Deformed Wavelet Transform and Related Uncertainty Principles

Abstract

The deformed wavelet transform is a new addition to the class of wavelet transforms that heavily rely on a pair of generalized translation and dilation operators governed by the well-known Dunkl transform. In this study, we adapt the symmetrical properties of the Dunkl Laplacian operator to prove a class of quantitative uncertainty principles associated with the deformed wavelet transform, including Heisenberg’s uncertainty principle, the Benedick–Amrein–Berthier uncertainty principle, and the logarithmic uncertainty inequalities. Moreover, using the symmetry between a square integrable function and its Dunkl transform, we establish certain local-type uncertainty principles involving the mean dispersion theorems for the deformed wavelet transform.

1. Introduction

A finite subset R of ${\mathbb{R}}^d\backslash \left\{0\right\}$ is called a root system if $R\cap \mathbb{R}\theta =\{\pm \theta \}$, and for all $\theta \in R$, we have ${\sigma }_{\theta }\left(R\right)=R$, where ${\sigma }_{\theta }$ is the reflection in the hyperplane $H_{\theta }\subset {\mathbb{R}}^d,$ which is orthogonal to $\theta$.

Let $T_j$, $j=1,2,\dots ,d$ be the Dunkl operators [1], related with a root system R and a multiplicity function k that is invariant under the action of the reflection group $W\subset O\left(d\right)$ associated to R.

The Dunkl transform ${\mathcal{F}}_D$ (see [2]) is a weighted integral transform with kernel $K(·,·)$, that is,

$${\displaystyle {\mathcal{F}}_D\left(f\right)\left(\xi \right)={\int }_{{\mathbb{R}}^d}f\left(t\right)K(-it,\xi )d{\gamma }_k\left(t\right),\xi \in {\mathbb{R}}^d,}$$

(1)

where $d{\gamma }_k$ is a weight measure, and for all $\lambda \in {\mathbb{C}}^d$, the Dunkl kernel $K(\lambda ,·)$ satisfies the following system

$$T_{j}g\left(x\right)={\lambda }_{j}g\left(x\right),g\left(0\right)=1,j\in \{1,2,\dots ,d\}.$$

This transformation extends the Fourier transform (if $k=0$), given by

$$\widehat{f}\left(\xi \right)={\left(2\pi \right)}^{-d/2}{\int }_{{\mathbb{R}}^d}f\left(t\right)e^{-i⟨t,\xi ⟩}dt,\xi \in {\mathbb{R}}^d,$$

(2)

and realizes an isometry from $L_k^2\left({\mathbb{R}}^d\right)=L^2({\mathbb{R}}^d,d{\gamma }_k)$ onto itself, that is, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\mathcal{F}}_D^{-1}\left(f\right)(·)={\mathcal{F}}_D\left(f\right)(-·)\mathrm{and}{\int }_{{\mathbb{R}}^d}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right)={\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left(f\right)\left(y\right)\right|}^{2}d{\gamma }_k\left(y\right).}$$

(3)

For more details about Dunkl’s theory, authors can refer to [1,2,3].

Thanks to the importance of time-frequency analysis, the theory of uncertainty principle has attracted a considerable attention and it has been extended to a large class of integral transformations (see, e.g., [4,5,6,7,8] and the references therein). The famous Heisenberg’s uncertainty principle asserts that a non-trivial function cannot be precisely concentrated at the same time in the time and frequency domains, and in this case the concentration (or localization) is measured by means of dispersions, that is,

$${\parallel \parallel x\parallel f\parallel }_2\parallel \parallel \xi \parallel \widehat{f}{\parallel }_2⩾d/2{\parallel f\parallel }_2^2,$$

(4)

where $\parallel ·\parallel ={⟨·⟩}^{1/2}$ is the Euclidean norm on ${\mathbb{R}}^d$, and its analog in the Dunkl setting (see [8]) is,

$${\parallel \parallel x\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\parallel \parallel \xi \parallel {\mathcal{F}}_D{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}⩾(\gamma +d/2){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

(5)

Several extensions of Heisenberg’s uncertainty principle appear in the literature, for example, the Beckner-type logarithmic uncertainty principles [9,10], Benedicks’ uncertainty principle [11], the entropy uncertainty principles, the local uncertainty inequalities, and many others (see, e.g., [12,13]). These inequalities are stronger than Heisenberg’s uncertainty principle, in the sense that from each of these extensions we can derive a Heisenberg-type uncertainty inequality.

In this paper, we are interested in finding some uncertainty principles for the new transformation called the deformed wavelet transform ${\mathcal{N}}_h^D$ defined by

$${\mathcal{N}}_h^D\left(f\right)(a,x)={⟨f,{\tau }_x{\mathrm{\Delta }}_{a}h⟩}_{L_k^2\left({\mathbb{R}}^d\right)},a,x\in {\mathbb{R}}^d,$$

(6)

where h is a deformed wavelet, ${\tau }_x$ is the Dunkl translation operator, and ${\mathrm{\Delta }}_a$ is a dilation operator (see Section 2.3 for details about the deformed wavelet transform). The first of our results is the Heisenberg-type inequality. We will prove this principle via different techniques (see Section 3 for details). In particular, we have the following inequality: For all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${∥\parallel x\parallel {\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}{∥\parallel a\parallel {\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}⩾C{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2,$$

(7)

where the constant $C>0$ does not depend on f and $L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)=L^2({\mathbb{R}}^{2d},d{\gamma }_{k}d{\gamma }_k)$. Then, we will prove a Beckner-type and logarithmic uncertainty inequalities. The proofs of these principles are based on the well-known Pitt’s and Sobolev’s inequalities (see Section 4 for details). Finally, we will show some uncertainty inequalities, in which the localization is measured by means of smallness of supports, such as Faris’s local uncertainty inequalities and the Benedicks’ uncertainty principle (see Section 5 for details).

This article is structured as follows: In Section 2, we recall some preliminary results about the Dunkl and deformed wavelet transforms. Section 3 is devoted to proving some new Heisenberg-type uncertainty inequalities for the deformed wavelet transform. In Section 4, we will show Pitt-type, Sobolev-type, and Beckner-type inequalities. Finally, in Section 5, we obtain some concentration uncertainty principles for sets of finite measures.

2. Preliminaries

In this section, we give some preliminary results about the Dunkl theory. The main references are [1,2,3,7,14,15,16].

2.1. The Dunkl Operators

Let $\{e_i,i=1,\dots ,d\}$ be the orthonormal basis of ${\mathbb{R}}^d$ and $\parallel t\parallel =\sqrt{⟨t,t⟩}$ be the Euclidean norm of ${\mathbb{R}}^d$, where $⟨,⟩$ is the usual scalar product. If $\theta \in {\mathbb{R}}^d\backslash \left\{0\right\}$, then we define ${\sigma }_{\theta }$ as the reflection in the hyperplane $H_{\theta }\subset {\mathbb{R}}^d$ that is orthogonal to $\theta$,

$${\sigma }_{\theta }\left(t\right):=t-2\frac{⟨\theta ,t⟩}{{\parallel \theta \parallel }^2}\theta .$$

(8)

Let R be a finite subset of ${\mathbb{R}}^d\backslash \left\{0\right\}$. Then, R is called a root system if $R\cap \mathbb{R}\theta =\{\pm \theta \}$ and for all $\theta \in R$, we have ${\sigma }_{\theta }\left(R\right)=R$.

The reflections ${\sigma }_{\theta }$, $\alpha \in R$ generate a finite group W of $O\left(d\right)$, which is called the reflection group associated with R.

Let ${\displaystyle \beta \in {\mathbb{R}}_{reg}^d:={\mathbb{R}}^d\backslash \bigcup _{\theta \in R}{H}_{\theta }}$. Then, we fix the positive root system $R_{+}=\left\{\theta \in R:⟨\theta ,\beta ⟩>0\right\}$, and we assume that for all $\theta \in R_{+}$: $\parallel \theta \parallel =\sqrt{2}$.

A multiplicity function $k:R\to [0,\infty )$ is a function that is invariant under the action of W. Then, we define the index $\gamma \left(k\right)$ by

$${\displaystyle \gamma \left(k\right):=\gamma =\sum _{\theta \in R_{+}}k\left(\theta \right).}$$

(9)

Moreover, we define the weight function ${\omega }_k$ by

$${\omega }_k\left(t\right)=\prod _{\theta \in R_{+}}{\left|⟨\theta ,t⟩\right|}^{2k\left(\theta \right)}.$$

(10)

In particular, ${\omega }_k$ is homogeneous to degree $2\gamma$ and W-invariant.

If $W={\mathbb{Z}}_2^d$, then the weight function ${\omega }_k$ is defined by

$${\omega }_k\left(t\right)=\prod _{i=1}^d{\left|t_i\right|}^{2{\theta }_i},{\theta }_j\ge 0,\forall 1\le i\le d.$$

It is well known that the Mehta-type constant is defined by

$${\displaystyle c_k={\int }_{{\mathbb{R}}^d}{e}^{-{\parallel t\parallel }^2/2}{\omega }_k\left(t\right)dt.}$$

(11)

Now, we shall fix some notation. Let

(1)

$B_d(0,r)=\{x\in {\mathbb{R}}^d:\parallel x\parallel \le r\}$ be the ball of ${\mathbb{R}}^d$,

(2)

$\mathcal{E}\left({\mathbb{R}}^d\right)=\{f:C^{\infty }\to {\mathbb{R}}^d$},

(3)

$D\left({\mathbb{R}}^d\right)$ be the subspace of $\mathcal{E}\left({\mathbb{R}}^d\right)$ with compact supports,

(4)

$\mathcal{S}\left({\mathbb{R}}^d\right)$ be the Schwartz space, and ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right)$ its topological dual.

The Dunkl operators on ${\mathbb{R}}^d$ associated with W and k are defined for a given $f\in C^1\left({\mathbb{R}}^d\right)$ by

$${\displaystyle T_{i}f\left(t\right):=\frac{\partial f}{\partial t_j}\left(t\right)+\sum _{\theta \in R_{+}}k\left(\theta \right)⟨\theta ,e_i⟩\frac{f\left(t\right)-f\left({\sigma }_{\theta }\left(t\right)\right)}{⟨\theta ,t⟩},t\in {\mathbb{R}}_{reg}^d,i=1,\dots ,d,}$$

(12)

where $C^p\left({\mathbb{R}}^d\right)=\{f:C^p\to {\mathbb{R}}^d\}$ is the space of $C^p$-functions on ${\mathbb{R}}^d$, $1\le p<\infty$.

The Dunkl–Laplacian operator ${▵}_k$ on ${\mathbb{R}}^d$ is defined for a given $C^2$-function f, by

$${\displaystyle {▵}_{k}f\left(t\right):=\sum _{i=1}^d{T}_i^{2}f\left(t\right)=▵f\left(t\right)+2\sum _{\theta \in R_{+}}k\left(\theta \right)\left(\frac{⟨\nabla f\left(t\right),\theta ⟩}{⟨\theta ,t⟩}-\frac{f\left(t\right)-f\left({\sigma }_{\theta }\left(t\right)\right)}{{⟨\theta ,t⟩}^2}\right),t\in {\mathbb{R}}_{reg}^d,}$$

where Δ and ∇ are, respectively, the usual Euclidean Laplacian and the gradient operators on ${\mathbb{R}}^d$.

