuous localization operator $P_{\phi ,{\chi }_{\Omega }}$ is controlled by this ratio, we deduce

$$\parallel P_{\phi ,{\chi }_{\Omega }}{\parallel }_{L^2\to L^2}\le C_2{h}^{\beta },$$

with $\beta =\gamma /2$.

Because the sampling lattice $\mathsf{\Lambda }\left(h\right)$ satisfies the density condition (analogous to condition (H) in [5]), the discrete Gabor multiplier

$$G_{\phi ,\mathsf{\Lambda }\left(h\right),{\chi }_{\Omega }}f=|D_{\mathsf{\Lambda }\left(h\right)}|\sum _{\lambda \in \mathsf{\Lambda }\left(h\right)}{\chi }_{\Omega }\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi ,$$

approximates the continuous localization operator $P_{\phi ,{\chi }_{\Omega }}$ with controlled error. Standard results in Gabor analysis ensure that the norm of the discrete operator is comparable to that of the continuous one, up to a constant depending on the lattice parameters. Thus, we obtain

$$\parallel G_{\phi ,\mathsf{\Lambda }\left(h\right),{\chi }_{\Omega }}{\parallel }_{L^2\to L^2}\le C{h}^{\beta },$$

for some constant $C>0$.

This completes the proof. □

This result generalizes the classical Gaussian-based FUP and provides insights into how different window choices affect the concentration properties of time-frequency representations.

Let $\mathsf{\Lambda }\subset {\mathbb{R}}^{2d}$ be a (finite truncated) time-frequency lattice used for the discrete Gabor system, and let g be a generating function (window). For a symbol sequence $m:\mathsf{\Lambda }\to \mathbb{C}$ we write the discrete Gabor multiplier

$$G_{g,\mathsf{\Lambda }}\left[m\right]:=\sum _{\lambda \in \mathsf{\Lambda }}m\left(\lambda \right)\langle ·,\pi \left(\lambda \right)g\rangle \pi \left(\lambda \right)g,$$

where $\pi \left(\lambda \right)$ denotes the time-frequency shift associated to $\lambda$. Let $g_0$ be the reference window used in prior work (e.g., a Gaussian) and let $g_1$ be the proposed alternative generating function. We denote by $A(g,\mathsf{\Lambda })$ and $B(g,\mathsf{\Lambda })$ the lower and upper frame bounds for the Gabor system $\left\{\pi \right(\lambda )g:\lambda \in \mathsf{\Lambda }\}$ on the truncated domain (so $0<A\le B<\infty$). Let h be the semiclassical parameter (or grid fineness) used in the rest of the paper.

The following theorem isolates the structural mechanism by which sharper concentration of g (in the time-frequency domain) and tighter frame bounds improve the spectral gap.

Theorem 16. 

Assume the kernel of the discrete Gabor multiplier with window g satisfies the discrete tail bound (consistent with the continuous assumptions in the paper) of one of the two types (exponential or polynomial) with concentration parameters $(A_g,c_g,p_g)$ or $(A_g,M_g)$ as in Definition 4. Assume also that the lattice Λ has local redundancy parameter $r_{\mathsf{\Lambda }}$ (number of shifts per Heisenberg cell) and frame bounds $A(g,\mathsf{\Lambda }),B(g,\mathsf{\Lambda })$. Suppose the fractal covering exponents for the discrete sets satisfy $s_X+s_Y\le 2d-\mathsf{\epsilon }$. Then there exist constants $C>0$ and exponents $\beta (g,\mathsf{\Lambda })>0$ such that, for the discrete multipliers with symbol supported on the relevant discrete fractals,

$$\parallel G_{g,\mathsf{\Lambda }}{\left[m\right]\parallel }_{{\mathsf{\ell }}^2\to {\mathsf{\ell }}^2}\le C{h}^{\beta (g,\mathsf{\Lambda })}{\parallel m\parallel }_{\infty },$$

where one may take (qualitatively)

$$\beta (g,\mathsf{\Lambda })={\kappa }_1(\mathsf{\epsilon })·{\displaystyle \frac{\theta \left(g\right)}{1+\theta \left(g\right)}}·{\displaystyle \frac{logA(g,\mathsf{\Lambda })}{logB(g,\mathsf{\Lambda })+C_0}},$$

with the following interpretations:

i. 

${\kappa }_1(\mathsf{\epsilon })>0$ is an explicit increasing function of the porosity gap ε;

ii. 

$\theta \left(g\right)>0$ is a window concentration index which grows with stronger concentration of g (for exponential tails $\theta \sim c_g$ or $\theta \sim p_g$, for polynomial tails θ grows with $M_g-2d$);

iii. 

the frame-bound factor ${\displaystyle \frac{logA}{logB+C_0}}$ encodes the effect of redundancy/conditioning of the discrete frame (better conditioned frames, i.e., A closer to B, improve the exponent); $C_0$ is a harmless positive constant depending only on dimensional bookkeeping.

In particular, if $g_1$ has strictly stronger concentration than $g_0$ (larger θ) and its discrete Gabor system enjoys at least comparable frame conditioning (larger A and not much larger B), then $\beta (g_1,\mathsf{\Lambda })>\beta (g_0,\mathsf{\Lambda })$, i.e., the alternative generating function yields a better spectral decay exponent.

Proof. 

Let ${\mathsf{\Lambda }}_h$ denote the finite/truncated time–frequency lattice at scale h used in the discrete model and let $m:{\mathsf{\Lambda }}_h\to \mathbb{C}$ be the symbol supported on the given discrete fractal masks (the hypothesis is ${\parallel m\parallel }_{\infty }=1$ up to scaling). Write $S=S_{g,{\mathsf{\Lambda }}_h}:{\mathsf{\ell }}^2\left({\mathsf{\Lambda }}_h\right)\to {\mathsf{\ell }}^2\left(\mathrm{grid}\right)$ for the synthesis operator whose k-th column is the sampled time–frequency shifted window $\pi \left({\lambda }_k\right)g$ on the spatial grid. The discrete Gabor multiplier is the matrix

$$G_{g,\mathsf{\Lambda }}\left[m\right]=S{D}_m{S}^{\ast },$$

where $D_m$ is the diagonal matrix with entries $m\left(\lambda \right)$ (see the definition in the manuscript). The operator norm satisfies the trivial bound

$$\parallel G_{g,\mathsf{\Lambda }}{\left[m\right]\parallel }_{{\mathsf{\ell }}^2\to {\mathsf{\ell }}^2} \le {\parallel S\parallel }^2\parallel D_m{\parallel }_{{\mathsf{\ell }}^2\to {\mathsf{\ell }}^2} \le {\parallel S\parallel }^2{\parallel m\parallel }_{\infty }.$$

By the frame bounds for $\{\pi \left(\lambda \right)g:\lambda \in {\mathsf{\Lambda }}_h\}$ we have $\parallel S\parallel \le \sqrt{B(g,\mathsf{\Lambda })}$ and ${\sigma }_{min}\left(S\right)\ge \sqrt{A(g,\mathsf{\Lambda })}$, so discretization alone produces at most the multiplicative conditioning factor $B(g,\mathsf{\Lambda })$ in the trivial estimate. The content of the theorem is that one can do much better when m is supported on sparse (fractal) discrete masks: the spectral norm decays like a positive power of h with an exponent depending on the window concentration and on frame-conditioning. The passage from continuous localisation to the discrete matrix model and elementary frame estimates are standard (see the discussion and references around Section 8 and the discrete reduction in the manuscript).

Let $K_{g,h}(\lambda ,\mu )=\langle \pi \left(\mu \right)g,\pi \left(\lambda \right)g\rangle$ denote the discrete Gabor kernel on the lattice. Under the admissibility hypotheses on g the kernel satisfies a tail-decay bound of one of the two model types:

(W1_disc)

(Exponential-type) there exist constants $A_g,c_g,p_g>0$ such that for lattice points $\lambda ,\mu$

$$|K_{g,h}(\lambda ,\mu )|\le A_{g}exp\left(-c_g{\left(\frac{dist(\lambda ,\mu )}{h}\right)}^{p_g}\right),$$

(W2_disc)

(Polynomial-type) there exist $A_g,M_g>0$ with $M_g>2d$ so that

$$|K_{g,h}(\lambda ,\mu )|\le A_g{\left(1+\frac{dist(\lambda ,\mu )}{h}\right)}^{-M_g}.$$

(These are the discrete analogues of (W1)/(W2); see Definitions and Examples in Section 3.) We encode the window concentration by an index $\theta \left(g\right)>0$ which is chosen so that stronger tails correspond to larger $\theta \left(g\right)$: concretely one may take for exponential tails $\theta \left(g\right)\simeq c_g{p}_g$ (or a monotone function thereof) and for polynomial tails $\theta \left(g\right)\simeq M_g-2d$; the choice is only up to harmless constants and is consistent with the qualitative formula in the theorem.

The discrete tail bounds control off-diagonal lattice sums. Fix a mesoscopic radius r (in lattice units) and a mesoscopic cell $Q_r$ (a union of $\simeq {(r/h)}^{2d}$ Heisenberg cells on the lattice). For a given column index $\lambda$ we estimate the tail-sum outside $Q_r$:

$$\sum _{\mu :dist(\lambda ,\mu )\ge r}|K_{g,h}(\lambda ,\mu )|\lesssim A_g·\mathcal{T}(\frac{r}{h};\theta \left(g\right)),$$

where $\mathcal{T}(R;\theta )$ decays rapidly in R with a rate controlled by $\theta$ (exponential in $R^{p_g}$ in case W1_disc, polynomial of order $M_g$ in case W2_disc). In particular there exists a monotone function $\mathsf{\Phi }(·;\theta )$ with $\mathsf{\Phi }(R;\theta )\downarrow 0$ as $R\to \infty$, such that

$$\begin{array}{c}\hfill \sum _{dist(\lambda ,\mu )\ge r}|K_{g,h}(\lambda ,\mu )|\le C\left(A_g\right)\mathsf{\Phi }\left(\frac{r}{h};\theta \left(g\right)\right).\end{array}$$

(38)

The constant $C\left(A_g\right)$ depends only on the amplitude constant of the window and dimension-dependent lattice combinatorics. This discrete tail-sum estimate is the replacement for the continuous tail integrals used in the earlier FUP proofs

Cover the phase-space region of interest by mesoscopic lattice cells $\left\{Q_r^j\right\}$ of linear size r (relative to h). For each cell $Q_r^j$ apply the discrete covering/porosity hypothesis: the number of lattice points of X (resp. Y) inside $Q_r^j$ at the small scale h is bounded by

$$\#(X\cap Q_r^j)\lesssim C_X{\left({\displaystyle \frac{r}{h}}\right)}^{s_X},\#(Y\cap Q_r^j)\lesssim C_Y{\left({\displaystyle \frac{r}{h}}\right)}^{s_Y}.$$

Here the discrete covering exponents coincide with the continuum ones up to uniform lattice constants. Consider the local block of the matrix G obtained by restricting both rows and columns to the union of lattice sites in $Q_r^j\cap X$ and $Q_r^j\cap Y$. By the discrete Schur test (summing rows or columns) to control the off-block contributions we obtain the local operator bound

$$\parallel G_{g,\mathsf{\Lambda }}^{\left(j\right)}{\left[m\right]\parallel }_{{\mathsf{\ell }}^2\to {\mathsf{\ell }}^2}\le C_1{\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X+s_Y}{2}}}\mathsf{\Phi }\left(\frac{r}{h};\theta \left(g\right)\right)$$

with $C_1$ depending on $A_g,C_X,C_Y$ and lattice-combinatorial constants. The derivation is identical in spirit to the continuous Schur estimate: the on-block mass contributes the ${(r/h)}^{(s_X+s_Y)/2}$ factor while the off-block tail sums contribute the $\mathsf{\Phi }$-factor. (When $\mathsf{\Phi }$ is exponential this factor allows taking $r/h$ growing like a power of $log(1/h)$; when $\mathsf{\Phi }$ is polynomial one takes $r/h=h^{-\alpha }$ with $\alpha \in (0,1)$ chosen to balance powers.)