Let $t,s\in {\mathbb{R}}^d$. Then the Dunkl kernel $K(t,s)$ is the unique analytic solution of the following system:

$$\left\{\begin{array}{ccc}{T}_{i}u(t,s)\hfill & =\hfill & s_{i}u(t,s),i=1,...,d,\hfill \\ u(0,s)\hfill & =\hfill & 1,\hfill \end{array}\right.$$

(13)

This kernel admits a unique holomorphic extension to ${\mathbb{C}}^d\times {\mathbb{C}}^d$ and satisfies the following well-known relations:

(1)

For all $s,t\in {\mathbb{C}}^d$ and all $\lambda \in \mathbb{C}$,

$$K(t,s)=K(s,t);K(t,0)=1;K(\lambda t,s)=K(t,\lambda s).$$

(2)

For all $n\in {\mathbb{N}}^d$, $t\in {\mathbb{R}}^d$ and $s\in {\mathbb{C}}^d$,

$$|D_s^n{K(t,s)|\le \parallel t\parallel }^{\left|n\right|}exp\left(\parallel t\parallel \parallel \mathrm{Re}\left(s\right)\parallel \right),\left|n\right|=n_1+\cdots +n_d.$$

(14)

where

$$D_s^n=\frac{{\partial }^{\left|n\right|}}{\partial s_1^{n_1}\cdots \partial s_d^{n_d}}.$$

In particular, for all $t,s\in {\mathbb{R}}^d$:

$$\left|K\right(it,s\left)\right|\le 1.$$

(3)

For all $t\in {\mathbb{R}}^d$ and $s\in {\mathbb{C}}^d$, the kernel K has the following Laplace-type integral representation:

$${\displaystyle K(t,s)={\int }_{{\mathbb{R}}^d}{e}^{⟨x,s⟩}d{\nu }_t\left(x\right),}$$

(15)

where ${\nu }_t$ is a positive probability measure on ${\mathbb{R}}^d$, with support in the ball $B_d(0,\parallel t\parallel )$ (see [17]).

2.2. The Dunkl Transform

Let $L_k^p\left({\mathbb{R}}^d\right)$ be the space of measurable functions on ${\mathbb{R}}^d$ such that

$$\begin{array}{ccc}{\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^d\right)}\hfill & :=\hfill & {\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left|f\left(t\right)\right|}^{p}d{\gamma }_k\left(t\right)}\right)}^{1/p}<\infty ,if1\le p<\infty ,\hfill \\ {\parallel f\parallel }_{L_k^{\infty }\left({\mathbb{R}}^d\right)}\hfill & :=\hfill & \mathrm{ess}\underset{t\in {\mathbb{R}}^d}{sup}\left|f\left(t\right)\right|<\infty ,\hfill \end{array}$$

where

$$d{\gamma }_k\left(t\right):=c_k^{-1}{\omega }_k\left(t\right)dt.$$

In particular, $L_k^2\left({\mathbb{R}}^d\right)$ is the Hilbert space equipped with the scalar product

$${\displaystyle {⟨f_1,f_2⟩}_{L_k^2\left({\mathbb{R}}^d\right)}:={\int }_{{\mathbb{R}}^d}{f}_1\left(t\right)\overline{f_2\left(t\right)}d{\gamma }_k\left(t\right).}$$

Let $\mathcal{F}=\{g:{\mathbb{R}}^d\to \mathbb{C}\}$. Then, we set

$${\mathcal{F}}_{rad}:=\left\{g\in \mathcal{F}:g\circ A=g,\forall A\in O(d,\mathbb{R})\right\}$$

the subspace of radial functions in $\mathcal{F}$. Notice that if $g\in {\mathcal{F}}_{rad}$, then there exists a unique complex function G defined on $[0,\infty )$ such that $g\left(t\right)=G(\parallel t\parallel )$, for all $t\in {\mathbb{R}}^d$.

Remark 1. 

Since ${\omega }_k$ is homogeneous, for any radial function g in $L_k^1\left({\mathbb{R}}^d\right)$, the function G defined on ${\mathbb{R}}_{+}$ by $g\left(t\right)=G(\parallel t\parallel )$ is integrable with respect to the measure ${\lambda }^{2\gamma +d-1}d\lambda$, and satisfies (see [14]),

$${\displaystyle {\int }_{{\mathbb{R}}^d}g\left(t\right)d{\gamma }_k\left(t\right)=d_k{\int }_0^{\infty }G\left(\lambda \right){\lambda }^{2\gamma +d-1}d\lambda ,}$$

(16)

where

$$d_k:=\frac{1}{2^{\gamma +\frac{d}{2}-1}\mathrm{\Gamma }(\gamma +\frac{d}{2})}.$$

(17)

The Dunkl transform of an integrable function $f\in L_k^1\left({\mathbb{R}}^d\right)$ is defined by

$${\displaystyle {\mathcal{F}}_D\left(f\right)\left(\xi \right)={\int }_{{\mathbb{R}}^d}f\left(t\right)K(-it,\xi )d{\gamma }_k\left(t\right),\xi \in {\mathbb{R}}^d,}$$

(18)

and it satisfies the following properties (see [2,7]):

(1)

For any $f\in L_k^1\left({\mathbb{R}}^d\right)$,

$$\parallel {\mathcal{F}}_D{\left(f\right)\parallel }_{L_k^{\infty }\left({\mathbb{R}}^d\right)}\le {\parallel f\parallel }_{L_k^1\left({\mathbb{R}}^d\right)}.$$

(19)

(2)

If $f,{\mathcal{F}}_D\left(f\right)\in L_k^1\left({\mathbb{R}}^d\right)$, then we have the following inversion formula:

$${\mathcal{F}}_D^{-1}\left(f\right)(·)={\mathcal{F}}_D\left(f\right)(-·),a.e.$$

(20)

(3)

The Dunkl transform ${\mathcal{F}}_D$ is an isomorphism from $\mathcal{S}\left({\mathbb{R}}^d\right)$ onto itself, such that

$$\forall \in f\mathcal{S}\left({\mathbb{R}}^d\right),\overline{{\mathcal{F}}_D}\left(f\right)(·)={\mathcal{F}}_D\left(f\right)(-·)\mathrm{and}{\mathcal{F}}_D\overline{{\mathcal{F}}_D}=\overline{{\mathcal{F}}_D}{\mathcal{F}}_D=\mathrm{Id}.$$

(21)

(4)

For any $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$, the following Plancherel’s formula holds:

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right)={\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left(f\right)\left(y\right)\right|}^{2}d{\gamma }_k\left(y\right).}$$

(22)

(5)

The Dunkl transform can be uniquely extended to an isometric isomorphism on $L_k^2\left({\mathbb{R}}^d\right)$.

(6)

For all $f_1,f_2\in \mathcal{S}\left({\mathbb{R}}^d\right)$, the following Parseval’s formula holds:

$${\displaystyle {\int }_{{\mathbb{R}}^d}{f}_1\left(t\right)\overline{f_2\left(t\right)}d{\gamma }_k\left(t\right)={\int }_{{\mathbb{R}}^d}{\mathcal{F}}_D\left(f_1\right)\left(y\right)\overline{{\mathcal{F}}_D\left(f_2\right)\left(y\right)}d{\gamma }_k\left(y\right).}$$

(23)

(7)

For all $f\in D\left({\mathbb{R}}^d\right)$ (resp. $f\in \mathcal{S}\left({\mathbb{R}}^d\right))$,

$${\mathcal{F}}_D\left(\overline{f}\right)\left(\xi \right)=\overline{{\mathcal{F}}_D\left(\breve{f}\right)\left(\xi \right)},$$

(24)

and

$${\mathcal{F}}_D\left(f\right)\left(\xi \right)={\mathcal{F}}_D\left(\breve{f}\right)(-\xi ),\xi \in {\mathbb{R}}^d,$$

(25)

where $\breve{f}(·)=f(-·)$.

Now, we will define The Dunkl translation operator (see [14]).

Definition 1. 

For $x\in {\mathbb{R}}^d$, the Dunkl translation operator ${\tau }_x$ is defined on $L_k^2\left({\mathbb{R}}^d\right)$ by

$${\mathcal{F}}_D\left({\tau }_{x}f\right)=K(ix,.){\mathcal{F}}_D\left(f\right).$$

(26)

The Dunkl translation operator satisfies the following properties (see [14,16]).

Proposition 1. 

Let $x\in {\mathbb{R}}^d$ and let ${\mathrm{W}}_k\left({\mathbb{R}}^d\right):=\left\{f\in L_k^1\left({\mathbb{R}}^d\right):{\mathcal{F}}_D\left(f\right)\in L_k^1\left({\mathbb{R}}^d\right)\right\}$, the generalized Wigner space (see [16]). Then,

(1) 

For any function $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$$\parallel {\tau }_x{f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\le {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.$$

(27)

(2) 

For any function $f\in {\mathrm{W}}_k\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\tau }_{x}f\left(t\right)={\int }_{{\mathbb{R}}^d}K(ix,y)K(it,y){\mathcal{F}}_D\left(f\right)\left(y\right)d{\gamma }_k\left(y\right),t\in {\mathbb{R}}^d.}$$

(3) 

For any function $f\in {\mathrm{W}}_k\left({\mathbb{R}}^d\right)$,

$${\tau }_{x}f\left(t\right)={\tau }_{t}f\left(x\right),t\in {\mathbb{R}}^d.$$

(28)

The explicit formula of the Dunkl translation operator is still an open question, and it is known only in some special cases. In particular, if $d=1$ and $W={\mathbb{Z}}_2$, then for all $x\in \mathbb{R}$ and $f\in C\left(\mathbb{R}\right)$,

$$\begin{array}{ccc}{\tau }_{y}f\left(x\right)& =& {\displaystyle \frac{1}{2}{\int }_{-1}^{1}f\left(\sqrt{x^2+y^2-2xyt}\right)(1+\frac{x-y}{\sqrt{x^2+y^2-2xyt}}){\mathrm{\Phi }}_k\left(t\right)d{\gamma }_k\left(t\right)}\\ & +& {\displaystyle \frac{1}{2}{\int }_{-1}^{1}f(-\sqrt{x^2+y^2-2xyt})(1-\frac{x-y}{\sqrt{x^2+y^2-2xyt}}){\mathrm{\Phi }}_k\left(t\right)d{\gamma }_k\left(t\right),}\end{array}$$

where

$${\mathrm{\Phi }}_k\left(t\right)=\frac{\mathrm{\Gamma }(k+\frac{1}{2})}{\sqrt{\pi }\mathrm{\Gamma }\left(k\right)}(1+t){(1-t^2)}^{k-1}.$$

From this, one can also give an explicit formula for the translation operator in the case of $W={\mathbb{Z}}_2^d$. Moreover we have the following boundedness result (see [15,16]).

Proposition 2. 

Let $W={\mathbb{Z}}_2^d$ and let $1\le p\le \infty$. Then, for any function $f\in L_k^p\left({\mathbb{R}}^d\right)$,

$$\parallel {\tau }_y{f\parallel }_{L_k^p\left({\mathbb{R}}^d\right)}\le 2^{d|\frac{1}{p}-\frac{1}{2}|}{\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^d\right)}.$$

(29)

On the other hand, if f is a radial function in ${\mathrm{W}}_k\left({\mathbb{R}}^d\right)$, then

$${\displaystyle {\tau }_{y}f\left(x\right)={\int }_{B_d(0,\parallel x\parallel )}F\left(\sqrt{{\parallel x\parallel }^2+{\parallel y\parallel }^2+2⟨x,t⟩}\right)d{\nu }_x\left(t\right),}$$

where F is the function defined by $f\left(z\right)=F(\parallel z\parallel )$.

Let $L_{k,rad}^p\left({\mathbb{R}}^d\right)$ be the subspace of radial functions in $L_k^p\left({\mathbb{R}}^d\right)$. Then, we have the following results (see [16]).

Proposition 3. 

Let $x\in {\mathbb{R}}^d$ and $1\le p\le \infty$.

(1) 

If $f\in L_{k,rad}^1\left({\mathbb{R}}^d\right)$ is nonnegative, then

$${\tau }_{x}f\in L_k^1\left({\mathbb{R}}^d\right),{\tau }_{x}f\ge 0$$

and

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\tau }_{x}f\left(t\right)d{\gamma }_k\left(t\right)={\int }_{{\mathbb{R}}^d}f\left(t\right)d{\gamma }_k\left(t\right).}$$

(30)

(2) 

If $f\in L_{k,rad}^p\left({\mathbb{R}}^d\right)$, then

$$\parallel {\tau }_x{f\parallel }_{L_k^p\left({\mathbb{R}}^d\right)}\le {\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^d\right)}.$$

(31)

The Dunkl translation operator is an essential tool to define the Dunkl convolution product (see [3,16]).

Definition 2. 

Let $f_1,f_2\in D\left({\mathbb{R}}^d\right)$. Then, the Dunkl convolution product is defined by:

$$f_1{\ast }_D{f}_2\left(y\right)={\int }_{{\mathbb{R}}^d}{\tau }_y{f}_1(-t)f_2\left(t\right)d{\gamma }_k\left(t\right),y\in {\mathbb{R}}^d.$$

(32)

The Dunkl convolution operator is associative and commutative and verifies the following well-known relations (see [3,16]).

Proposition 4. 

Let $1\le p,q,r\le \infty$ such that $\frac{1}{p}+\frac{1}{q}-\frac{1}{r}=1.$

(1) 

If $f_1\in L_{k,rad}^p\left({\mathbb{R}}^d\right)$ and $f_2\in L_k^q\left({\mathbb{R}}^d\right)$, then $f_1{\ast }_D{f}_2\in L_k^r\left({\mathbb{R}}^d\right)$ such that,

$${∥f_1{\ast }_D{f}_2∥}_{L_k^r\left({\mathbb{R}}^d\right)}\le {∥f_1∥}_{L_k^p\left({\mathbb{R}}^d\right)}{∥f_2∥}_{L_k^q\left({\mathbb{R}}^d\right)}.$$

(33)

(2) 

If $W={\mathbb{Z}}_2^d$, then for all $f_1\in L_k^p\left({\mathbb{R}}^d\right)$ and $f_2\in L_k^q\left({\mathbb{R}}^d\right)$, the function $f_1{\ast }_D{f}_2$ belongs to $L_k^r\left({\mathbb{R}}^d\right)$ such that

$${∥f_1{\ast }_D{f}_2∥}_{L_k^r\left({\mathbb{R}}^d\right)}\le 2^{d|\frac{1}{p}-\frac{1}{2}|}{∥f_1∥}_{L_k^p\left({\mathbb{R}}^d\right)}{∥f_2∥}_{L_k^q\left({\mathbb{R}}^d\right)}.$$

(34)

(3) 

If $f_1,f_2\in L_k^2\left({\mathbb{R}}^d\right)$, then $f_1{\ast }_D{f}_2\in L_k^2\left({\mathbb{R}}^d\right)$ if and only if ${\mathcal{F}}_D\left(f\right){\mathcal{F}}_D\left(g\right)\in L_k^2\left({\mathbb{R}}^d\right)$, and in this case,

$${\mathcal{F}}_D\left(f_1{\ast }_D{f}_2\right)={\mathcal{F}}_D\left(f_1\right){\mathcal{F}}_D\left(f_2\right).$$

(35)

(4) 

If $f_1,f_2\in L_k^2\left({\mathbb{R}}^d\right)$, then

$${\displaystyle {\int }_{{\mathbb{R}}^d}|f_1{\ast }_D{f}_2{\left(t\right)|}^{2}d{\gamma }_k\left(t\right)={\int }_{{\mathbb{R}}^d}|{\mathcal{F}}_D\left(f_1\right)\left(y\right){|}^2{\left|{\mathcal{F}}_D\left(f_2\right)\left(y\right)\right|}^{2}d{\gamma }_k\left(y\right).}$$

(36)

For the remainder of this article, we will assume that $W={\mathbb{Z}}_2^d$.

2.3. Deformed Wavelet Transform

From the reference [18], we will recall some preliminaries results about the deformed wavelet transform and some of its properties. First, we will fix some notation.

The dilation operator ${\mathrm{\Delta }}_a$, $a:=(a_1,\dots ,a_d)\in {\mathbb{R}}^d$, of a measurable function f is defined by

$${\mathrm{\Delta }}_{a}f\left(t\right):=|a_1{|}^{\frac{2{\alpha }_1+1}{2}}\dots {\left|a_d\right|}^{\frac{2{\alpha }_d+1}{2}}f(a_1{t}_1,\dots ,a_d{t}_d),t\in {\mathbb{R}}^d.$$

(37)

This operator satisfies the following relations.

Proposition 5. 

Let $a,b\in {\mathbb{R}}^d\backslash \left\{0\right\}$.

(1) 

We have

$${\mathrm{\Delta }}_a{\mathrm{\Delta }}_b={\mathrm{\Delta }}_{(a_1{b}_1,\dots ,a_d{b}_d)}.$$

(38)

(2) 

If $f\in L_k^2\left({\mathbb{R}}^d\right)$, then ${\mathrm{\Delta }}_{a}f\in L_k^2\left({\mathbb{R}}^d\right)$ such that

$$\parallel {\mathrm{\Delta }}_a{f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}={\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)},$$

(39)

and

$${\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}f\right)\left(y\right)=|a_1{|}^{-\frac{2{\alpha }_1+1}{2}}\dots {\left|a_d\right|}^{-\frac{2{\alpha }_d+1}{2}}{\mathcal{F}}_D\left(f\right)\left(\frac{y_1}{a_1},\dots ,\frac{y_d}{a_d}\right),y\in {\mathbb{R}}^d.$$

(40)

(3) 

If $f\in L_k^p\left({\mathbb{R}}^d\right)$, $p\in [1,\infty ]$, then ${\mathrm{\Delta }}_{a}f\in L_k^p\left({\mathbb{R}}^d\right)$ such that

$$\parallel {\mathrm{\Delta }}_a{f\parallel }_{L_k^p\left({\mathbb{R}}^d\right)}=\prod _{j=1}^d|a_j{|}^{(2{\alpha }_j+1)(\frac{1}{p}-\frac{1}{2})}{\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^d\right)}.$$

(41)

(4) 

If $f,g\in L_k^2\left({\mathbb{R}}^d\right)$, then

$${⟨{\mathrm{\Delta }}_{a}f,g⟩}_{L_k^2\left({\mathbb{R}}^d\right)}={⟨f,{\mathrm{\Delta }}_{\left(1/a_1,\dots ,1/a_d\right)}g⟩}_{L_k^2\left({\mathbb{R}}^d\right)}.$$

(42)

(5) 

For all $x\in {\mathbb{R}}^d$,

$${\mathrm{\Delta }}_a{\tau }_x={\tau }_{\left(x_1/a_1,\dots ,x_d/a_d\right)}{\mathrm{\Delta }}_a.$$

(43)

Definition 3. 

A deformed wavelet on ${\mathbb{R}}^d$ is a measurable function h on ${\mathbb{R}}^d$ satisfying the following condition: For almost all $\xi \in {\mathbb{R}}^d\backslash \left\{0\right\}$,

$${\displaystyle 0<C_h:={\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}h\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(a\right)<\infty .}$$

(44)

Proposition 6. 

If h is a deformed wavelet on ${\mathbb{R}}^d$, then

$${\displaystyle C_h=c_k^{-1}{\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left(h\right)\left({\xi }_1/a_1,\dots ,{\xi }_d/a_d\right)\right|}^2\frac{d{a}_1}{|a_1|}\dots \frac{d{a}_d}{|a_d|}.}$$

(45)

Example 1. 

Let $s>0$. Then, the function ${\alpha }_s$ defined by

$${\alpha }_s\left(t\right)=\frac{1}{{\left(2s\right)}^{\gamma +\frac{d}{2}}}{e}^{-\frac{{\parallel t\parallel }^2}{4s}},t\in {\mathbb{R}}^d,$$

(46)

satisfies

$$\forall \xi \in {\mathbb{R}}^d,{\mathcal{F}}_D\left({\alpha }_s\right)\left(\xi \right)=e^{-{s\parallel \xi \parallel }^2}.$$

(47)

Thus, the function ${\displaystyle h=-\frac{d}{ds}{\alpha }_s}$ is a deformed wavelet on ${\mathbb{R}}^d$ that belongs to $\mathcal{S}\left({\mathbb{R}}^d\right)$.