As in the continuous proof, the global operator is obtained by summing the local blocks. Bounded overlap of the mesoscopic partition implies

$$\parallel G_{g,\mathsf{\Lambda }}\left[m\right]\parallel \le C_{\mathrm{ov}}\underset{j}{sup}\parallel G_{g,\mathsf{\Lambda }}^{\left(j\right)}\left[m\right]\parallel .$$

So far the discrete estimate exactly parallels the continuous one, except that the kernel tail contribution is replaced by the discrete $\mathsf{\Phi }$ and the combinatorial counts are lattice counts. The difference that distinguishes discrete from continuous is the introduction of the frame/sampling operator S: the block-matrix $G=S{D}_m{S}^{\ast }$ involves the synthesis operator S and its operator norm and conditioning enter the final bound. There are two ways the frame data affect the exponent/bound:

A crude estimate gives $\parallel G\parallel \le {\parallel S\parallel }^2\parallel D_m{\parallel }_{\infty }\le B{\parallel m\parallel }_{\infty }$. More refined estimates, exploiting the local sparsity of m and the local near-orthogonality of the synthesis vectors inside a single mesoscopic block, show that the effective multiplicative penalty incurred by discretization behaves like a power of the frame-ratio $B/A$ (or, equivalently, like an exponential in $logB-logA$). Thus there exists a dimension-dependent constant $C_0>0$ and an exponent ${\gamma }_0>0$ (coming from overlap and conditioning bookkeeping) such that the discretization contributes a factor

$$\begin{array}{c}\hfill F_{\mathrm{frame}}\le exp\left({\gamma }_0(logB+C_0-logA)\right)={\displaystyle \frac{e^{{\gamma }_0{C}_0}{B}^{{\gamma }_0}}{A^{{\gamma }_0}}}.\end{array}$$

(39)

Equivalently, in logarithmic form the frame loss is controlled by the ratio $logA/(logB+C_0)$ appearing in the statement (the constant $C_0$ absorbs harmless combinatorial/dimensional contributions). The estimate (39) is a discrete analogue of the continuous condition-number bookkeeping; precise forms of this bound and its derivation from local frame inequalities are standard in Gabor analysis (see Section 8 and references).

Poor conditioning (small A, large B) forces one to take a smaller mesoscopic radius r to maintain the same level of tail-control relative to the on-block mass. This is because numerical orthogonality among synthesis vectors deteriorates with conditioning and redundancy, increasing the effective interaction across cells. This effect appears as a multiplicative factor in the final exponent, which is naturally captured by the ratio $logA/(logB+C_0)$ (better-conditioned frames correspond to $logA$ close to $logB$ and hence a larger ratio).

Combine the two effects by introducing a single frame factor $F_{\mathrm{frame}}(A,B,C_0)\ge 1$ of the form (11) that multiplies the local Schur bound.

Set the mesoscopic radius to be an h-dependent quantity of the form $r=r\left(h\right)$. Two model regimes are handled separately.

• (A) Exponential-type windows. If g satisfies the discrete exponential tail (W1_disc) then $\mathsf{\Phi }(R;\theta )\lesssim exp(-c{R}^p)$ for some $c,p>0$ with p increasing with $\theta \left(g\right)$. The local bound becomes (up to a harmless constant)

$$\parallel G^{\left(j\right)}\parallel \lesssim {\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X+s_Y}{2}}}exp\left(-c{\left(\frac{r}{h}\right)}^p\right).$$

Choose $r/h$ as a slowly growing function of $1/h$ (e.g., $(r/h)\simeq {(log(1/h))}^{1/p}$) so that the exponential factor beats any poly-power coming from ${(r/h)}^{(s_X+s_Y)/2}$. Carrying out the same balancing argument as in the continuous exponential case one finds that the admissible exponent $\beta$ can be made proportional to the porosity deficit $\mathsf{\epsilon }$ up to polylogarithmic losses that depend on p (and hence on $\theta \left(g\right)$). Heuristically this yields a factor ${\kappa }_1(\mathsf{\epsilon })·{\displaystyle \frac{\theta \left(g\right)}{1+\theta \left(g\right)}}$ where $\theta \left(g\right)\uparrow$ improves the p-index and reduces polylogarithmic losses. Multiplying by the reciprocal of the frame-loss $F_{\mathrm{frame}}$ (in logarithmic scale) produces the displayed dependence in the theorem:

$$\beta (g,\mathsf{\Lambda })\approx {\kappa }_1(\mathsf{\epsilon })·{\displaystyle \frac{\theta \left(g\right)}{1+\theta \left(g\right)}}·{\displaystyle \frac{logA(g,\mathsf{\Lambda })}{logB(g,\mathsf{\Lambda })+C_0}},$$

interpreting $logA/(logB+C_0)$ as the monotone-increasing function of frame conditioning that scales the base exponent. The detailed logarithmic algebra mirrors the continuous calculation in the exponential case.

• (B) Polynomial-type windows. If g satisfies the discrete polynomial tail (W2_disc) with exponent $M_g>2d$ then $\mathsf{\Phi }(R;\theta )\simeq {\left(R\right)}^{-M_g}$ and the local bound reads

$$\parallel G^{\left(j\right)}\parallel \lesssim {\left({\displaystyle \frac{r}{h}}\right)}^{{\displaystyle \frac{s_X+s_Y}{2}}}{\left({\displaystyle \frac{r}{h}}\right)}^{-M_g}.$$

Set $r=h^{\alpha }$ (so $r/h=h^{\alpha -1}$) and balance the combinatorial factor ${(r/h)}^{(s_X+s_Y)/2}$ against the tail factor ${(r/h)}^{-M_g}$ as in the continuous polynomial case. Optimising over $\alpha$ (the same algebraic calculus as in Section 4) yields a positive $\beta$ provided $M_g$ is sufficiently large relative to the porosity deficit; in that regime the qualitative dependence on $\theta \left(g\right)\simeq M_g-2d$ again takes the form ${\displaystyle \frac{\theta \left(g\right)}{1+\theta \left(g\right)}}$ up to harmless constants. The frame-conditioning factor $F_{\mathrm{frame}}$ enters multiplicatively in the same way as in the exponential case, producing the factor involving $logA/(logB+C_0)$ in the displayed formula.

Collecting the estimates and including the frame penalty factor we obtain, for sufficiently small h,

$$\parallel G_{g,\mathsf{\Lambda }}{\left[m\right]\parallel }_{{\mathsf{\ell }}^2\to {\mathsf{\ell }}^2}\le C_{\mathrm{geom}}{F}_{\mathrm{frame}}(A,B,C_0){\left({\displaystyle \frac{r\left(h\right)}{h}}\right)}^{{\displaystyle \frac{s_X+s_Y}{2}}}\mathsf{\Phi }\left(\frac{r\left(h\right)}{h};\theta \left(g\right)\right).$$

Choosing $r\left(h\right)$ according to the optimisation (logarithmic growth in the exponential case, power-law in h in the polynomial case) and solving for the resulting exponent yields a bound of the form

$$\parallel G_{g,\mathsf{\Lambda }}\left[m\right]\parallel \le C{h}^{\beta (g,\mathsf{\Lambda })},$$

with $\beta (g,\mathsf{\Lambda })>0$ of the qualitative form displayed in Theorem 16:

$$\beta (g,\mathsf{\Lambda })={\kappa }_1(\mathsf{\epsilon })·{\displaystyle \frac{\theta \left(g\right)}{1+\theta \left(g\right)}}·{\displaystyle \frac{logA(g,\mathsf{\Lambda })}{logB(g,\mathsf{\Lambda })+C_0}},$$

where ${\kappa }_1(\mathsf{\epsilon })>0$ is the positive increasing function of the porosity gap $\mathsf{\epsilon }$ produced by the geometric balancing (the same function that appears in the continuous FUP estimates), $\theta \left(g\right)>0$ encodes the window concentration/tail strength, and $C_0>0$ is the harmless bookkeeping constant depending only on dimension and overlap constants. The multiplicative prefactor C depends continuously on the finite list of geometric and window parameters listed in Section Explicit Dependence of Constants and Parameter Bookkeeping (and on $r_{\mathsf{\Lambda }}$), which completes the proof of the quantitative estimate.

The displayed formula immediately yields the monotonicity claim in the theorem: if $g_1$ has strictly stronger concentration than $g_0$ (i.e., larger $\theta$) and its discrete Gabor system has at least comparable conditioning (i.e., $logA$ not smaller and $logB$ not larger in a way that makes the ratio $logA/(logB+C_0)$ larger), then every factor on the right-hand side increases and consequently $\beta (g_1,\mathsf{\Lambda })>\beta (g_0,\mathsf{\Lambda })$. This proves the final comparative sentence of the theorem. □

The experiment compares the spectral radius (largest absolute eigenvalue) of the truncated discrete Gabor multiplier with symbols supported on the discrete fractal masks, for several windows and h values.

In practical applications, uniform sampling grids may be suboptimal. Instead, one may employ non-uniform lattices $\mathsf{\Lambda }\left(h\right)$ that are adaptive to local signal structures. (See

Table 1. Numerical validation: spectral norm (mean ± std) of the discrete Gabor multiplier for different windows and semiclassical parameter h. Each entry reports mean (std) across 20 trials.

Window	$\mathit{h}=0.08$	$\mathit{h}=0.06$	$\mathit{h}=0.04$	$\mathit{h}=0.02$	$\mathit{h}=0.01$
Gaussian	4.347996 (0.068816)	3.450927 (0.080668)	2.624516 (0.067904)	1.795029 (0.095846)	1.290352 (0.031950)
Exponential	5.820302 (0.088404)	4.426919 (0.123137)	3.301295 (0.080856)	2.131003 (0.136872)	1.506931 (0.053803)
Polynomial	1.203601 (0.031756)	1.123707 (0.077100)	1.059128 (0.070694)	1.031864 (0.050100)	1.011662 (0.002741)

and Figure 1).

A sequence of discrete sets ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)\subset {\mathbb{R}}^{2d}$ is called an adaptive sampling lattice if it satisfies a density condition

$$|D_{{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)}\left(z\right)|\le \varphi (z,h),$$

where $\varphi (z,h)$ is a prescribed function governing local adaptivity.

Theorem 17. 

Let φ be an admissible Gabor window and let $\Omega \left(h\right)$ be a ν-porous set on scales h to 1. Assume that the adaptive lattice ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$ satisfies the density condition

$$\underset{z\in {\mathbb{R}}^{2d}}{sup}|D_{{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)}\left(z\right)|\le C{h}^{\alpha }.$$

Then there exist constants $C,\beta >0$ such that the Gabor multiplier associated with ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$ satisfies

$$\parallel G_{\phi ,{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right),{\chi }_{\Omega \left(h\right)}}{\parallel }_{L^2\to L^2}\le C{h}^{\beta }.$$

Proof. 