For $a\in {\mathbb{R}}^d\backslash \left\{0\right\}$ and h, a deformed wavelet in $L_k^2\left({\mathbb{R}}^d\right)$, we define the family $h_{a,x}$, $x\in {\mathbb{R}}^d$, of functions in $L_k^2\left({\mathbb{R}}^d\right)$ by

$$h_{a,x}:={\tau }_x{\mathrm{\Delta }}_{a}h.$$

(48)

Then, for all $a,x\in {\mathbb{R}}^d$ and $h\in L_k^2\left({\mathbb{R}}^d\right)$,

$$\parallel h_{a,x}{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\le {\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.$$

(49)

For $1⩽p⩽\infty$, we denote by $L_{{\mu }_k}^p\left({\mathbb{R}}^{2d}\right)$ the space of measurable functions f on ${\mathbb{R}}^{2d}$ such that

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{L_{{\mu }_k}^p\left({\mathbb{R}}^{2d}\right)}& :=& {\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|f(a,x)\right|}^{p}d{\mu }_k(a,x)}\right)}^{\frac{1}{p}}<\infty ,1\le p<\infty ,\hfill \\ \hfill {\parallel f\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2d}\right)}& :=& ess\underset{(a,x)\in {\mathbb{R}}^{2d}}{sup}\left|f(a,x)\right|<\infty ,\hfill \end{array}$$

where ${\mu }_k$ is the weight measure defined by

$$d{\mu }_k(a,x)=d{\gamma }_k\left(a\right)d{\gamma }_k\left(x\right).$$

Definition 4. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet. Then, the deformed continuous wavelet transform ${\mathcal{N}}_h^D$ is defined for a regular function f by

$${\mathcal{N}}_h^D\left(f\right)(a,x)={⟨f,h_{a,x}⟩}_{L_k^2\left({\mathbb{R}}^d\right)}={⟨f,{\tau }_x{\mathrm{\Delta }}_{a}h⟩}_{L_k^2\left({\mathbb{R}}^d\right)},a,x\in {\mathbb{R}}^d,$$

(50)

or equivalently

$${\mathcal{N}}_h^D\left(f\right)(a,x)=\breve{f}{\ast }_D\overline{{\mathrm{\Delta }}_{a}h}\left(x\right).$$

(51)

Remark 2. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet. Then,

(1) 

For all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$$\parallel {\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2d}\right)}\le {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.$$

(52)

(2) 

For all $f\in L_k^2\left({\mathbb{R}}^d\right)$ and all $\lambda >0$,

$${\mathcal{N}}_h^D\left(f_{\lambda }\right)(a,x)={\mathcal{N}}_h^D\left(f\right)\left(\lambda a,\frac{x}{\lambda }\right),(a,x)\in {\mathbb{R}}^{2d}$$

(53)

where

$$g_t\left(x\right):=t^{-\frac{2\gamma +d}{2}}g(x/t),t>0,x\in {\mathbb{R}}^d.$$

Lemma 1. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\mathcal{F}}_D\left({\mathcal{N}}_h^D\left(f\right)(a,.)\right)\left(\xi \right)={\mathcal{F}}_D\left(\overline{{\mathrm{\Delta }}_{a}h}\right)\left(\xi \right){\mathcal{F}}_D\left(f\right)(-\xi ),\xi \in {\mathbb{R}}^d.$$

(54)

Theorem 1 (Parseval-type formula for ${\mathcal{N}}_h^D$).

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet. Then, for all $f_1,f_2\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{f}_1\left(x\right)\overline{f_2\left(x\right)}d{\gamma }_k\left(x\right)=C_h^{-1}{\int }_{{\mathbb{R}}^{2d}}{\mathcal{N}}_h^D\left(f_1\right)(a,x)\overline{{\mathcal{N}}_h^D\left(f_2\right)(a,x)}d{\mu }_k(a,x).}$$

(55)

Corollary 1 (Plancherel-type formula for ${\mathcal{N}}_h^D$).

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right)=C_h^{-1}{\int }_{{\mathbb{R}}^{2d}}{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^{2}d{\mu }_k(a,x).}$$

(56)

Then, the Riesz–Thorin interpolation theorem implies this result.

Proposition 7. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet and $2\le p\le \infty$. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$$\parallel {\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^p\left({\mathbb{R}}^{2d}\right)}\le C_h^{p^{-1}}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{p^{-1}(p-2)}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.$$

(57)

Theorem 2 (An inversion formula for ${\mathcal{N}}_h^D$).

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet. Then for all f in $L_k^1\left({\mathbb{R}}^d\right)$ (resp. $L_k^2\left({\mathbb{R}}^d\right)$) such that ${\mathcal{F}}_D\left(f\right)\in L_k^1\left({\mathbb{R}}^d\right)$ (resp. ${\mathcal{F}}_D\left(f\right)\in L_k^1\left({\mathbb{R}}^d\right)\cap L_k^{\infty }\left({\mathbb{R}}^d\right)$),

$${\displaystyle f\left(t\right)=C_h^{-1}{\int }_{{\mathbb{R}}^{2d}}{\mathcal{N}}_h^D\left(f\right)(a,x)h_{a,x}\left(t\right)d{\mu }_k(a,x),a.e.t\in {\mathbb{R}}^d.}$$

(58)

Proof. 

Using similar ideas as in the proof for Theorem 6.III.3 of [19] p. 99, we obtain the relation (58). □

3. Heisenberg-Type Uncertainty Principles for the Deformed Wavelet Transform

In this section, we prove several versions of the Heisenberg-type uncertainty inequalities for the deformed wavelet transform via different techniques, including the generalized entropy, the contraction semigroup method of the homogeneous integral transform, and others.

3.1. Heisenberg-Type Uncertainty Inequalities for Functions in $L_k^2\left({\mathbb{R}}^d\right)$

First, we recall the following Heisenberg-type uncertainty inequality for the Dunkl transform, first proved by Rösler [8] and generalized in [20].

Proposition 8. 

For all $s,t>0$, there exists a constant ${\mathcal{C}}_k(s,t)>$, such that for all $f\in L_k^2\left({\mathbb{R}}^d\right)$, the following holds:

$$\begin{array}{c}\hfill {\parallel \parallel \xi \parallel }^t{\mathcal{F}}_D\left(f\right){\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{s}{s+t}}{\parallel \parallel x\parallel }^{s}f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{t}{s+t}}\ge {\mathcal{C}}_k(s,t){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.\end{array}$$

(59)

For $s,t\ge 1$, ${\mathcal{C}}_k(s,t)={\left(\frac{2\gamma +d}{2}\right)}^{\frac{st}{s+t}}$.

Our first result is the following Heisenberg-type uncertainty inequality for ${\mathcal{N}}_h^D(a,.)$.

Theorem 3. 

Let $s,t>0$. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$ we have

$$\begin{array}{c}\hfill {{\parallel \parallel x\parallel }^s{\mathcal{N}}_h^D\left(f\right)(a,x)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{t}{t+s}}{\parallel \parallel \xi \parallel }^t{\mathcal{F}}_k\left(f\right)\left(\xi \right){\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{s}{s+t}}\ge {\mathcal{C}}_k(s,t){\left(C_h\right)}^{\frac{t}{2(s+t)}}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)},\end{array}$$

(60)

where ${\mathcal{C}}_k(s,t)$ is the same constant given in Proposition 8.

Proof. 

By (44), we have

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}|{\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}u\right)\left(\xi \right){|}^2|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)d{\gamma }_k\left(a\right)=C_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

Involving (54), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]{\left(\xi \right)|}^{2}d{\mu }_k(a,\xi )=C_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

(61)

On the other hand, Inequality (59) implies that for all $a\in {\mathbb{R}}^d$,

$${\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)}\right)}^{\frac{s}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|x\left|\right|}^{2s}{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^{2}d{\gamma }_k\left(x\right)}\right)}^{\frac{t}{s+t}}$$

$${\displaystyle \ge {\left({\mathcal{C}}_k(s,t)\right)}^2{\int }_{{\mathbb{R}}^d}{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\gamma }_k\left(x\right).}$$

Integrating both sides with respect to the measure $d{\gamma }_k\left(a\right)$, we obtain, by Hölder’s inequality and Plancherel’s formula,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)}\right)}^{\frac{s}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|x\left|\right|}^{2s}{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\gamma }_k\left(x\right)}\right)}^{\frac{t}{s+t}}d{\gamma }_k\left(a\right)}$$

$${\displaystyle \ge {\left({\mathcal{C}}_k(s,t)\right)}^2{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\gamma }_k\left(x\right)d{\gamma }_k\left(a\right).}$$

Therefore, from (61), we obtain,

$${\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{|{\mathcal{F}}_k[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right)|}^{2}d{\mu }_k(a,\xi )}\right)}^{\frac{s}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|\right|x\left|\right|}^{2s}{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}\right)}^{\frac{t}{s+t}}$$

$$=C_h^{\frac{s}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_k\left(f\left(\xi \right)\right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\right)}^{\frac{s}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|\right|x\left|\right|}^{2s}{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}\right)}^{\frac{t}{s+t}}$$

$$\ge {\left({\mathcal{C}}_k(s,t)\right)}^2{C}_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

This proves the result. □

Now, we will prove another version of Heisenberg’s uncertainty principle for the deformed wavelet transform. To do so, we have first to prove prove an uncertainty inequality involving entropy. Let $\rho$ be a probability density function on ${\mathbb{R}}^{2d},$ i.e.,

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}\rho (a,x)d{\mu }_k(a,x)=1.}$$

Following [21], the k-entropy of $\rho$ is defined by

$${\displaystyle E_k\left(\rho \right):=-{\int }_{{\mathbb{R}}^{2d}}ln\left(\rho (a,x)\right)\rho (a,x)d{\mu }_k(a,x).}$$

Then, we have the following entropy uncertainty inequality for the deformed wavelet transform.

Proposition 9. 

For all $f\in L_k^2\left({\mathbb{R}}^d\right),$ we have

$$E_k(|{\mathcal{N}}_h^D{\left(f\right)|}^2)⩾-2C_h{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{2}ln\left({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\right).$$

(62)

Proof. 

Assume that ${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}=1.$ By (52),

$$|{\mathcal{N}}_h^D{\left(f\right)(a,x)|⩽\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}=1.$$

(63)

Particularly $E_k(|{\mathcal{N}}_h^D\left(f\right){|}^2)⩾0.$ Next, let

$$\phi :=\frac{f}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}\mathrm{and}\psi :=\frac{h}{{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}.$$

Then, $\phi ,\psi \in L_k^2\left({\mathbb{R}}^d\right)$ and ${\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel \psi \parallel }_{L_k^2\left({\mathbb{R}}^d\right)}=1.$

Therefore, $E_k(|{\mathcal{N}}_{\psi }^D\left(\phi \right){|}^2)⩾0.$ Moreover,

$${\mathcal{N}}_{\psi }^D\left(\phi \right)=\frac{1}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}{\mathcal{N}}_h^D\left(f\right),$$

which implies

$$E_k(|{\mathcal{N}}_{\psi }^D{\left(\phi \right)|}^2)=\frac{1}{{\parallel f\parallel }_{L_k^2({\mathbb{R}}^d)}^2{\parallel h\parallel }_{L_k^2({\mathbb{R}}^d)}^2}{E}_k\left(\right|{\mathcal{N}}_h^D\left(f\right){|}^2)+ln({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)})\frac{2C_h}{{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2}.$$

Using the fact that $E_k(|{\mathcal{N}}_{\psi }^D\left(\phi \right){|}^2)⩾0,$ we deduce that

$$E_k(|{\mathcal{N}}_h^D{\left(f\right)|}^2)⩾-2C_h{\left|\right|f\left|\right|}_{L_k^2({\mathbb{R}}^d)}^{2}ln({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}).$$

This completes the proof. □

From the last result, we can derive a new version of Heisenberg’s uncertainty inequality for ${\mathcal{N}}_h^D$.

Theorem 4. 

For all $p,q>0$, there exists a constant $M_{p,q}\left(k\right)$ such that for all $f\in L_k^2\left({\mathbb{R}}^d\right)$, we have

$$\begin{array}{c}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\parallel x\parallel }^p{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}\right)}^{\frac{q}{p+q}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\parallel a\parallel }^q{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}\right)}^{\frac{p}{p+q}}\\ ⩾M_{p,q}\left(k\right)C_h{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2,\end{array}$$

where

$${\displaystyle M_{p,q}\left(k\right)=\frac{2\gamma +d}{p^{\frac{q}{p+q}}{q}^{\frac{p}{p+q}}}exp\left(\frac{pq}{(p+q)(2\gamma +d)}ln\left(\frac{pq{\left(d_k\right)}^2}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{p}\right)\mathrm{\Gamma }\left(\frac{2\gamma +d}{q}\right)}\right)-1\right).}$$

Proof. 