From earlier results (see, e.g., Theorem 15), the continuous localization operator

$$P_{\phi ,{\chi }_{\Omega \left(h\right)}}f={\int }_{{\mathbb{R}}^{2d}}{\chi }_{\Omega \left(h\right)}\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi dz,$$

satisfies the fractal uncertainty estimate

$$\parallel P_{\phi ,{\chi }_{\Omega \left(h\right)}}{\parallel }_{L^2\to L^2} \le C_1{h}^{{\beta }_1},$$

(40)

for some constants $C_1,{\beta }_1>0$ that depend on $\phi$, the porosity parameter $\nu$, and the dimension.

The adaptive lattice ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$ is designed so that its sampling cells are small in the sense that

$$|D_{{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)}\left(z\right)|\le C_0{h}^{\alpha }.$$

This condition guarantees that the discrete sum

$$G_{\phi ,{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right),{\chi }_{\Omega \left(h\right)}}f=|D_{{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)}|\sum _{\lambda \in {\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)}{\chi }_{\Omega \left(h\right)}\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi ,$$

serves as an accurate Riemann-sum approximation of the continuous integral defining $P_{\phi ,{\chi }_{\Omega \left(h\right)}}f$. In particular, standard results in Gabor analysis (see, e.g., [6,19]) imply that the operator norms of the discrete multiplier and the continuous localization operator are comparable, up to a constant that depends on the density of the lattice.

By (40), we have

$$\parallel P_{\phi ,{\chi }_{\Omega \left(h\right)}}{f\parallel }_{L^2}\le C_1{h}^{{\beta }_1}{\parallel f\parallel }_{L^2}.$$

The adaptive lattice condition ensures that the discretization error is controlled uniformly as $h\to 0$. Therefore, there exists a constant $C_2>0$ (depending on the lattice density, i.e., on $C_0$ and $\alpha$) such that

$$\parallel G_{\phi ,{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right),{\chi }_{\Omega \left(h\right)}}f-P_{\phi ,{\chi }_{\Omega \left(h\right)}}{f\parallel }_{L^2}\le C_2{h}^{\delta }{\parallel f\parallel }_{L^2},$$

for some $\delta >0$. Thus, by the triangle inequality,

$$\parallel G_{\phi ,{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right),{\chi }_{\Omega \left(h\right)}}{f\parallel }_{L^2} \le \parallel P_{\phi ,{\chi }_{\Omega \left(h\right)}}{f\parallel }_{L^2}+C_2{h}^{\delta }{\parallel f\parallel }_{L^2}\le \left(C_1{h}^{{\beta }_1}+C_2{h}^{\delta }\right){\parallel f\parallel }_{L^2}.$$

By choosing $\beta =min\{{\beta }_1,\delta \}$ and absorbing constants into a single constant C, we obtain

$$\parallel G_{\phi ,{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right),{\chi }_{\Omega \left(h\right)}}{f\parallel }_{L^2}\le C{h}^{\beta }{\parallel f\parallel }_{L^2},$$

for all $0<h\le 1$.

The argument above is formulated for the $L^2$ norm. In many cases, one may also verify corresponding endpoint estimates (e.g., in $L^1$ and $L^{\infty }$) and then interpolate using the Riesz-Thorin theorem. However, since our main interest is in the $L^2$ bound, the above estimate suffices to conclude that

$$\parallel G_{\phi ,{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right),{\chi }_{\Omega \left(h\right)}}{\parallel }_{L^2\to L^2} \le C{h}^{\beta }.$$

The adaptive lattice condition ensures that the discrete sampling approximates the continuous time-frequency localization operator with controlled error. Combining this with the fractal uncertainty principle for the continuous operator yields the desired norm decay for the discrete Gabor multiplier. This completes the proof. □

This result is particularly useful in applications where non-uniform sampling is required, such as medical imaging and radar signal processing.

Beyond norm estimates, it is important to understand the spectral properties of Gabor multipliers in this generalized setting.

Proposition 6. 

Let $G_{\phi ,\mathsf{\Lambda },b}$ be a Gabor multiplier with an admissible window φ and a symbol $b\left(\lambda \right)$ supported on a ν-porous set Ω. Then the singular values ${\sigma }_n\left(G_{\phi ,\mathsf{\Lambda },b}\right)$ satisfy

$${\sigma }_n\left(G_{\phi ,\mathsf{\Lambda },b}\right)\le C{n}^{-\gamma },$$

for some exponent $\gamma >0$ that depends on ν and the smoothness of φ.

Proof. 

Using the theory of Gabor frames (see, e.g., [6,19]), one can identify the range of the STFT with a reproducing kernel Hilbert space (often the Bargmann-Fock space when $\phi$ is Gaussian, and a generalized Fock space in the non-Gaussian case). More precisely, by applying a generalized Bargmann transform $B_{\phi }$, one may represent the operator $G_{\phi ,\mathsf{\Lambda },b}$ as a Toeplitz operator

$$T_{b}F\left(z\right)={\int }_{{\mathbb{C}}^d}b\left(z\right)K_{\phi }(z,\xi )F\left(\xi \right)d{\mu }_{\phi }\left(\xi \right),$$

acting on the corresponding Fock space ${\mathcal{F}}_{\phi }^2\left({\mathbb{C}}^d\right)$, with reproducing kernel $K_{\phi }(z,\xi )$ and measure $d{\mu }_{\phi }$. In this framework, the singular values of $G_{\phi ,\mathsf{\Lambda },b}$ coincide (up to constants) with the singular values of $T_b$. Hence, it suffices to prove the decay estimate for $T_b$.

Since the symbol $b\left(\lambda \right)$ is supported on a $\nu$-porous set $\Omega$, the effective “size” (or measure) of the support of b is small in a fractal sense. More precisely, due to the porosity condition, one can show that the measure (or the counting function) of $\Omega$ obeys a power law bound. For example, one may define the maximal Nyquist density for $\Omega$ by

$$\rho (\Omega ,R)=\underset{z\in {\mathbb{R}}^{2d}}{sup}{\displaystyle \frac{|\Omega \cap B_R\left(z\right)|}{|B_R\left(z\right)|}},$$

and then use the porosity condition to deduce that for appropriate scales (related to h),

$$\rho (\Omega ,R)\le C_1{h}^{\gamma },$$

for some $\gamma >0$. This implies that the effective number of lattice points $\lambda \in \mathsf{\Lambda }$ in $\Omega$ is significantly lower than the total number one would expect in a full ball of radius R. In spectral terms, this reduced density implies that the operator $T_b$ (and hence $G_{\phi ,\mathsf{\Lambda },b}$) has a rapidly decaying spectrum.

In many settings, one can relate the counting function $N\left(ϵ\right)$ of eigenvalues (or singular values) exceeding $ϵ$ to the phase-space volume where the symbol is nonzero. For Toeplitz operators in reproducing kernel Hilbert spaces, a Weyl law often takes the form

$$N\left(ϵ\right)\sim C{ϵ}^{-\delta },$$

where $\delta >0$ is determined by the effective dimension of the support. In our case, the fractal (porous) structure of $\Omega$ implies that the effective dimension is smaller than the full $2d$. Consequently, one can derive an asymptotic bound on the singular values such that

$${\sigma }_n\left(T_b\right)\le C{n}^{-\gamma },$$

for some $\gamma >0$. The precise value of $\gamma$ depends on the fractal (porosity) parameter $\nu$ and on the smoothness and decay properties of $\phi$ (which control the behavior of the reproducing kernel $K_{\phi }(z,\xi )$).

By transferring the Weyl law estimate from $T_b$ back to the original operator $G_{\phi ,\mathsf{\Lambda },b}$ (using the equivalence provided by the generalized Bargmann transform), we deduce that the singular values ${\sigma }_n\left(G_{\phi ,\mathsf{\Lambda },b}\right)$ decay as

$${\sigma }_n\left(G_{\phi ,\mathsf{\Lambda },b}\right)\le C{n}^{-\gamma }.$$

The constant C and the exponent $\gamma >0$ depend on the porosity constant $\nu$, the regularity (or decay) properties of the window $\phi$, and the geometric properties of the lattice $\mathsf{\Lambda }$.

Thus, by representing the Gabor multiplier as a Toeplitz operator in a generalized Fock space, employing fractal measure estimates on the support of the symbol b, and using Weyl-type asymptotics for localization operators, we conclude that

$${\sigma }_n\left(G_{\phi ,\mathsf{\Lambda },b}\right)\le C{n}^{-\gamma },$$

for some $\gamma >0$. This completes the proof. □

Explicit Dependence of Constants and Parameter Bookkeeping

Theorems in the main text assert inequalities of the form

$$\parallel {\mathbf{1}}_X{\mathcal{T}}_h{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le C{h}^{\beta },$$

but for clarity and applicability we now record the precise finite list of parameters on which the constants C and $\beta$ depend, give a short lemma collecting this dependence, and state a representative quantitative bound with the dependence written explicitly.

Let d denote the spatial dimension and fix an ambient radius $R>0$ used in the local covering hypotheses. We summarize the quantities that enter constants below:

i.

Covering constant $C_{\mathrm{cov}}$ and local covering exponents $s_X,s_Y$ (or anisotropic versions $s_X^{\mathbf{a}},s_Y^{\mathbf{a}}$); the porosity gap $\mathsf{\epsilon }$ defined by $s_X+s_Y\le 2d-\mathsf{\epsilon }$.

ii.

A finite collection of seminorms of the analysing window g (denote these collectively by $\mathcal{S}\left(g\right)$), and, when convenient, tail parameters (for example exponential-type constants $(A,c,p)$ or polynomial-type constants $(A,M)$) appearing in Definition 4 and Assumption 2.

iii.

Lattice redundancy/local density $r_{\mathsf{\Lambda }}$ and discrete frame bounds $A(g,\mathsf{\Lambda }),B(g,\mathsf{\Lambda })$.

iv.

The ambient dimension d (and anisotropy vector $\mathbf{a}$ when applicable), which enters combinatorial volume factors.

For compactness we introduce the notation

$$\begin{array}{cc}& \mathsf{Geom}:=(d,R,C_{\mathrm{cov}},s_X,s_Y,\mathsf{\epsilon }),\hfill \\ & \mathsf{Win}:=\mathcal{S}\left(g\right)(\mathrm{or}(A,c,p)\mathrm{or}(A,M)),\hfill \\ & \mathsf{Frame}:=(r_{\mathsf{\Lambda }},A(g,\mathsf{\Lambda }),B(g,\mathsf{\Lambda })).\hfill \end{array}$$

Lemma 14 

(Constants bookkeeping). There exist (computable) functions

$$\mathcal{C}(\mathsf{Geom},\mathsf{Win},\mathsf{Frame}),\mathcal{B}(\mathsf{Geom},\mathsf{Win},\mathsf{Frame})>0$$

such that the main FUP bound may be written in the explicit form

$$\parallel {\mathbf{1}}_X{\mathcal{T}}_h{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le \mathcal{C}(\mathsf{Geom},\mathsf{Win},\mathsf{Frame})h^{\mathcal{B}(\mathsf{Geom},\mathsf{Win},\mathsf{Frame})},$$

for all sufficiently small $h\in (0,h_0]$. The functions $\mathcal{C}$ and $\mathcal{B}$ depend only on the finite collection of parameters listed above and on universal numeric constants depending on d.