Let ${\eta }_{t,p,q}$, $t,p,q>0$ be the function defined on ${\mathbb{R}}^{2d}$ by

$${\displaystyle {\eta }_{t,p,q}(a,x):=\frac{pq}{{\left(d_k\right)}^2\mathrm{\Gamma }\left(\frac{2\gamma +d}{p}\right)\mathrm{\Gamma }\left(\frac{2\gamma +d}{q}\right)}{t}^{-\frac{(p+q)(2\gamma +d)}{pq}}exp\left(-\frac{{\parallel a\parallel }^p+{\parallel x\parallel }^q}{t}\right).}$$

By simple computation, we see that

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\eta }_{t,p,q}(a,x)d{\mu }_k(a,x)=1.}$$

It is clear that $d{\sigma }_{t,p,q}^k(a,x):={\eta }_{t,p,q}(a,x)d{\mu }_k(a,x)$ is a probability measure on ${\mathbb{R}}^{2d}$. Then, by Jensen’s inequality we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}\frac{|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^2}{{\eta }_{t,p,q}(a,x)}ln\left(\frac{|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^2}{{\eta }_{t,p,q}(a,x)}\right)d{\sigma }_{t,p,q}^k(a,x)⩾0.}$$

Therefore,

$$\begin{array}{c}{E}_k(|{\mathcal{N}}_h^D{\left(f\right)|}^2)+C_{h}ln\left(\frac{pq}{{\left(d_k\right)}^2\mathrm{\Gamma }\left(\frac{2\gamma +d}{p}\right)\mathrm{\Gamma }\left(\frac{2\gamma +d}{q}\right)}\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2\hfill \\ \hfill {\displaystyle ⩽C_{h}ln\left(t^{\frac{(p+q)(2\gamma +d)}{pq}}\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2+t^{-1}{\int }_{{\mathbb{R}}^{2d}}{(\parallel a\parallel }^p+{\parallel x\parallel }^q)|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x).}\end{array}$$

If ${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}=1$, then, by Proposition 9, we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{(\parallel a\parallel }^p+{\parallel x\parallel }^q)|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)⩾t\left(ln\left(\frac{pq}{{\left(d_k\right)}^2\mathrm{\Gamma }\left(\frac{2\gamma +d}{p}\right)\mathrm{\Gamma }\left(\frac{2\gamma +d}{q}\right)}\right)-ln\left(t^{\frac{(2\gamma +d)(p+q)}{pq}}\right)\right)C_h{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.}$$

However, the expression

$$t\left(ln\left(\frac{pq}{{\left(d_k\right)}^2\mathrm{\Gamma }\left(\frac{2\gamma +d}{p}\right)\mathrm{\Gamma }\left(\frac{2\gamma +d}{q}\right)}\right)-ln\left(t^{\frac{(2\gamma +d)(p+q)}{pq}}\right)\right)$$

attains its upper bound at

$${\displaystyle t_0=exp\left(\frac{pq}{(2\gamma +d)(p+q)}ln\left(\frac{pq}{{\left(d_k\right)}^2\mathrm{\Gamma }\left(\frac{2\gamma +d}{p}\right)\mathrm{\Gamma }\left(\frac{2\gamma +d}{q}\right)}\right)-1\right),}$$

, and consequently,

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{(\parallel a\parallel }^p+{\parallel x\parallel }^q)|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)⩾C_h{C}_{p,q}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2,}$$

where

$$C_{p,q}=\frac{(p+q)(2\gamma +d)}{pq}exp\left(\frac{pq}{(p+q)(2\gamma +d)}ln\left(\frac{pq}{{\left(d_k\right)}^2\mathrm{\Gamma }\left(\frac{2\gamma +d}{p}\right)\mathrm{\Gamma }\left(\frac{2\gamma +d}{q}\right)}\right)-1\right).$$

Therefore, for every $f\in L_k^2\left({\mathbb{R}}^d\right)$ and $h\in L_k^2\left({\mathbb{R}}^d\right)$ such that ${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}=1,$ we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\parallel x\parallel }^p|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^{2d}}{\parallel a\parallel }^q|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^d{\mu }_k(a,x)⩾C_h{C}_{p,q}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.}$$

Now, replacing f by $f_{\lambda }$ in the last inequality, we obtain,

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^p|{\mathcal{N}}_h^D\left(f_{\lambda }\right)(a,x){|}^{2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^{2d}}{\left|\right|a\left|\right|}^q|{\mathcal{N}}_h^D\left(f_{\lambda }\right)(a,x){|}^{2}d{\mu }_k(a,x)⩾C_h{C}_{p,q}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.}$$

Using (53), we deduce that

$${\displaystyle {\lambda }^p{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^p|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)+\frac{1}{{\lambda }^q}{\int }_{{\mathbb{R}}^{2d}}{\left|\right|a\left|\right|}^q|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)⩾C_h{C}_{p,q}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.}$$

In particular, the inequality holds at the point

$${\displaystyle \lambda ={\left(\frac{{\displaystyle q{\int }_{{\mathbb{R}}^{2d}}{\left|\right|a\left|\right|}^q{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}}{{\displaystyle p{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^p{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}}\right)}^{\frac{1}{p+q}},}$$

which implies that

$${\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^p{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}\right)}^{\frac{q}{p+q}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|a\left|\right|}^q{|{\mathcal{N}}_h^D\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}\right)}^{\frac{p}{p+q}}⩾C_h{M}_{p,q}\left(k\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2,$$

where

$$M_{p,q}\left(k\right)=C_{p,q}\frac{q^{\frac{q}{p+q}}{p}^{\frac{p}{p+q}}}{q+p}.$$

Now, the desired result follows from above, by substituting f by $f/{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}$ and h by $h/{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.$ □

Remark 3. 

If $p=q=2$, then

$$\begin{array}{cc}\hfill {∥\parallel x\parallel {\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}{∥\parallel a\parallel {\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}& ⩾\frac{2\gamma +d}{2e}{\left(\frac{2}{d_k\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}\right)}^{\frac{2}{2\gamma +d}}{C}_h{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.\hfill \end{array}$$

3.2. $L^p$–Heisenberg’s Uncertainty Principles for the Deformed Wavelet Transform

Let $\lambda >0$. We put

$$G_{\lambda }(b,y):=e^{-{\lambda \left|\right|(b,y)\left|\right|}^2},\mathrm{for}\mathrm{all}(b,y)\in {\mathbb{R}}^{2d}.$$

By simple calculations, it is easy to prove the following.

Lemma 2. 

Let $1\le q<\infty$ and $\lambda >0$. There exists a positive constant C such that we have

$$\left|\right|G_{\lambda }{\left|\right|}_{L_{{\mu }_k}^q\left({\mathbb{R}}^{2d}\right)}=C{\lambda }^{-\frac{2\gamma +d}{q}}.$$

Lemma 3. 

We assume that $h\in L_k^2\left({\mathbb{R}}^d\right)$. Let $1<p\le 2$ and $0<s<\frac{2\gamma +d}{2p^{\prime }}$. Then, there exists a positive constant C such that, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$ and $\lambda >0$,

$$\left|\right|G_{\lambda }(b,y){\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\le C{\left(C_h\right)}^{\frac{1}{p^{\prime }}}{({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)})}^{\frac{p^{\prime }-2}{p^{\prime }}}{\lambda }^{-2s}\left[{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right].$$

(64)

Proof. 

Inequality (64) holds if ${\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}=\infty .$ Assume that

$${\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}<\infty .$$

For $r>0$ let $f_r=f{\mathbf{1}}_{B_d(0,r)}$ and $f^r=f-f_r$. Then, since

$$|f^r\left(y\right)|\le r^{-s}{\left|\right|\left|y\right||}^{s}f\left(y\right)|$$

the relation (57) implies

$$\begin{array}{ccc}\left|\right|G_{\lambda }(b,y){\mathcal{N}}_h^D(f{\mathbf{1}}_{{}_{B_d^c(0,r)}}){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill & \le \hfill & \left|\right|G_{\lambda }{\left|\right|}_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2d}\right)}\left|\right|{\mathcal{N}}_h^D(f{\mathbf{1}}_{{}_{B_d^c(0,r)}}){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill \\ & \le \hfill & {\left(C_h\right)}^{\frac{1}{p^{\prime }}}{({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)})}^{\frac{p^{\prime }-2}{p^{\prime }}}\left|\right|f{\mathbf{1}}_{{}_{B_d^c(0,r)}}{\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}\hfill \\ & \le \hfill & {\left(C_h\right)}^{\frac{1}{p^{\prime }}}{({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)})}^{\frac{p^{\prime }-2}{p^{\prime }}}{r}^{-s}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}.\hfill \end{array}$$

On the other hand, by (52) and Hölder’s inequality

$$\begin{array}{ccc}\left|\right|G_{\lambda }(b,y){\mathcal{N}}_h^D\left(f{\mathbf{1}}_{B_d(0,r)}\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill & \le \hfill & \left|\right|G_{\lambda }{\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\left|\right|{\mathcal{N}}_h^D\left(f{\mathbf{1}}_{B_d(0,r)}\right){\left|\right|}_{L_k^{\infty }\left({\mathbb{R}}^d\right)}\hfill \\ & \le \hfill & {\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\left|\right|G_{\lambda }{\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\left|\right|f{\mathbf{1}}_{B_d(0,r)}{\left|\right|}_{L^2\left({\mathbb{R}}^d\right)}\hfill \\ & \le \hfill & {\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\left|\right|G_{\lambda }{\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}{\left|\right|\left|\right|y\left|\right|}^{-s}{\mathbf{1}}_{B_d(0,r)}{\left|\right|}_{L_k^{2p^{\prime }}\left({\mathbb{R}}^d\right)}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}.\hfill \end{array}$$

A simple computation gives:

$${\left|\right|\left|\right|y\left|\right|}^{-s}{\mathbf{1}}_{B_d(0,r)}{\left|\right|}_{L_k^{2p^{\prime }}\left({\mathbb{R}}^d\right)}=C{r}^{-s+\frac{2\gamma +d}{2p^{\prime }}}.$$

So

$$\begin{array}{ccc}\left|\right|G_{\lambda }(b,y){\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill & \le \hfill & \left|\right|G_{\lambda }(b,y){\mathcal{N}}_h^D\left(f_r\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}+\left|\right|G_{\lambda }(b,y){\mathcal{N}}_h^D\left(f^r\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill \\ & \le \hfill & C{r}^{-s}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}[{(\frac{C_h}{{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2})}^{\frac{1}{p^{\prime }}}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}\hfill \\ & +\hfill & r^{\frac{2\gamma +d}{2p^{\prime }}}\left|\right|G_{\lambda }{\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}].\hfill \end{array}$$

Choosing $r={(\frac{C_h}{{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2})}^{\frac{2}{2\gamma +d}}{\lambda }^2$, we obtain (64). □

Theorem 5. 

Let $1<p\le 2$ and $0<s<\frac{2\gamma +d}{2p^{\prime }}$ and $t>0$. Then, there exists a positive constant C such that, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$$\begin{array}{ccc}\parallel {\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill & \le \hfill & C{\left({\left(C_h\right)}^{\frac{1}{p^{\prime }}}{\left({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\right)}^{\frac{p^{\prime }-2}{p^{\prime }}}\right)}^{\frac{t}{s+t}}{\left({\parallel \parallel y\parallel }^{s}f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}+\parallel {\parallel y\parallel }^{s}f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right)}^{\frac{t}{s+t}}\hfill \\ \hfill & \hfill & {\parallel \parallel (b,y)\parallel }^{4t}{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{\frac{s}{s+t}}.\hfill \end{array}$$

(65)

Proof. 

Let $1<p\le 2$ and $0<s<\frac{2\gamma +d}{2p^{\prime }}$. Assume that $t\le \frac{1}{2}$. From the previous lemma, for all $\lambda >0$,

$$\begin{array}{ccc}\left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill & \le \hfill & \left|\right|G_{\lambda }(b,y){\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}+\left|\right|(1-G_{\lambda }(b,y)){\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill \\ & \le \hfill & C{\left(C_h\right)}^{\frac{1}{p^{\prime }}}{({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)})}^{\frac{p^{\prime }-2}{p^{\prime }}}{\lambda }^{-2s}\left[{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right]\hfill \\ & +\hfill & \left|\right|(1-G_{\lambda }(b,y)){\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}.\hfill \end{array}$$

On the other hand,

$$\left|\right|(1-G_{\lambda }(b,y)){\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}={\lambda }^{2t}{\left|\right|\left(\lambda \right|\left|(b,y)\right||}^2{)}^{-2t}(1-G_{\lambda }(b,y)){\left|\right|(b,y)\left|\right|}^{4t}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}.$$

Since $(1-e^{-u})u^{-2t}$ is bounded for $u\ge 0$ if $t\le \frac{1}{2}$, we obtain

$$\begin{array}{ccc}\left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill & \le \hfill & C{\left(C_h\right)}^{\frac{1}{p^{\prime }}}{({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)})}^{\frac{p^{\prime }-2}{p^{\prime }}}{\lambda }^{-2s}\left[{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right]\hfill \\ & +\hfill & C{\lambda }^{2t}{\left|\right|\left|\right|(b,y)\left|\right|}^{4t}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)},\hfill \end{array}$$

from which, optimizing in $\lambda$, we obtain (65) for $0<s<\frac{2\gamma +d}{2p^{\prime }}$ and $t\le \frac{1}{2}$.

If $t>\frac{1}{2}$, let $t^{\prime }\le \frac{1}{2}$. For $u\ge 0$, we have $u^{4t^{\prime }}\le 1+u^{4t}$, which for $u=\frac{\left|\right|(b,y)\left|\right|}{\epsilon }$ gives the inequality

$${\left(\frac{\left|\right|(b,y)\left|\right|}{\epsilon }\right)}^{4t^{\prime }}<1+{\left(\frac{\left|\right|(b,y)\left|\right|}{\epsilon }\right)}^{4t},\mathrm{for}\mathrm{all}\epsilon >0.$$

It follows that

$${\left|\right|\left|\right|(b,y)\left|\right|}^{4t^{\prime }}{\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\le {\epsilon }^{4t^{\prime }}\left|\right|{\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}+{\epsilon }^{4(t^{\prime }-t)}{\left|\right|\left|\right|(b,y)\left|\right|}^{4t}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}.$$

Optimizing in $\epsilon$, we obtain

$${\left|\right|\left|\right|(b,y)\left|\right|}^{4t^{\prime }}{\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\le \left|\right|{\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{\frac{t-t^{\prime }}{t}}{\left|\right|\left|\right|(b,y)\left|\right|}^{4t}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{\frac{t^{\prime }}{t}}.$$

Together with (65) for $t^{\prime }$, we obtain the result for $t>\frac{1}{2}$. □

Corollary 2. 

Let $0<s<\frac{2\gamma +d}{4}$ and $t>0$. Then, there exists a positive constant C such that, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$, we have

$${\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}\le C{\left(C_h\right)}^{\frac{s}{2(s+t)}}{\left[{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}\right]}^{\frac{t}{s+t}}\left|\right|\left|\right|(b,y){\left|\right|}^{4t}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{s}{s+t}}.$$

(66)

Proof. 

Using the previous theorem for $p=2$, and applying Plancherel’s Formula (56), we obtain the result. □

4. Weighted Inequalities for the Deformed Wavelet Transform

In this section, our motive is to formulate some weighted uncertainty inequalities for the deformed wavelet transform. To accomplish this motive, firstly, we begin our study with the following.

4.1. Pitt’s Uncertainty Principle

The Pitt inequality in the Dunkl setting expresses a fundamental relationship between a sufficiently smooth function and the corresponding Dunkl transform. This subject was first studied in [22]; then, Gorbachev et al. in [23] improved it and provided in the Dunkl setting a sharp Pitt-type inequality and a logarithmic uncertainty inequality. Specifically, for all f in the Schwartz space $\mathcal{S}\left({\mathbb{R}}^d\right)$ and $0\le \alpha <\gamma +d/2$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\parallel \xi \parallel }^{-2\alpha }|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\le C_k\left(\alpha \right){\int }_{{\mathbb{R}}^d}{\parallel x\parallel }^{2\alpha }{\left|f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right),}$$

(67)

where

$$C_k\left(\alpha \right):=\frac{1}{4^{\alpha }}{\left(\frac{\mathrm{\Gamma }\left(\frac{2\gamma +d-2\alpha }{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d+2\alpha }{4}\right)}\right)}^2$$

(68)

and $\mathrm{\Gamma }$ denotes the well known Euler’s Gamma function.

The first main objective of this Subsection is to formulate an analogue of Pitt’s inequality (67) for the deformed wavelet transform.

Theorem 6. 

For any arbitrary $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$, the Pitt-type inequality for the deformed wavelet transform is given by

$${\displaystyle C_h{\int }_{{\mathbb{R}}^d}{\parallel \xi \parallel }^{-2\alpha }|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le C_k\left(\alpha \right){\int }_{{\mathbb{R}}^{2d}}{\parallel x\parallel }^{2\alpha }|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x),0\le \alpha <\gamma +d/2.}$$

(69)

Proof. 

From (67), we have, for all $a\in {\mathbb{R}}^d$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{-2\alpha }|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le C_k\left(\alpha \right){\int }_{{\mathbb{R}}^d}{\left|\right|x\left|\right|}^{2\alpha }|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\gamma }_k\left(x\right)}$$

(70)

which upon integration with respect to the Haar measure $d{\gamma }_k\left(a\right)$ yields

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{-2\alpha }|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\mu }_k(a,\xi )\le C_k\left(\alpha \right){\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2\alpha }|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x).}$$

(71)

Invoking Lemma 1, we can express the inequality (71) in the following manner:

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{-2\alpha }|{\mathcal{F}}_D{\left(f\right)(-\xi )|}^2|{\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}h\right)\left(\xi \right){|}^{2}d{\mu }_k(a,\xi )\le C_k\left(\alpha \right){\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2\alpha }|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x).}$$

(72)

Equivalently, we have

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{-2\alpha }|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^2\left\{{\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}h\right)(-\xi )\right|}^{2}d{\gamma }_k\left(a\right)\right\}d{\gamma }_k\left(\xi \right)\le C_k\left(\alpha \right){\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2\alpha }|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x).}$$

(73)

Thus

$${\displaystyle C_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{-2\alpha }|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le C_k\left(\alpha \right){\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2\alpha }|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x),}$$

(74)

which proves the desired result. □

In the following theorem, we derive the Beckner’s inequality for the deformed wavelet transform by using the logarithmic estimate obtained from (67).

Theorem 7. 

For any function $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$, the following inequality holds:

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}log\parallel x\parallel |{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d}log\parallel \xi \parallel |{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\ge }\hfill \\ \\ \left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right)C_h{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.\hfill \end{array}$$

(75)

Proof. 

For $0\le s<\frac{2\gamma +d}{2},$ let

$${\displaystyle Q\left(s\right)=C_h{\int }_{{\mathbb{R}}^d}{\parallel \xi \parallel }^{-2s}|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-C_k\left(s\right){\int }_{{\mathbb{R}}^{2d}}{\parallel x\parallel }^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x).}$$

(76)

Taking the derivative of (76), we obtain

$$\begin{array}{c}{\displaystyle Q^{\prime }\left(s\right)=-2C_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{-2s}log\left|\right|\xi \left|\right||{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ {\displaystyle -2C_k\left(s\right){\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}log\left|\right|x\left|\right||{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)-C_k^{\prime }\left(s\right){\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x),}\hfill \end{array}$$

where

$$\begin{array}{ccc}{C}_k^{\prime }\left(s\right)\hfill & =\hfill & -C_k\left(s\right)\left(2log2+\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d-2s}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d-2s}{4}\right)}+\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d+2s}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d+2s}{4}\right)}\right).\hfill \end{array}$$

(77)

For $s=0,$ we have

$$C_k^{\prime }\left(0\right)=-2\left(log2+\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}\right).$$

(78)

By virtue of deformed Pitt’s inequality (69), it follows that $Q\left(s\right)\le 0$, for all $s\in [0,\frac{2\gamma +d}{2})$ and

$$\begin{array}{ccc}\hfill Q\left(0\right)& =& {\displaystyle C_h{\int }_{{\mathbb{R}}^d}|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-C_k\left(0\right){\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)}\hfill \end{array}$$

(79)

$$\begin{array}{ccc}& =& C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2-C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2=0.\hfill \end{array}$$

(80)

Therefore

$$Q^{\prime }\left(0^{+}\right):=\underset{s\to 0^{+}}{lim}\frac{Q\left(s\right)}{s}=\underset{s\to 0^{+}}{lim}{Q}^{\prime }\left(s\right)\le 0,$$

which is equivalent to

$$\begin{array}{c}{\displaystyle -2C_h{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-2C_k\left(0\right){\int }_{{\mathbb{R}}^{2d}}log\left|\right|x\left|\right||{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)}\hfill \\ {\displaystyle -C_k^{\prime }\left(0\right){\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\le 0.}\hfill \end{array}$$

(81)

From (56) and (78) we have,

$$\begin{array}{c}\hfill {\displaystyle -2C_h{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-2{\int }_{{\mathbb{R}}^{2d}}log\left|\right|x\left|\right||{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)}\\ \hfill +2\left(log2+\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}\right)C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2\le 0,\end{array}$$

or equivalently,

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}log\left|\right|x\left|\right||{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\ge }\hfill \\ \\ \left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right)C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.\hfill \end{array}$$

(82)

The proof is complete. □

4.2. Beckner-Type uncertainty principle

The Beckner-type inequality (see [23]) for the Dunkl transform states that for all $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{log\left|\right|x\left|\right|\left|f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right)+{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\ge \left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right){\int }_{{\mathbb{R}}^d}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right).}$$

(83)

This inequality is also known in the literature as the logarithmic uncertainty inequality.