Proof. 

Every time an inequality used an unspecified constant one can (and should) replace it by a symbolic name that is then traced back to one of the finitely many parameters listed above. The Schur and interpolation steps use only finitely many volumes and seminorms and hence involve only these parameters. □

Proposition 7. 

Assume the kernel satisfies the exponential tail bound (1) with constants $(A,c)$ (use $p=1$ for simplicity), and assume the local covering bounds hold with constant $C_{\mathrm{cov}}$ and exponents $s_X,s_Y$ on an ambient radius R. Define the porosity gap $\mathsf{\epsilon }:=2d-(s_X+s_Y)>0$. Then there are universal numeric constants $C_d,C_d^{\prime }>0$ depending only on d such that for $0<h\le h_1$ one has

$$\parallel {\mathbf{1}}_X{\mathcal{T}}_h{\mathbf{1}}_Y{\parallel }_{L^2\to L^2}\le C_{d}A{C}_{\mathrm{cov}}{R}^{(s_X+s_Y)/2}{h}^{\mathsf{\epsilon }/2}{\left(log(1/h)\right)}^{{\displaystyle \frac{2d-(s_X+s_Y)}{2}}}.$$

(41)

Equivalently, one can write $\mathcal{C}(\mathsf{Geom},\mathsf{Win})\le C_d^{\prime }A{C}_{\mathrm{cov}}{R}^{(s_X+s_Y)/2}$ and $\mathcal{B}(\mathsf{Geom},\mathsf{Win})\ge \mathsf{\epsilon }/2$ (with the displayed polylog factor isolated explicitly).

Remarks. 

The estimate (41) is obtained by the Schur-test decomposition and the tail integral estimate of Lemma 6 using the mesoscopic choice $r\left(h\right)=hlog(1/h)$. The prefactor $A{C}_{\mathrm{cov}}{R}^{(s_X+s_Y)/2}$ comes from (i) the pointwise kernel amplitude A, (ii) the number of covering boxes $\lesssim C_{\mathrm{cov}}{(R/r)}^{s_Y}$ (or symmetric), and (iii) the volumes $r^{d-s_Y}$ appearing in the near-term estimates; collect these factors to produce the displayed dependence. □

This result in

Table 2. Sample parameter summary and numerical estimates of prefactors.

Window	Tail Type	Tail Params	$\mathcal{S}\left(\mathit{g}\right)$ (Seminorms)	A	Estimated C	Notes
Gaussian	exponential	$c\approx 1.2$	explicit Gaussian seminorms	$A_G$	$C_G$	analytic values computable
Compact $C^6$	polynomial	$M\approx 6$	finitely many derivatives	$A_c$	$C_c$	computed by integration by parts
DPSS	exponential-like	$c\approx 0.9$	numeric seminorms	$A_D$	$C_D$	estimated numerically

shows that the spectral concentration properties of Gabor multipliers persist beyond the Gaussian setting, reinforcing the universality of the FUP framework.

In this section, we extended the FUP beyond the standard Gaussian Gabor multiplier framework. These results provide a flexible framework for time-frequency analysis in both theoretical and applied settings, opening pathways for further investigation into optimal window choices and adaptive signal representations.

9. Applications

In this section, we bridge our theoretical advances with computational practice. In particular, we introduce a discrete version of the fractal localization operators and Gabor multipliers, analyze the convergence and error estimates of their approximations, and illustrate potential applications in signal processing and quantum mechanics.

To approximate the continuous operators introduced in earlier sections, we define a discrete analogue based on sampling the time-frequency plane on an adaptive lattice.

Definition 9. 

Let $\mathsf{\Lambda }\left(h\right)\subset {\mathbb{R}}^{2d}$ be an h-dependent lattice satisfying a density condition (see condition (H) in [5]). Given a bounded symbol $S:\mathsf{\Lambda }\left(h\right)\to \{0,1\}$ that mimics the characteristic function of a ν-porous set, we define the discrete localization operator

$$P_{h,S}f=|D_{\mathsf{\Lambda }\left(h\right)}|\sum _{\lambda \in \mathsf{\Lambda }\left(h\right)}S\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi ,$$

where $\phi \in L^2\left({\mathbb{R}}^d\right)$ is an admissible window and $\pi \left(\lambda \right)$ denotes the time-frequency shift.

This operator can be viewed as a discrete counterpart to Daubechies’ continuous localization operator. In the case where $\phi ={\phi }_0$ (the Gaussian), the operator corresponds to a Gaussian Gabor multiplier.

We now study the convergence of the discrete operator $P_{h,S}$ as the lattice becomes finer (i.e., as $h\to 0$) and provide an error estimate in terms of h.

Theorem 18. 

Let $\mathsf{\Lambda }\left(h\right)\subset {\mathbb{R}}^{2d}$ be an h–dependent lattice with fundamental cell $D_{\mathsf{\Lambda }\left(h\right)}$ of volume $|D_{\mathsf{\Lambda }\left(h\right)}|$ and diameter $diam{D}_{\mathsf{\Lambda }\left(h\right)}=:r\left(h\right)\to 0$ as $h\to 0$. Let $\phi \in \mathcal{S}\left({\mathbb{R}}^d\right)$ be a Schwartz window (in particular φ and all its translates/modulations are uniformly Lipschitz in phase space). Let $S:\mathsf{\Lambda }\left(h\right)\to \{0,1\}$ be a discrete symbol that models the characteristic function of a ν–porous set $\Omega \subset {\mathbb{R}}^{2d}$ in the sense that the symmetric difference between Ω and the cellular approximation

$${\Omega }_h:=\bigcup _{\lambda \in \mathsf{\Lambda }\left(h\right):S\left(\lambda \right)=1}{D}_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)(\mathit{here}{D}_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)=\lambda +D_{\mathsf{\Lambda }\left(h\right)})$$

satisfies

$$meas\left(\Omega ▵{\Omega }_h\right)\le C_0{h}^{\gamma }$$

for some constants $C_0,\gamma >0$ determined by the porosity data and the lattice construction.

Write the continuous localization operator

$$P_{S}f={\int }_{{\mathbb{R}}^{2d}}S\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi dz,$$

and the discrete operator

$$P_{h,S}f=|D_{\mathsf{\Lambda }\left(h\right)}|\sum _{\lambda \in \mathsf{\Lambda }\left(h\right)}S\left(\lambda \right)\langle f,\pi \left(\lambda \right)\phi \rangle \pi \left(\lambda \right)\phi .$$

Then there exist constants $C,\beta >0$ (depending only on φ, the lattice geometry and the porosity constants) and $h_0>0$ such that for all $0<h\le h_0$ and all $f\in L^2\left({\mathbb{R}}^d\right)$

$$\parallel (P_{h,S}-P_S){f\parallel }_{L^2\left({\mathbb{R}}^d\right)}\le C{h}^{\beta }{\parallel f\parallel }_{L^2\left({\mathbb{R}}^d\right)}.$$

Consequently $\parallel P_{h,S}-P_S{\parallel }_{L^2\to L^2}\le C{h}^{\beta }$.

Proof. 

The proof is a quantitative Riemann–sum approximation combined with two elementary continuity estimates for the integrand. Fix $f\in L^2\left({\mathbb{R}}^d\right)$. Partition phase space by the disjoint cells ${\left\{D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)\right\}}_{\lambda \in \mathsf{\Lambda }\left(h\right)}$ and write

$$P_{S}f=\sum _{\lambda \in \mathsf{\Lambda }\left(h\right)}{\int }_{D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)}S\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi dz.$$

By definition of $P_{h,S}$ (and using $V_{\phi }f\left(\lambda \right)=\langle f,\pi \left(\lambda \right)\phi \rangle$)

$$P_{h,S}f=\sum _{\lambda \in \mathsf{\Lambda }\left(h\right)}S\left(\lambda \right)V_{\phi }f\left(\lambda \right)\left(|D_{\mathsf{\Lambda }\left(h\right)}|\pi \left(\lambda \right)\phi \right).$$

Therefore

$$\begin{array}{cc}\hfill (P_{h,S}-P_S)f& =\sum _{\lambda \in \mathsf{\Lambda }\left(h\right)}{R}_{\lambda }\left(f\right),\hfill \\ \hfill R_{\lambda }\left(f\right)& :={\int }_{D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)}\left[S\left(\lambda \right)V_{\phi }f\left(\lambda \right)\pi \left(\lambda \right)\phi -S\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi \right]dz.\hfill \end{array}$$

We estimate the operator norm by bounding $\parallel (P_{h,S}-P_S){f\parallel }_{L^2}$ for ${\parallel f\parallel }_{L^2}=1$ and then homogeneity gives the general estimate.

Fix $\lambda$ and split the integrand as

$$\begin{array}{cc}& S\left(\lambda \right)V_{\phi }f\left(\lambda \right)\pi \left(\lambda \right)\phi -S\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi \hfill \\ & =S\left(\lambda \right)V_{\phi }f\left(\lambda \right)\left(\pi \left(\lambda \right)\phi -\pi \left(z\right)\phi \right)+\left(S\left(\lambda \right)V_{\phi }f\left(\lambda \right)-S\left(z\right)V_{\phi }f\left(z\right)\right)\pi \left(z\right)\phi .\hfill \end{array}$$

Correspondingly write $R_{\lambda }\left(f\right)=A_{\lambda }\left(f\right)+B_{\lambda }\left(f\right)$ according to the two terms above.

Using the triangle inequality and Cauchy–Schwarz in the integral,

$$\parallel A_{\lambda }{\left(f\right)\parallel }_{L^2}\le {\int }_{D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)}|V_{\phi }{f\left(\lambda \right)|\parallel \pi \left(\lambda \right)\phi -\pi \left(z\right)\phi \parallel }_{L^2}dz.$$

Since $\phi \in \mathcal{S}\left({\mathbb{R}}^d\right)$, the map $z\mapsto \pi \left(z\right)\phi$ is uniformly Lipschitz on bounded regions of phase space; precisely there exists $C_1>0$ (depending only on a finite number of seminorms of $\phi$ and on the diameter of the region where the symbol is supported) such that

$${\parallel \pi \left(\lambda \right)\phi -\pi \left(z\right)\phi \parallel }_{L^2}\le C_1|z-\lambda |\mathrm{for}z\in D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right).$$

Also $|V_{\phi }{f\left(\lambda \right)| \le \parallel f\parallel }_{L^2}{\parallel \phi \parallel }_{L^2}$ by Cauchy–Schwarz. Hence, for ${\parallel f\parallel }_{L^2}=1$,

$$\parallel A_{\lambda }{\left(f\right)\parallel }_{L^2}\le {\parallel \phi \parallel }_{L^2}{C}_1{\int }_{D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)}|z-\lambda |dz\le C_2|D_{\mathsf{\Lambda }\left(h\right)}|r\left(h\right),$$

where $r\left(h\right)=diam{D}_{\mathsf{\Lambda }\left(h\right)}$ and $C_2$ depends on $\phi$ and the local geometry. Summing over those $\lambda$ with $D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)$ intersecting the (fixed, bounded) phase-space region of interest yields

$$\sum _{\lambda }{\parallel A_{\lambda }\left(f\right)\parallel }_{L^2}\le C_{3}r\left(h\right),$$

for a constant $C_3$ independent of h and f (we use that the union of the relevant cells has volume uniformly bounded in h). Thus the first contribution is $O\left(r\right(h\left)\right)$:

$$\parallel \sum _{\lambda }{A}_{\lambda }\left(f\right){\parallel }_{L^2}\le C_{3}r\left(h\right).$$

We write

$$S\left(\lambda \right)V_{\phi }f\left(\lambda \right)-S\left(z\right)V_{\phi }f\left(z\right)=\left(S\left(\lambda \right)-S\left(z\right)\right)V_{\phi }f\left(\lambda \right)+S\left(z\right)\left(V_{\phi }f\left(\lambda \right)-V_{\phi }f\left(z\right)\right).$$

The second piece is handled as well by smoothness of the STFT (again a standard fact for Schwartz windows) there is $C_4>0$ with

$$|V_{\phi }f\left(\lambda \right)-V_{\phi }f\left(z\right)|\le C_4{|z-\lambda |\parallel f\parallel }_{L^2},$$

so integrating as before gives a contribution bounded by $C_{5}r\left(h\right)$.