Now, we will give an other proof of Theorem 7 by using the Beckner-type inequality (83).

Proof of Theorem 7.

Let $a\in {\mathbb{R}}^d$. We replace f in (83) with ${\mathcal{N}}_h^D\left(f\right)(a,.)$, so that

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^d}log\left|\right|x\left|\right||{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\gamma }_k\left(x\right)+{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D\left[{\mathcal{N}}_h^D\left(f\right)(a,.)\right]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\ge }\hfill \\ {\displaystyle \left(\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2\right){\int }_{{\mathbb{R}}^d}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\gamma }_k\left(x\right).}\hfill \end{array}$$

(84)

Integrating (84) with respect to the measure $d{\gamma }_k\left(a\right)$,

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}log\left|\right|x\left|\right||{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\mu }_k(a,\xi )\ge }\hfill \\ {\displaystyle \left(\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2\right){\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x).}\hfill \end{array}$$

(85)

Using Plancherel’s formula (56), we obtain

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}log\left|\right|x\left|\right||{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^{2d}}log\left|\right|\xi \left|\right||{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]{\left(\xi \right)|}^{2}d{\mu }_k(a,x)\ge }\hfill \\ \left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right)C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.\hfill \end{array}$$

(86)

We shall now simplify the second integral of (86). By using Lemma 1, we infer that

$$\begin{array}{ccc}\hfill {\displaystyle {\int }_{{\mathbb{R}}^{2d}}log\left|\right|\xi \left|\right||{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]{\left(\xi \right)|}^{2}d{\mu }_k(a,\xi )}& =& {\displaystyle {\int }_{{\mathbb{R}}^d}\left({\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\right)d{\gamma }_k\left(a\right)}\hfill \\ & =& {\displaystyle C_h{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right).}\hfill \end{array}$$

(87)

Plugging the estimate (87) in (86) gives the desired inequality for the deformed transforms as

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}log\left|\right|x\left|\right||{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right||{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\ge }\hfill \\ \left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right)C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.\hfill \end{array}$$

The proof is complete. □

Corollary 3. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet such that $C_h=1$. Then, for all $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$,

$$\begin{array}{c}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^2|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)}\right)}^{1/2}{\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^2{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\right)}^{1/2}\hfill \\ \ge exp\left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right){\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.\hfill \end{array}$$

Proof. 

From (75) and Jensen’s inequality, we obtain

$$\begin{array}{ccc}& & {\displaystyle log{\left\{{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^2\frac{|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\mu }_k(a,x){\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^2\frac{|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\gamma }_k\left(\xi \right)\right\}}^{1/2}}\hfill \\ & =& {\displaystyle log{\left\{{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^2\frac{|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\mu }_k(a,x)\right\}}^{1/2}+log{\left\{{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^2\frac{|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\gamma }_k\left(\xi \right)\right\}}^{1/2}}\hfill \\ & \ge & {\displaystyle {\int }_{{\mathbb{R}}^{2d}}log\left|\right|x\left|\right|\frac{|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^d}log\left|\right|\xi \left|\right|\frac{|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & \left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right),\hfill \end{array}$$

which gives the desired result. □

Remark 4. 

(1) 

From the approximation

$${\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(z\right)}{\mathrm{\Gamma }\left(z\right)}=logz-\frac{1}{2z}-2{\int }_{{\mathbb{R}}^d}\frac{t}{(t^2+z^2)(e^{2\pi t}-1)}dt}$$

(88)

we obtain

$$exp\left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right)\approx \frac{2\gamma +d}{2}\mathrm{for}2\gamma +d\gg 1,$$

(89)

which is the same constant given in Theorem 3.

(2) 

Proceeding as above in logarithmic uncertainty inequality (83) we deduce the following Heisenberg uncertainty inequality:

$${\displaystyle \parallel \parallel t\parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\parallel \parallel \xi \parallel {\mathcal{F}}_D\left(f\right){\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\ge exp\left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right){\int }_{{\mathbb{R}}^d}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right).}$$

(90)

(3) 

Using the approximation relation (88) we deduce that the constant in the right-hand side of (90),

$$exp\left({\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{4}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{4}\right)}+log2}\right)\approx \gamma +d/2\mathrm{if}d\gg 1,$$

which is the same constant in Proposition 8.

4.3. Dunkl Logarithmic Sobolev Inequalities

In this subsection, we will prove some logarithmic Sobolev-type uncertainty principle for the deformed wavelet transform. First, we will recall some preliminary results.

Definition 5. 

Let $u\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right)$ be a distribution. Then

(1) 

Its The Dunkl transform is defined by

$$⟨{\mathcal{F}}_D\left(u\right),vs.⟩=⟨u,{\mathcal{F}}_D^{-1}\left(v\right)⟩,\mathrm{for}\mathrm{all}v\in \mathcal{S}\left({\mathbb{R}}^d\right).$$

(91)

(2) 

For all $j=1,...,d$,

$${\mathcal{F}}_D\left(T_{j}u\right)=i{y}_j{\mathcal{F}}_D\left(u\right).$$

(92)

Definition 6. 

The Dunkl Sobolev space $H_k^s\left({\mathbb{R}}^d\right)$ of order $s\in \mathbb{R}$ is defined by

$$H_k^s\left({\mathbb{R}}^d\right)=\left\{\phi \in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right){:(1+\parallel y\parallel }^2{)}^{\frac{s}{2}}{\mathcal{F}}_D\left(\phi \right)\in L_k^2\left({\mathbb{R}}^d\right)\right\}.$$

(93)

Remark 5. 

From (22) and (92), we have

$$H_k^1\left({\mathbb{R}}^d\right)=\left\{\phi \in L_k^2\left({\mathbb{R}}^d\right):{\nabla }_k\phi \in L_k^2\left({\mathbb{R}}^d\right)\right\},$$

(94)

where ${\nabla }_k=\left(T_1,...,T_d\right)$ denotes the Dunkl gradient operator.

Definition 7. 

Let $b>0$ and $1\le p<\infty$. Then, we define the following weighted Lebesgue space by

$$L_{k,b}^p\left({\mathbb{R}}^d\right)=\left\{f\in L_k^p\left({\mathbb{R}}^d\right):{⟨x⟩}^{b}f\in L_k^p\left({\mathbb{R}}^d\right)\right\},$$

(95)

where for $x\in {\mathbb{R}}^d$, we have $⟨x⟩={(1+\parallel x\parallel }^2{)}^{1/2}$.

We recall in the following the logarithmic uncertainty inequality in the Dunkl setting.

Theorem 8. 

There exists a positive constant $K_{d,k}$ such that for all $f\in H_k^1\left({\mathbb{R}}^d\right)\cap L_{k,1}^1\left({\mathbb{R}}^d\right)$,

$${\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2\le {\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{2}log(C(k,d){⟨x⟩}^2)d{\gamma }_k\left(x\right)+{\int }_{{\mathbb{R}}^d}log\left(K(k,d)\right|\left|\xi \right|\left|\right)|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right),}$$

(96)

where

$${\displaystyle C(k,d):={\left(\frac{d_k{\mathrm{\Gamma }}^2\left(\frac{d+2\gamma }{2}\right)}{2\mathrm{\Gamma }(2\gamma +d)}\right)}^{\frac{1}{2\gamma +d}}.}$$

In the next theorem, we give another version of uncertainty inequality for the deformed wavelet transform.

Theorem 9. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet. Then, for all $f\in H_k^1\left({\mathbb{R}}^d\right)\cap L_{k,1}^1\left({\mathbb{R}}^d\right)$,

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}log\left(C(k,d)(1+|\left|x\right|{|}^2)\right)d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d}|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}log\left(K(k,d)\right|\left|\xi \right|\left|\right)d{\gamma }_k\left(\xi \right)\ge }\hfill \\ \\ \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}{C}_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.\hfill \end{array}$$

Proof. 

From (96),

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^d}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}log\left(C(k,d)(1+|\left|x\right|{|}^2)\right)d{\gamma }_k\left(x\right)+{\int }_{{\mathbb{R}}^d}|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}log\left(K(k,d)\right|\left|\xi \right|\left|\right)d{\gamma }_k\left(\xi \right)\ge }\\ {\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}{\int }_{{\mathbb{R}}^d}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\gamma }_k\left(x\right),\mathrm{for}\mathrm{all}a\in {\mathbb{R}}^d.}\end{array}\end{array}$$

Integration the last inequality with respect to the measure $d{\gamma }_k\left(a\right)$, we obtain

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}log\left(C(k,d)(1+|\left|x\right|{|}^2)\right)d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^{2d}}log\left(K(k,d)\right|\left|\xi \right|\left|\right)|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\mu }_k(\xi ,a)}\\ {\displaystyle \ge \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}{\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x).}\end{array}$$

(97)

By (97) and Lemma 1, we have

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}log\left(C(k,d)(1+|\left|x\right|{|}^2)\right)d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d}|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}log\left(K(k,d)\right|\left|\xi \right|\left|\right)d{\gamma }_k\left(\xi \right)\ge }\\ \\ \frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}{C}_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.\end{array}$$

The proof is complete. □

From Inequality (9), we can derive the following new uncertainty principle for the deformed wavelet transform.

Theorem 10. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ be a deformed wavelet with $C_h=1$. Then, for all $f\in H_k^1\left({\mathbb{R}}^d\right)\cap L_{k,1}^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\}$,

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^2|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\ge \frac{exp\left(\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}\right)}{C(k,d)K(k,d)\parallel {\nabla }_k{f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^3-{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.}$$

(98)

Proof. 