The first piece, involving $S\left(\lambda \right)-S\left(z\right)$, is nonzero only for those cells $D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)$ that intersect the boundary of $\Omega$ (i.e., where the discrete symbol value differs from the continuous symbol). Let ${\mathcal{B}}_h$ denote the union of such boundary cells. Then

$$\parallel \sum _{\lambda }{\int }_{D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)}\left(S\left(\lambda \right)-S\left(z\right)\right)V_{\phi }f\left(\lambda \right)\pi \left(z\right)\phi dz{\parallel }_{L^2} \le \underset{z}{sup}|V_{\phi }f\left(z\right)|·meas{\left({\mathcal{B}}_h\right)}^{1/2}·C_6,$$

by Cauchy–Schwarz and boundedness of ${\parallel \pi \left(z\right)\phi \parallel }_{L^2}={\parallel \phi \parallel }_{L^2}$ (the constant $C_6$ depends only on $\phi$). Using the general bound ${sup}_z|V_{\phi }{f\left(z\right)| \le \parallel \phi \parallel }_{L^2}{\parallel f\parallel }_{L^2}$ and our modeling assumption $meas(\Omega ▵{\Omega }_h)\le C_0{h}^{\gamma }$, we get

$$\mathrm{boundary}\mathrm{contribution}\le C_7{h}^{\gamma /2}·{\parallel f\parallel }_{L^2}.$$

Combining the two pieces in $B_{\lambda }$ yields

$$\parallel \sum _{\lambda }{B}_{\lambda }\left(f\right){\parallel }_{L^2} \le C_8\left(r\left(h\right)+h^{\gamma /2}\right){\parallel f\parallel }_{L^2}.$$

Adding the provided results, we obtain, for all $f\in L^2$,

$$\parallel (P_{h,S}-P_S){f\parallel }_{L^2}\le C\left(r\left(h\right)+h^{\gamma /2}\right){\parallel f\parallel }_{L^2}.$$

Finally, note that the lattice construction gives $r\left(h\right)\to 0$ as $h\to 0$; in the usual semiclassical/adaptive lattices one may choose $r\left(h\right)\lesssim h^{\alpha }$ for some $\alpha >0$ determined by the lattice density hypothesis. Hence there exists $\beta >0$ (for instance $\beta =min(\alpha ,\gamma /2)$) and $C^{\prime }>0$ such that for all sufficiently small h

$$\parallel P_{h,S}-P_S{\parallel }_{L^2\to L^2}\le C^{\prime }{h}^{\beta }.$$

This completes the proof. □

Example 5. 

We illustrate Theorem 18 by a simple, concrete model in dimension $d=1$ (so phase space is ${\mathbb{R}}^2$).

Take the normalized Gaussian

$$\phi \left(t\right)=2^{1/4}{e}^{-\pi t^2},{\parallel \phi \parallel }_{L^2}=1,$$

so $|V_{\phi }{\phi \left(u\right)|}^2=e^{-{\pi |u|}^2}$ and $\mathsf{\epsilon }\left(R\right)={\int }_{|u|>R}{e}^{-{\pi |u|}^2}du$ has Gaussian decay (see Example 1).

Let $S={\mathbf{1}}_{\mathcal{D}}$ be the indicator of the disc

$$\mathcal{D}:=\{(x,\xi )\in {\mathbb{R}}^2:x^2+{\xi }^2\le R_0^2\},$$

with a fixed radius $R_0>0$. The continuous localization operator is

$$P_{S}f={\int }_{{\mathbb{R}}^2}{\mathbf{1}}_{\mathcal{D}}\left(z\right)V_{\phi }f\left(z\right)\pi \left(z\right)\phi dz.$$

Fix a lattice spacing exponent $\alpha \in (0,1)$ and set

$$\mathsf{\Lambda }\left(h\right)=r\left(h\right){\mathbb{Z}}^2,r\left(h\right):=h^{\alpha }.$$

The fundamental cell has diameter $diam{D}_{\mathsf{\Lambda }\left(h\right)}\lesssim r\left(h\right)$. Define the discrete symbol $S_h:\mathsf{\Lambda }\left(h\right)\to \{0,1\}$ by the cellular rule

$$S_h\left(\lambda \right)=1⟺\mathrm{center}\left(D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)\right)\in \mathcal{D},$$

and set $P_{h,S}$ as in Definition 9 (Riemann sum with cell volume $|D_{\mathsf{\Lambda }\left(h\right)}{|\asymp r\left(h\right)}^2$).

By Lemma 16 we have

$$meas\left(\mathcal{D}▵{\mathcal{D}}_{r\left(h\right)}\right)\le C^{\prime }r{\left(h\right)}^{2-s},$$

where $s\in [0,2)$ is the effective box-counting exponent of the boundary (for a smooth disc $\partial \mathcal{D}$ we may take $s=1$). Using the estimate in the proof of Theorem 18 one obtains the quantitative bound

$$\parallel P_{h,S}-P_S{\parallel }_{L^2\to L^2} \le C\left(r\left(h\right)+r{\left(h\right)}^{\frac{2-s}{2}}\right).$$

Recalling $r\left(h\right)=h^{\alpha }$, this can be written as

$$\parallel P_{h,S}-P_S{\parallel }_{L^2\to L^2} \le C{h}^{\beta },\beta =\alpha ·min\left\{1,{\displaystyle \frac{2-s}{2}}\right\}.$$

(42)

Two remarks:

1. 

For a smooth boundary (disc) we may set $s=1$, so $\beta =\alpha min\{1,\frac{1}{2}\}=\alpha /2$.

2. 

For fractal/porous boundaries with larger s the exponent β decreases in a computable way via (42).

This example makes explicit how the cellular approximation error (Lemma 16) and the continuity/Riemann-sum error combine to give the power-law estimate in Theorem 18.

Lemma 15. 

Let $D\subset {\mathbb{R}}^2$ be compact and let $\{D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right):\lambda \in \mathsf{\Lambda }\left(h\right)\}$ be a partition of ${\mathbb{R}}^2$ into fundamental cells (for instance translates of a fixed fundamental domain for the lattice $\mathsf{\Lambda }\left(h\right)$) used in the center-in-D rule

$$S_h\left(\lambda \right)=1⟺\mathrm{center}\left(D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)\right)\in D.$$

Assume there exist constants $0<a\le b<\infty$ (independent of h) such that every cell satisfies the uniform in/out ball comparability

$$B\left(\mathrm{center}\left(D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)\right),ar\left(h\right)\right)\subset D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)\subset B\left(\mathrm{center}\left(D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)\right),br\left(h\right)\right),$$

(43)

where $r\left(h\right)>0$ is the sampling/cell-scale parameter used in Example 5.

Define

$$U_h:=\bigcup _{\lambda :S_h\left(\lambda \right)=1}{D}_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)$$

the union of cells whose centers lie in D. Then with $c:=max\{a^{-1},b\}$ (so in particular one may take $c=b$), we have the inclusions

$$D-cr\left(h\right)\subset U_h\subset D+cr\left(h\right),$$

where for a set E and $\rho >0$ we write $E+\rho :=\{x:dist(x,E)\le \rho \}$ and $E-\rho :=\{x:dist(x,{\mathbb{R}}^2\setminus E)>\rho \}$.

Consequently, if D has covering/Hausdorff exponent $s\in [0,2]$ (equivalently there exists $C_1>0$ with $N_D\left(\rho \right)\le C_1{\rho }^{-s}$ for $0<\rho \le {\rho }_0$), then the symmetric difference obeys the uniform bound

$$meas\left(D▵U_h\right)\le meas\left((D+cr(h\left)\right)\setminus (D-cr(h\left)\right)\right)\lesssim r{\left(h\right)}^{2-s},$$

with implicit constant depending only on $C_1$, s, $a,b$ and geometric constants.

Proof. 

The inclusions in (43) imply the following two simple implications.

• (1) If $x\in D$ and y is any point with $dist(y,x)\le ar\left(h\right)$ then the ball $B(y,ar(h\left)\right)$ meets D (it contains x). Hence the cell $D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)$ whose center is y satisfies $B(y,ar\left(h\right))\subset D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)$ and therefore $D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)$ intersects D. By the definition of $U_h$ (center-in-D rule) the cell is included in $U_h$. Thus every point of $D-ar\left(h\right)$ (points of D at distance $\ge ar\left(h\right)$ from the complement) is contained in $U_h$. Equivalently,

$$D-ar\left(h\right)\subset U_h.$$

(If one prefers the slightly coarser constant $c\ge a$ as in the statement, replace a by c here.)

• (2) Conversely, suppose a cell $D_{\mathsf{\Lambda }\left(h\right)}\left(\lambda \right)$ is included in $U_h$. By definition its center $z_{\lambda }$ lies in D, and by the outer inclusion in $(\ast )$ the whole cell is contained in the ball $B(z_{\lambda },br\left(h\right))$. Hence every point of $U_h$ lies within distance $br\left(h\right)$ of D, i.e.,

$$U_h\subset D+br\left(h\right).$$

Combining (1) and (2) yields the claimed two-sided inclusion with $c=max\{a,b\}$ (or with $c=b$ if one only records the outward radius).

Finally, the symmetric difference is contained in the boundary layer

$$D▵U_h\subset (D+cr\left(h\right))\setminus (D-cr\left(h\right)),$$

so its Lebesgue measure is bounded by the measure of the $cr\left(h\right)$–neighbourhood of $\partial D$. Standard estimates for the volume of tubular neighbourhoods (see e.g., Mattila, Geometry of Sets and Measures in Euclidean Spaces, or Federer) give that if $N_D\left(\rho \right)\le C_1{\rho }^{-s}$ for $\rho \le {\rho }_0$, then

$$meas\left(\{x:dist(x,D)\le \rho \}\right)\lesssim {\rho }^{2-s},0<\rho \le {\rho }_0,$$

with implicit constants depending only on $C_1$ and s. Applying this with $\rho =cr\left(h\right)$ yields

$$meas\left(D▵U_h\right)\lesssim {\left(cr\left(h\right)\right)}^{2-s}\asymp r{\left(h\right)}^{2-s},$$

which is the desired uniform bound. □

Remark 9. 

The uniform cell-shape assumption $(\ast )$ is satisfied in the usual lattice constructions used in Section 9: for instance if the cells are translates of a fixed fundamental parallelogram of side-lengths comparable to $r\left(h\right)$, then one may take $a,b$ equal to half the inradius/outradius of that parallelogram (independent of h). If the cells are chosen as Euclidean discs of radius $r\left(h\right)$ the constants simplify to $a=b=1$. The covering-number hypothesis $N_D\left(\rho \right)\le C_1{\rho }^{-s}$ is equivalent (up to constants) to the usual covering/exponent assumption used elsewhere in the manuscript (see Example 2 and the discussion following Lemma 2).

Lemma 16. 

Let $\Omega \subset {\mathbb{R}}^{2d}$ be a bounded measurable set. Assume there exist constants $C_{\mathrm{cov}}>0$ and $s\in [0,2d)$ such that for every $0<\rho \le {\rho }_0$ the covering number of Ω by Euclidean balls of radius ρ satisfies

$$N_{\Omega }\left(\rho \right)\le C_{\mathrm{cov}}{\rho }^{-s}.$$

Let ${\left\{D_{\lambda }\right\}}_{\lambda \in \mathsf{\Lambda }}$ be a partition of ${\mathbb{R}}^{2d}$ into measurable cells (for instance the translates of a fundamental cell of a lattice) with the property that each cell $D_{\lambda }$ has diameter $diam{D}_{\lambda }\le r$ and volume

$$c_1{r}^{2d}\le |D_{\lambda }|\le c_2{r}^{2d}$$

for constants $c_1,c_2>0$ independent of r. For a given $r>0$ define the cellular approximation

$${\Omega }_r:=\bigcup _{\lambda :\mathrm{center}\left(D_{\lambda }\right)\in \Omega }{D}_{\lambda },$$

i.e., the union of those cells whose centers lie in Ω. Then there exists a constant $C^{\prime }>0$ (depending only on $C_{\mathrm{cov}},c_1,c_2$ and the ambient dimension $2d$) and ${\rho }_1>0$ such that for all $0<r\le {\rho }_1$ one has

$$meas\left(\Omega ▵{\Omega }_r\right)\le C^{\prime }{r}^{2d-s}.$$

In particular, if in your semiclassical notation $r=r\left(h\right)$ and $r\left(h\right)\lesssim h^{\alpha }$ for some $\alpha >0$, then

$$meas\left(\Omega ▵{\Omega }_{r\left(h\right)}\right)\le C^{″}{h}^{\alpha (2d-s)}$$

for some $C^{″}>0$.

Proof. 

Let $\partial \Omega$ denote the topological boundary of $\Omega$. The symmetric difference $\Omega ▵{\Omega }_r$ is contained in a uniform neighbourhood of $\partial \Omega$ of radius $C_{0}r$ (with $C_0$ depending only on the choice of the cell centers—for example $C_0=1$ if each cell is contained in the Euclidean ball of radius r about its center). Indeed, if $x\in \Omega ▵{\Omega }_r$ then either

1.

$x\in \Omega$ but the center of the cell containing x lies in ${\Omega }^c$, or

2.

$x\notin \Omega$ but the center of its cell lies in $\Omega$.

In both cases the center of the cell containing x lies within distance at most $diam{D}_{\lambda }\le r$ of a boundary point of $\Omega$, hence x lies in the r–neighbourhood of $\partial \Omega$. Thus

$$\Omega ▵{\Omega }_r\subset {(\partial \Omega )}_r,$$

where ${(\partial \Omega )}_r:=\{y\in {\mathbb{R}}^{2d}:dist(y,\partial \Omega )\le r\}$.

It remains to estimate the Lebesgue measure of ${(\partial \Omega )}_r$. Cover $\partial \Omega$ by at most $N_{\partial \Omega }\left(r\right)$ balls of radius r, where $N_{\partial \Omega }\left(r\right)$ denotes the minimal such covering number. Clearly

$${|(\partial \Omega )}_r| \le N_{\partial \Omega }\left(r\right)·|B(0,r)|\le C_{\mathrm{geom}}{N}_{\partial \Omega }\left(r\right)r^{2d},$$

where $|B(0,r)|={\omega }_{2d}{r}^{2d}$ and $C_{\mathrm{geom}}$ is an absolute geometric constant (we may take $C_{\mathrm{geom}}={\omega }_{2d}$).

To relate $N_{\partial \Omega }\left(r\right)$ to $N_{\Omega }\left(r\right)$ note that any r–ball which meets $\partial \Omega$ must intersect either $\Omega$ or ${\Omega }^c$, hence

$$N_{\partial \Omega }\left(r\right)\le N_{\Omega }\left(r\right)+N_{{\Omega }^c}\left(r\right).$$

Because $\Omega$ is bounded, we may apply the covering hypothesis to both $\Omega$ and ${\Omega }^c\cap B(0,R)$ (for a sufficiently large fixed ball $B(0,R)$ containing $\Omega$) and find the same power law bound; in particular there is a constant $C_1$ (depending only on $C_{\mathrm{cov}}$ and the chosen bounding ball) such that for all sufficiently small r,

$$N_{\partial \Omega }\left(r\right)\le C_1{r}^{-s}.$$

Combining the last two displays yields

$$|\Omega ▵{\Omega }_r|\le |{(\partial \Omega )}_r|\le C_{\mathrm{geom}}{C}_1{r}^{-s}{r}^{2d}=C^{\prime }{r}^{2d-s},$$

with $C^{\prime }:=C_{\mathrm{geom}}{C}_1$. This proves the first claim.

Finally, if $r=r\left(h\right)\lesssim h^{\alpha }$ then $r^{2d-s}\lesssim h^{\alpha (2d-s)}$, giving the stated corollary with $C^{″}$ equal to the implied constant times $C^{\prime }$. □

Porosity (as used elsewhere in the manuscript) implies an upper bound on the Minkowski/box dimension of $\Omega$, and hence a covering-number bound of the form $N_{\Omega }\left(\rho \right)\le C_{\mathrm{cov}}{\rho }^{-s}$ for some $s<2d$. Thus, the hypothesis of Lemma 16 is satisfied for the porous sets treated in the paper; the exponent $2d-s$ which appears in the volume estimate is the standard exponent describing how the r–neighbourhood volume of a fractal set decays as $r\downarrow 0$.

Corollary 4. 

Under the same assumptions as in Theorem 18, the discrete Gaussian Gabor multiplier $G_{{\phi }_0,\mathsf{\Lambda }\left(h\right),S}$ satisfies

$$\parallel G_{{\phi }_0,\mathsf{\Lambda }\left(h\right),S}{\parallel }_{L^2\to L^2}\le C{h}^{\beta },$$

for $0<h\le 1$, where C and β are as in Theorem 18.

This result extends the continuous fractal uncertainty estimate to a fully discrete setting, which is essential for computational applications.

We now illustrate how these discrete operators can be employed in practice. Two applications are presented: one in sparse signal recovery and one in the numerical study of resonance phenomena in quantum systems.

A signal $f\in L^2\left({\mathbb{R}}^d\right)$ is said to be fractal sparse if its time-frequency representation $V_{\phi }f$ is predominantly supported on a $\nu$-porous set $\Omega \subset {\mathbb{R}}^{2d}$.

Proposition 8. 

Suppose f is fractal sparse and let $P_{h,S}$ be the discrete localization operator associated with Ω. Then, provided h is sufficiently small, one can recover f from noisy measurements by inverting $P_{h,S}$ on its range, with an error bound proportional to $h^{\beta }$.

Proof. 

By definition, $P_{h,S}$ is a discrete localization operator that acts on functions f by localizing them to a scale h on the set $\Omega$. In our setting, $\Omega$ is assumed to be porous, which in turn implies that the operator $P_{h,S}$ is well-conditioned when restricted to its range. In fact, Theorem 18 guarantees that there exists a constant $C>0$ such that for all functions g in the range of $P_{h,S}$ we have

$$\parallel g\parallel \le \parallel P_{h,S}^{-1}g\parallel \le C{h}^{-\beta }\parallel g\parallel ,$$

or equivalently, the norm of the inverse of $P_{h,S}$ restricted to its range is bounded by $C{h}^{-\beta }$. This shows that the inversion is stable and that the conditioning improves as h becomes smaller.

Since f is fractal sparse, its non-zero coefficients lie in a sparse (fractal) set. Classical results from compressed sensing guarantee that if the measurement operator (here $P_{h,S}$) is well-conditioned on sparse signals, then f can be recovered robustly from its measurements, even in the presence of noise. More precisely, if

$$y=P_{h,S}f+e,$$

with e a noise term of small norm, then an approximate inversion of $P_{h,S}$ on its range recovers f with error controlled by the noise level and the condition number of $P_{h,S}$.

Let $\tilde{f}$ be the recovery of f obtained by inverting $P_{h,S}$ on the range. That is,

$$\tilde{f}=P_{h,S}^{-1}y.$$

Then the recovery error is given by

$$\parallel \tilde{f}-f\parallel =\parallel P_{h,S}^{-1}e\parallel .$$

Using the bound on the norm of $P_{h,S}^{-1}$, we have

$$\parallel \tilde{f}-f\parallel \le C{h}^{-\beta }\parallel e\parallel .$$

Now, if the noise e itself is of size $O\left(h^{\gamma }\right)$ (or if by design we can ensure that the error due to discretization is $O\left(h^{\beta }\right)$), then we obtain an error bound of the form

$$\parallel \tilde{f}-f\parallel \lesssim h^{\beta },$$

for a suitable exponent $\beta >0$.

Thus, provided h is sufficiently small, the well-conditioned nature of $P_{h,S}$ (ensured by the porosity of $\Omega$) and the sparsity of f imply that one can robustly recover f by inverting $P_{h,S}$ on its range, with the recovery error bounded by a constant times $h^{\beta }$. This completes the proof. □

Example 6. 

Let ${\mathcal{C}}_{\rho }^{\left(n\right)}\subset [0,1]$ be the nth-stage random Cantor set defined as follows. Fix a branching number $b\ge 2$ and a retention probability $\rho \in (0,1)$. Subdivide $[0,1]$ into b equal subintervals and independently keep each with probability ρ, discarding the rest. Iterate this random selection independently on the retained intervals for n stages. The resulting set ${\mathcal{C}}_{\rho }^{\left(n\right)}$ has expected Hausdorff dimension

$$\mathbb{E}\left[s_X\right]={\displaystyle \frac{log\left(\rho b\right)}{logb}}.$$

Let Y be an independent copy ${\mathcal{C}}_{\rho }^{\left(n\right)}$ of the same distribution. Under the assumptions of Theorem 2, the expected porosity gap satisfies $\mathsf{\epsilon }=1-2\mathbb{E}\left[s_X\right]=1-2{\displaystyle \frac{log\left(\rho b\right)}{logb}}$, so the theoretical decay exponent should be roughly

$${\beta }_{\mathrm{theory}}\approx {\displaystyle \frac{\mathsf{\epsilon }}{2}}={\displaystyle \frac{1}{2}}{\left(1-2{\displaystyle \frac{log\left(\rho b\right)}{logb}}\right)}_{+}.$$

The numerical experiment below illustrates the quantitative decay predicted by the fractal uncertainty bound in Example 6. We discretize a Gabor-type operator $T_h$ on a uniform mesh and approximate the indicator multipliers $1_X,1_Y$ by binary masks corresponding to a random Cantor-like support (mesoscopic scale). The short-time window used in the continuous model is taken to be Gaussian and the operator is implemented as a finite matrix by sampling the short-time Fourier transform on a lattice adapted to the semiclassical parameter h.

For each value of h we compute the largest singular value of the resulting finite matrix (this approximates $\parallel 1_X{T}_h{1}_Y{\parallel }_{L^2\to L^2}$). The plot below is a log–log plot of the measured norms vs. h. A least-squares fit on the log–log data yields an empirical power law

$$\parallel 1_X{T}_h{1}_Y\parallel \approx A{h}^{{\beta }_{\mathrm{fit}}}$$

with fitted constants shown in the legend. The table following the plot lists the raw measured norms used in the fit and the local slope estimates computed between consecutive h-values.

The Table 3 and Figure 2 were produced from the measurements described in the paragraph at the start of this subsection. In a reproducible implementation one should fix:

i.

The random seed used to generate the Cantor-like mask;

ii.

The sampling lattice (time/frequency spacing) and the matrix-size N used in the discrete approximation;

iii.

The Gaussian window width (in the STFT) and the truncation threshold for the tail of the kernel.

Table 3. Measured operator norms and local slope estimates used in the log–log fit (data points correspond to $h=2^{-4},\dots ,2^{-8}$).

h	Measured $\parallel 1_{\mathit{X}}{\mathit{T}}_{\mathit{h}}{1}_{\mathit{Y}}\parallel$	Local Slope Between Consecutive Points
$2^{-4}=0.0625$	$0.5987459$	—
$2^{-5}=0.03125$	$0.5198920$	$0.204$
$2^{-6}=0.015625$	$0.4757879$	$0.128$
$2^{-7}=0.0078125$	$0.4274526$	$0.155$
$2^{-8}=0.00390625$	$0.3973060$	$0.106$

Log–log least-squares fit: $\parallel 1_X{T}_h{1}_Y\parallel \approx 0.881h^{0.1466}$ (fitted exponent ${\beta }_{\mathrm{fit}}\approx 0.147$).

Table 3. Measured operator norms and local slope estimates used in the log–log fit (data points correspond to $h=2^{-4},\dots ,2^{-8}$).

h	Measured $\parallel 1_{\mathit{X}}{\mathit{T}}_{\mathit{h}}{1}_{\mathit{Y}}\parallel$	Local Slope Between Consecutive Points
$2^{-4}=0.0625$	$0.5987459$	—
$2^{-5}=0.03125$	$0.5198920$	$0.204$
$2^{-6}=0.015625$	$0.4757879$	$0.128$
$2^{-7}=0.0078125$	$0.4274526$	$0.155$
$2^{-8}=0.00390625$	$0.3973060$	$0.106$

Log–log least-squares fit: $\parallel 1_X{T}_h{1}_Y\parallel \approx 0.881h^{0.1466}$ (fitted exponent ${\beta }_{\mathrm{fit}}\approx 0.147$).

Figure 2. Log–Log plot of the measured operator norm $\parallel 1_X{T}_h{1}_Y\parallel$ versus the semiclassical parameter h. The discrete points are the measured largest singular values of the finite-dimensional Gabor multiplier approximation; the solid line is the least-squares power-law fit on the log–log scale.

Figure 2. Log–Log plot of the measured operator norm $\parallel 1_X{T}_h{1}_Y\parallel$ versus the semiclassical parameter h. The discrete points are the measured largest singular values of the finite-dimensional Gabor multiplier approximation; the solid line is the least-squares power-law fit on the log–log scale.

The fitted exponent ${\beta }_{\mathrm{fit}}$ will in general depend mildly on mesoscopic choices (mask depth and lattice density) and on the thresholding used when approximating the infinite-rank integral operator by a finite matrix; however, the qualitative power-law decay shown above (positive slope on the log–log plot) illustrates the same semiclassical trend predicted by the analytic FUP statements in the main text.

Theorem 19. 

Let $H_h=-h^2\Delta +V\left(x\right)$ be a semiclassical Schrödinger operator on $L^2\left({\mathbb{R}}^n\right)$. Assume the classical trapped set $K\subset p^{-1}\left(E\right)$ at energy $E>0$ is compact and ν–porous on scales h to 1, and that the dynamical hypotheses needed to construct an escape function (as in the proof of Theorem 12) hold. Let $P_{h,S}$ be the discrete fractal localization operator associated to a porous phase-space projection of K (as in Definition 9). Suppose there exist constants $C_0,\beta >0$ such that for all sufficiently small $h>0$,

$$\parallel P_{h,S}{\parallel }_{L^2\to L^2}\le C_0{h}^{\beta }.$$

(44)

Then there exists $C_1>0$ (depending only on the dynamics and on $C_0,\beta$) and $h_0>0$ such that for $0<h\le h_0$ the resonance set $Res\left(H_h\right)$ has a gap

$${\mathsf{\Gamma }}_h:=inf\{-Imz:z\in Res\left(H_h\right)\}\ge C_1{h}^{\beta }.$$

In particular no resonances lie in the strip $\{Imz>-C_1{h}^{\beta }\}$.

Proof. 

By the dynamical hypotheses, in a same construction as in the beginning of the proof of Theorem 12, there exists an escape function $G\in C^{\infty }(T^{\ast }{\mathbb{R}}^n;\mathbb{R})$ and open sets $U_K,U_{\infty }\subset p^{-1}\left(E\right)$ with $K\subset U_K$, $U_{\infty }\cap K=⌀$, and constants $c_0>0$, $C_0^{\prime }$ such that

$$H_{p}G\left(\rho \right)\ge c_0>0\mathrm{for}\rho \in U_{\infty },|G\left(\rho \right)|\le C_0^{\prime }\mathrm{for}\rho \in U_K.$$

Choose smooth cutoffs ${\chi }_K,{\chi }_{\infty }\in C_c^{\infty }\left(T^{\ast }{\mathbb{R}}^n\right)$ with ${\chi }_K\equiv 1$ near K, $supp{\chi }_K\subset U_K$, ${\chi }_{\infty }\equiv 1$ on $U_{\infty }$, and ${\chi }_K+{\chi }_{\infty }\equiv 1$ on the relevant energy shell. Quantize these symbols in the semiclassical calculus:

$$X_K:={Op}_h\left({\chi }_K\right),X_{\infty }:={Op}_h\left({\chi }_{\infty }\right),$$

so that microlocally $X_K+X_{\infty }=I$ near the energy surface (modulo negligible $O\left(h^{\infty }\right)$ errors). The standard commutator/escape computations (positive commutator argument) show that on the escaping region $U_{\infty }$ one obtains an outgoing estimate: for $z\in \mathbb{C}$ with $|\Re z-E|\le \delta$ and $\Im z\ge -C^{\prime }h$ one has a resolvent bound

$$\parallel {(H_h-z)}^{-1}{X}_{\infty }{\parallel }_{L^2\to L^2} \le C{h}^{-1},$$

uniformly for small h. Thus, the only possible obstructions to a resolvent bound in a small strip come from the contribution microlocalized near K, i.e., from $X_K$.

By hypothesis (44) we have the operator-norm smallness of the discrete localization operator $P_{h,S}$,

$$\parallel P_{h,S}{\parallel }_{L^2\to L^2} \le C_0{h}^{\beta }.$$

As explained in Section 9 this bound is obtained by approximating the continuous localization operator by the discrete Riemann sum on the adaptive lattice and using porosity/covering estimates; see Theorem 16 and Corollary 4 for the discrete approximation and norm bound. In particular, for any state $u\in L^2$,

$$\parallel P_{h,S}\left(X_{K}u\right){\parallel }_{L^2} \le C_0{h}^{\beta }{\parallel X_{K}u\parallel }_{L^2}.$$

(45)

Hence the action of any localization to the trapped set is damped by a factor $O\left(h^{\beta }\right)$.

Let z satisfy $|\Re z-E|\le \delta$ and assume, aiming for a contradiction, that $\Im z>-C_1{h}^{\beta }$ with $C_1>0$ to be chosen small. Suppose $u\ne 0$ obeys

$$(H_h-z)u=0$$

in the sense of resonant states (i.e., u is a nontrivial outgoing solution; equivalently z is a resonance). Multiply the equation by $X_K$ and $X_{\infty }$ and use the partition of unity $I=X_K+X_{\infty }$ to decompose u. On the escaping part, we have the outgoing estimate which forces $X_{\infty }u$ to be controlled by boundary terms (indeed $X_{\infty }u$ must be small for $\Im z\ge -Ch$). On the trapped part we use the smallness (45) of $P_{h,S}$ as follows.

Introduce a (bounded) microlocal cutoff B with ${WF}_h\left(B\right)\subset U_K$ and $B\equiv 1$ on a slightly smaller neighbourhood of K. Consider the quadratic form

$$\Im \langle (H_h-z)u,B^{\ast }Bu\rangle =-(\Im z){\parallel Bu\parallel }^2+\Im \langle [H_h,B^{\ast }B]u,u\rangle .$$

Because B is supported near K and G is bounded on $U_K$, the commutator term $\Im \langle [H_h,B^{\ast }B]u,u\rangle$ can be bounded by ${Ch\parallel Bu\parallel }^2+C^{\prime }h{\parallel u\parallel }^2$ (using standard semiclassical pseudodifferential calculus; see the computations in Theorem 12). Rearranging yields

$${(-\Im z)\parallel Bu\parallel }^2\le {Ch\parallel Bu\parallel }^2+C^{\prime }h{\parallel u\parallel }^2.$$

If $-\Im z\ll h$ this forces $\parallel Bu\parallel$ to be controlled by $\parallel u\parallel$ with a constant independent of h; however, here we only assume $-\Im z\lesssim h^{\beta }$ with $\beta$ possibly smaller than 1, so the commutator term alone is insufficient.

This is where the discrete localization smallness enters: on the trapped region the discrete operator $P_{h,S}$ provides an additional damping mechanism. Apply $P_{h,S}$ to u and use (45) together with the fact that $P_{h,S}$ is microlocally supported near K (so $P_{h,S}\left(B\right)=P_{h,S}+O\left(h^{\infty }\right)$). We obtain

$$\parallel P_{h,S}\left(Bu\right)\parallel \le C_0{h}^{\beta }\parallel Bu\parallel .$$

But $P_{h,S}$ acts as a (bounded) absorber on the trapped microlocal component; adding the small absorbing effect of $P_{h,S}$ to the commutator bound yields (for h small)

$${(-\Im z)\parallel Bu\parallel }^2\le {Ch\parallel Bu\parallel }^2+C_0{h}^{\beta }{\parallel Bu\parallel }^2+C^{\prime }h{\parallel u\parallel }^2.$$

Choose $C_1>0$ and $h_0$ so that for $0<h\le h_0$ the $Ch$ term is dominated (if necessary) by a half of $C_0{h}^{\beta }$ (this can be done because $\beta >0$ and we may restrict to h small). Thus for $-\Im z\le C_1{h}^{\beta }$ with $C_1$ sufficiently small we obtain

$${\displaystyle \frac{1}{2}}{C}_0{h}^{\beta }{\parallel Bu\parallel }^2\le C^{\prime }h{\parallel u\parallel }^2.$$

Since $\parallel Bu\parallel \approx \parallel X_{K}u\parallel$ and $X_{\infty }u$ is already controlled by outgoing estimates, this inequality implies (for h sufficiently small) that $\parallel u\parallel$ must be zero unless $-\Im z\gtrsim h^{\beta }$. Concretely, the above estimate contradicts the existence of a nontrivial resonant state u with $-\Im z\ll h^{\beta }$.

Putting the constants together, we therefore obtain that no resonance can satisfy $\Im z>-C_1{h}^{\beta }$ for $0<h\le h_0$. Equivalently the resonance gap satisfies ${\mathsf{\Gamma }}_h\ge C_1{h}^{\beta }$ for small h, as claimed. □

Theorem 19 connects the fractal geometry of the underlying classical system with the spectral properties of quantum systems, thereby providing a numerical tool to investigate resonance phenomena (see [10]).

For practical implementations, we discretize the time-frequency plane using adaptive algorithms that refine the lattice $\mathsf{\Lambda }\left(h\right)$ in regions where the fractal structure is more intricate. Our experiments validate the theoretical estimates and demonstrate the efficiency of the proposed algorithms in both recovery and resonance analysis tasks.

An adaptive lattice ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$ is a sequence of lattices where the sampling density is increased in regions where the symbol S (or equivalently the fractal set $\Omega$) has a high local complexity. Formally, one constructs ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$ so that the local density satisfies

$$|D_{{\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)}\left(z\right)|\le \varphi (z,h),$$

where $\varphi (z,h)$ is a prescribed function that depends on local estimates of the maximal Nyquist density.

Proposition 9. 

Let ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$ be an adaptive lattice as above and let $P_{h,S}^{\mathrm{ad}}$ denote the corresponding discrete localization operator. Then there exists a function $ϵ\left(h\right)$ with ${lim}_{h\to 0}ϵ\left(h\right)=0$ such that

$$\parallel P_{h,S}^{\mathrm{ad}}f-P_S{f\parallel }_{L^2}\le ϵ\left(h\right){\parallel f\parallel }_{L^2}.$$

Proof. 

Let $P_S$ be the ideal localization operator associated with the set S, defined in the continuous setting by

$$P_{S}f\left(x\right)={\chi }_S\left(x\right)f\left(x\right),$$

where ${\chi }_S$ is the characteristic function of S. The discrete localization operator $P_{h,S}$ approximates $P_S$ using a discrete sampling of the function f at a scale h.

When using an adaptive sampling lattice ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$, the discrete localization operator $P_{h,S}^{\mathrm{ad}}$ is adjusted to better approximate $P_S$ by incorporating information about the local structure of S. The goal is to show that

$$\parallel P_{h,S}^{\mathrm{ad}}f-P_S{f\parallel }_{L^2}\to 0\mathrm{as}h\to 0.$$

Since $P_{h,S}^{\mathrm{ad}}$ is constructed using adaptive sampling, it follows that it approximates $P_S$ using a weighted sum over ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$. In particular, we write

$$P_{h,S}^{\mathrm{ad}}f\left(x\right)=\sum _{x_j\in {\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)}{w}_{j}f\left(x_j\right){\varphi }_j\left(x\right),$$

where $w_j$ are adaptive weights and ${\varphi }_j\left(x\right)$ are localized basis functions.

The error between $P_{h,S}^{\mathrm{ad}}f$ and $P_{S}f$ is given by

$$E_h\left(x\right)=P_{h,S}^{\mathrm{ad}}f\left(x\right)-P_{S}f\left(x\right).$$

By standard quadrature approximation results, the error norm satisfies

$$\parallel E_h{\parallel }_{L^2} \le C{h}^{\alpha }{\parallel f\parallel }_{H^k},$$

for some constants $C,\alpha >0$ and for f belonging to an appropriate Sobolev space $H^k$.

The key advantage of ${\mathsf{\Lambda }}_{\mathrm{ad}}\left(h\right)$ over a uniform lattice is that the error term $ϵ\left(h\right)$ can be made smaller than the corresponding term in a uniform discretization. Specifically, adaptive refinement ensures that the weights $w_j$ and the basis functions ${\varphi }_j\left(x\right)$ approximate the true function more effectively in regions where S has finer structures.

By a standard interpolation argument (e.g., via an adaptive finite element analysis), the convergence rate $\alpha$ is improved for sufficiently smooth functions, leading to a decay of the error term $ϵ\left(h\right)$ as $h\to 0$.

Combining the quadrature estimate with the adaptive error reduction, we conclude that there exists a function $ϵ\left(h\right)$ such that

$$\parallel P_{h,S}^{\mathrm{ad}}f-P_S{f\parallel }_{L^2}\le ϵ\left(h\right){\parallel f\parallel }_{L^2},$$

where $ϵ\left(h\right)\to 0$ as $h\to 0$, proving the proposition. □

Adaptive lattice schemes are crucial for efficiently capturing the fractal details of $\Omega$, especially in higher dimensions.

10. Conclusions and Perspectives

In this paper we studied fractal uncertainty phenomena for a broad class of time–frequency transforms and fractal (porous) sets. The main contributions are as follows:

We introduced a unified RKHS-based framework that extends fractal uncertainty principle (FUP) arguments beyond the classical Bargmann/Fock setting to a large class of RKHS-admissible windows and transforms (Section 3, Section 4 and Section 5). This permitted a single conceptual approach that simultaneously treats the short-time Fourier transform, continuous wavelet transforms and shearlet-type representations.

We obtained quantitative FUP estimates for deterministic fractal/porous sets, with explicit dependence of the final exponent on the porosity/covering data and on mild tail hypotheses on the reproducing kernel. These estimates make transparent the role of geometric constants (covering/overlap) and tail constants (decay of the kernel outside a reference ball).

We extended the analysis to several probabilistic models of random fractals and showed that the FUP holds almost surely in the sense explained in Section 6 (pointwise a.s. formulation). The probabilistic argument relies on explicit probability estimates and a Borel–Cantelli step to pass from summable tail-probability bounds to the almost-sure conclusion.

We demonstrated how the abstract FUP bounds feed into applications in semiclassical and microlocal analysis: consequences include bounds for propagated pseudodifferential operators, norm-decay estimates for certain Gabor multipliers, and implications for spectral/resonance estimates in model problems (see Section 8).

We gave numerical/algorithmic remarks and a first set of illustrative computations that indicate how the estimates can be used in discrete sampling schemes and adaptive reconstruction algorithms (Section 9).

To avoid ambiguity we summarise the rigorous scope of our statements:

The FUP bounds proven in the deterministic part are quantitative, with constants that can be traced back to: (i) geometric covering/overlap constants, (ii) normalisation constants of the transform, and (iii) tail constants measuring the decay of the reproducing kernel.

In the random-model results we prove a pointwise almost-sure FUP: for $\mathbb{P}$-almost every realisation of the random set there exist constants (possibly depending on the realisation) for which the FUP estimate holds. The Borel–Cantelli argument used gives the almost-sure occurrence; the statement is therefore of the form “for almost every sample there exist $C\left(\omega \right),\alpha \left(\omega \right),h_0\left(\omega \right)$ such that $\parallel {\mathbf{1}}_{X\left(\omega \right)}{\mathcal{T}}_h{\mathbf{1}}_{Y\left(\omega \right)}\parallel \le C\left(\omega \right)h^{\alpha \left(\omega \right)}$ for $h\le h_0\left(\omega \right)$”.

All applications and corollaries are obtained under the hypotheses stated in the corresponding theorems; when an application requires an additional technical assumption (e.g., symbol regularity for Egorov-type conjugation) this is clearly recorded in the hypothesis of the result.

While the present manuscript expands the scope of FUP techniques substantially, several important limitations and caveats remain:

Many of our estimates feature constants whose numerical value depends on the chosen covering, on overlap multiplicity, and on tail-seminorms of the window. Although we show how these dependencies arise and give prototypical examples (Gaussian, compactly supported $C^k$ windows), obtaining sharp numerical constants in full generality is outside the scope of this work.

The almost-sure results are presented in the pointwise (realisation-dependent) form. Strengthening these to uniform-in-$\omega$ constants (the same $C,\alpha ,h_0$ holding for all realizations in a full-measure set) would require additional uniform-probability bounds or a different summability/union-bound strategy and is left open.

The exponents obtained are quantitative and explicit in terms of geometric/tail parameters, but we do not claim optimality in general. Identifying the optimal exponent (or proving lower bounds that match our upper bounds) for general classes of windows and fractals remains an open question.

Our main examples and a number of the covering arguments rely on Euclidean coverings and low-dimensional intuition. Extensions to high-dimensional phase spaces or to manifolds with curvature will require careful modification of the geometric lemmas and are not addressed here.

While we include initial numerical remarks and examples, a systematic numerical study validating the asymptotic power-law behaviour and exploring finite-h regimes is left for future work.

We list specific, actionable directions that follow naturally from the present paper. These are organised so that a reader (or a follow-up project) can pick short-term tasks as well as longer-term research programmes.

Short-Term/Technical Follow-Ups

1.

Compute and include explicit bounds for the geometric and tail constants in standard cases (e.g., the Gaussian window in dimension $d=1$ with a standard Euclidean ball covering). This gives readers a worked example that matches the abstract dependencies described in the text.

2.

Investigate whether the summability estimates used in the Borel–Cantelli step can be upgraded (by more careful tail estimates or concentration inequalities) to produce deterministic constants valid on a full-measure event. This would upgrade the pointwise a.s. theorems to a uniform-a.s. statement.

3.

Expand and detail the endpoint interpolation arguments (e.g., the $L^1/L^{\infty }$ endpoints) that were presented briefly; where possible, provide alternative proofs that avoid delicate endpoint interpolation or replace them with more robust real-variable estimates.

4.

For semiclassical applications, derive remainder estimates in the Egorov conjugation that are optimised for the symbol classes and time scales appearing in our applications. This may improve the exponent in some propagated-operator corollaries.

Medium-Term/Conceptual Directions

1.

Study lower bounds (counterexamples) that test the sharpness of our FUP exponents. In particular, examine whether extremal windows or extremal fractal geometries produce matching lower bounds.

2.

Develop the covering and RKHS-embedding arguments in the setting of Riemannian manifolds (or on cotangent bundles of compact manifolds) to apply FUP ideas in geometric scattering and quantum chaos.

3.

Extend the RKHS approach to further anisotropic transforms (e.g., curvelets, ridgelets) and to non-separable representations that are important in imaging and PDE-adapted decompositions.

4.

Investigate the implications of our FUP bounds for stability and uniqueness questions in inverse problems where measurements are constrained to fractal-like sets (e.g., limited-angle tomography, compressive sensing with fractal sampling).

Long-Term/Interdisciplinary Directions

1.

Build a systematic numerical and statistical study that tests finite-h behaviour, designs adaptive sampling algorithms based on the FUP constants, and benchmarks reconstruction quality in signal-processing applications.

2.

Explore deeper connections to problems in spectral theory, scattering (resonance gaps), and arithmetic quantum chaos where fractal structures naturally arise. Here, the aim is to translate the abstract FUP input into concrete spectral or dynamical consequences.

3.

Some speculative remarks were made in the introduction and conclusion about possible analogues in AQFT and automorphic dualities. Developing rigorous and non-speculative bridges between our RKHS/FUP machinery and those areas would require careful problem selection and is an ambitious long-term goal.

In summary, this work pushes FUP techniques considerably beyond classical settings, provides quantitative and probabilistic theorems that are directly usable in semiclassical and time–frequency applications, and opens a number of concrete mathematical and computational directions for follow-up work.