Let f be in $H_k^1\left({\mathbb{R}}^d\right)\cap L_{k,1}^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\}$. For $C_h=1$, we infer from (9) that

$$\begin{array}{ccc}\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2& \le & {\displaystyle {\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}log\left(C(k,d)\right(1+{\left|\right|x\left|\right|}^2\left)\right)d{\mu }_k(a,x)}\\ & +& {\displaystyle {\int }_{{\mathbb{R}}^d}log\left(K(k,d)\right|\left|\xi \right|\left|\right)|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right).}\end{array}$$

(99)

Using Jensen’s inequality in (99), we can deduce that

$$\begin{array}{ccc}\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}& \le & {\displaystyle logC(k,d)\left({\int }_{{\mathbb{R}}^{2d}}\frac{|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{(1+|\left|x\right||}^2)d{\mu }_k(a,x)\right)}\\ & +& {\displaystyle \frac{1}{2}{\int }_{{\mathbb{R}}^d}log(K^2{(k,d)\left|\right|\xi \left|\right|}^2)\frac{|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\gamma }_k\left(\xi \right).}\end{array}$$

(100)

To obtain a fruitful estimate of the second integral of (100), we set

$${\displaystyle d{\varrho }_k\left(\xi \right)=\frac{|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^2}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}d{\gamma }_k\left(\xi \right),\mathrm{so}\mathrm{that}{\int }_{{\mathbb{R}}^d}d{\varrho }_k\left(\xi \right)=1.}$$

(101)

Jensen’s inequality implies,

$$\begin{array}{ccc}\hfill {\displaystyle {\int }_{{\mathbb{R}}^d}log(K^2{(k,d)\left|\right|\xi \left|\right|}^2\left)\right|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)}& =& {\displaystyle {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2{\int }_{{\mathbb{R}}^d}log(K^2{(k,d)\left|\right|\xi \left|\right|}^2\left)\right|d{\rho }_k\left(\xi \right)}\hfill \\ & \le & {\displaystyle {\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{2}log\left\{K^2(k,d){\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^2|d{\rho }_k\left(\xi \right)\right\}}\hfill \\ & \le & {\displaystyle {\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{2}log\left\{\frac{K^2(k,d)}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2}{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^2||{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\right\}}\hfill \\ & \le & {\displaystyle {\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{2}log\left\{\frac{K^2(k,d)}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{\int }_{{\mathbb{R}}^d}{\left|{\nabla }_{k}f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right)\right\}.}\hfill \end{array}$$

(102)

Using (102) in (100), we have

$$\begin{array}{ccc}\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}\hfill & \le \hfill & {\displaystyle log\left(\frac{C(k,d)K(k,d)}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^3}\left\{{\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^2{(1+|\left|x\right||}^2)d{\mu }_k(a,x)\right\}{\parallel {\nabla }_{k}f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\right),}\hfill \end{array}$$

(103)

which can be rewritten as follows

$${\displaystyle \left\{{\int }_{{\mathbb{R}}^{2d}}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^2\left(1+{\left|\right|x\left|\right|}^2\right)d{\mu }_k(a,x)\right\}{\left\{{\int }_{{\mathbb{R}}^d}{\left|{\nabla }_{k}f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right)\right\}}^{1/2}\ge \frac{exp\left(\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}\right)}{C(k,d)K(k,d)}{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^3.}$$

(104)

By (56), we obtain

$$\begin{array}{c}{\displaystyle \left\{{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^2|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\right\}{\left\{{\int }_{{\mathbb{R}}^d}{\left|{\nabla }_{k}f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right)\right\}}^{1/2}\ge \frac{exp\left(\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}\right)}{C(k,d)K(k,d)}{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^3}\hfill \\ -{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2{\parallel {\nabla }_{k}f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.\hfill \end{array}$$

Then,

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^2|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\ge \frac{exp\left(\frac{{\mathrm{\Gamma }}^{\prime }\left(\frac{2\gamma +d}{2}\right)}{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}\right)}{C(k,d)K(k,d)\parallel {\nabla }_k{f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^3-{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.}$$

This completes the proof of the theorem. □

5. Concentration Uncertainty Principles for the Deformed Wavelet Transform

In this section, we shall derive some concentration-based uncertainty principles for the deformed wavelet transform such as the Benedick–Amrein–Berthier and local-type uncertainty principles.

5.1. Benedick–Amrein–Berthier’s Uncertainty Principle

In [20], the authors proved a Benedicks-type uncertainty principle in the Dunkl setting, that is, if $E_1,E_2\subset {\mathbb{R}}^d$ are of finite measures, then there exists a constant $C_k(E_1,E_2)$ (called the annihilating constant), such that for all $f\in L_k^2\left({\mathbb{R}}^d\right)$

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right)\le C_k(E_1,E_2)\left({\displaystyle {\int }_{{\mathbb{R}}^d\backslash E_1}{\left|f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right)+{\int }_{{\mathbb{R}}^d\backslash E_2}{\left|{\mathcal{F}}_D\left(f\right)\left(y\right)\right|}^{2}d{\gamma }_k\left(y\right)}\right).}$$

(105)

In the next theorem, we state the analog of (105) for the deformed wavelet transforms.

Theorem 11. 

For all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d\backslash E_1}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d\backslash (-E_2)}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C_k(E_1,E_2)}.}$$

(106)

Proof. 

Let $f\in L_k^2\left({\mathbb{R}}^d\right)$; then, for any $a\in {\mathbb{R}}^d$, the function ${\mathcal{N}}_h^D\left(f\right)(a,.)$ belongs in $L_k^2\left({\mathbb{R}}^d\right)$. Therefore replacing f with ${\mathcal{N}}_h^D\left(f\right)(a,.)$ in (105), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^d}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\gamma }_k\left(x\right)\le C_k(E_1,E_2)\left\{{\int }_{{\mathbb{R}}^d\backslash E_1}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\gamma }_k\left(x\right)+{\int }_{{\mathbb{R}}^d\backslash E_2}|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\right\}.}$$

(107)

By integrating (107) with respect to measure $d{\gamma }_k\left(a\right)$,

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\le }\hfill \\ {\displaystyle C_k(E_1,E_2)\left\{{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d\backslash E_1}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d\backslash E_2}{\mathcal{F}}_D\left[{\mathcal{N}}_h^D\left(f\right)(a,x)\right]\left(\xi \right){|}^{2}d{\mu }_k(a,\xi )\right\}.}\hfill \end{array}$$

Using Lemma 1 together with Plancherel’s Formula (56), the above inequality becomes

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d\backslash E_1}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^d\backslash (-E_2)}{\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^2|{\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}h\right)(-\xi ){|}^{2}d{\mu }_k(a,\xi )\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C_k(E_1,E_2)}}$$

which further implies

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d\backslash E_1}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+{\int }_{{\mathbb{R}}^d\backslash (-E_2)}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^2\left\{{\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}h\right)(-\xi )\right|}^{2}d{\gamma }_k\left(a\right)\right\}d{\gamma }_k\left(\xi \right)\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C_k(E_1,E_2)}.}$$

Since h is a deformed wavelet, we have

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d\backslash E_1}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d\backslash (-E_2)}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C_k(E_1,E_2)}}$$

which is the desired Benedicks–Amrein–Berthier’s uncertainty principle for the deformed wavelet transform. □

From Theorem 11, we derive the following general uncertainty inequality.

Corollary 4. 

For all $s,t>0$, there exists a constant ${\mathcal{C}}_k(s,t)>0$ such that for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\left({\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\right)}^{\frac{t}{2}}{\left({\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\right)}^{\frac{s}{2}}\ge {\mathcal{C}}_k(s,t){\left(C_h\right)}^{\frac{t}{2}}{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{s+t}.}$$

Proof. 

Let $s,t>0$ and let $f\in L_k^2\left({\mathbb{R}}^d\right)$. Take $E_1=E_2=B_d(0,1)$, the unit ball in ${\mathbb{R}}^d$. Then, by (106),

$${\displaystyle {\int }_{B_d^c(0,1)}{\int }_{{\mathbb{R}}^d}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+C_h{\int }_{B_d^c(0,1)}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C\left(k\right)}.}$$

Here, $C\left(k\right):=C_k(E_1,E_2)$.

It follows that

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+C_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C\left(k\right)}.}$$

Replacing f by $f_{\lambda }$ and using (53),

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D\left(f\right)(\lambda a,\frac{x}{\lambda }){|}^{2}d{\mu }_k(a,x)+{\lambda }^{2\gamma +d}{C}_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_D\left(f\right)\left(\lambda \xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C\left(k\right)}.}$$

Thus

$${\displaystyle {\lambda }^{2s}{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)+{\lambda }^{-2t}{C}_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\ge \frac{C_h{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}{C\left(k\right)}.}$$

The desired result follows by minimizing the right-hand side over $\lambda >0.$ □

5.2. $L^2$ Local-Type Uncertainty Principle

The classical Heisenberg uncertainty principle states that if a signal f is well concentrated in the natural domain, the corresponding Fourier transform cannot be properly localised at a point in the spectral domain. However, it does not preclude f from being localised within a small neighbourhood of two or more widely separated points. In fact, the latter phenomenon cannot occur either, and it is the motive of local uncertainty inequalities to make this precise. In this subsection, our goal is to derive some local uncertainty principles for the deformed wavelet transform. We begin this subsection by recalling the local uncertainty principle for the Dunkl transform [20].

Proposition 10. 

Let E be a subset of ${\mathbb{R}}^d$ with finite measure ${\displaystyle 0<{\gamma }_k\left(E\right):={\int }_{E}d{\gamma }_k\left(x\right)<\infty }$. For $0<s<\gamma +d/2$, there exists a constant $\mathfrak{C}=\mathfrak{C}(k,s)>0$ such that for all $f\in L_k^2\left({\mathbb{R}}^d\right)$

$$\parallel {\mathcal{F}}_D{\left(f\right)\parallel }_{L_k^2\left(E\right)}^2\le \mathfrak{C}{\left({\gamma }_k\left(E\right)\right)}^{\frac{2s}{2\gamma +d}}{\parallel \parallel y\parallel }^s{f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

(108)

We are now ready to obtain the local uncertainty principle for the deformed wavelet transform by employing the inequality (108).

Theorem 12. 

Let $E\subset {\mathbb{R}}^d$ with finite measure and let $0<s<\gamma +d/2$. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\parallel x\parallel }^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\ge \frac{{\mathfrak{C}}^{-1}{C}_h}{{\left({\gamma }_k\left(E\right)\right)}^{\frac{2s}{2\gamma +d}}}{\parallel {\mathcal{F}}_D\left(f\right)\parallel }_{L_k^2\left(E\right)}^2.}$$

(109)

Proof. 

Let $f\in L_k^2\left({\mathbb{R}}^d\right)$ and $a\in {\mathbb{R}}^d$. Then, by replacing f with ${\mathcal{N}}_h^D\left(f\right)(a,.)$ in (108), we obtain

$${\displaystyle {\int }_E|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le \mathfrak{C}{\left({\gamma }_k\left(E\right)\right)}^{\frac{2s}{2\gamma +d}}\parallel {\parallel x\parallel }^s{\mathcal{N}}_h^D\left(f\right)(a,.){\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.}$$

(110)

Now, integrating the last inequality with respect to the measure $d{\gamma }_k\left(a\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_E|{\mathcal{F}}_D[{\mathcal{N}}_h^D\left(f\right)(a,.)]\left(\xi \right){|}^{2}d{\mu }_k(a,\xi )\le \mathfrak{C}{\left({\gamma }_k\left(E\right)\right)}^{\frac{2s}{2\gamma +d}}{\int }_{{\mathbb{R}}^{2d}}{\parallel x\parallel }^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)}$$

which, together with Lemma 1 and Fubini’s theorem, give

$${\displaystyle {\int }_E|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^2\left({\int }_{{\mathbb{R}}^d}{\left|{\mathcal{F}}_D\left({\mathrm{\Delta }}_{a}h\right)(-\xi )\right|}^{2}d{\gamma }_k\left(a\right)\right)d{\gamma }_k\left(\xi \right)\le \mathfrak{C}{\left({\gamma }_k\left(E\right)\right)}^{\frac{2s}{2\gamma +d}}{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x).}$$

(111)

Using the hypothesis h, Inequality (111) reduces to

$${\displaystyle C_h{\int }_E|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le \mathfrak{C}(k,s){\left({\gamma }_k\left(E\right)\right)}^{\frac{2s}{2\gamma +d}}{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D{\left(f\right)(a,x)|}^{2}d{\mu }_k(a,x).}$$

The proof is complete. □

Corollary 5. 

Let $0<s<\gamma +d/2$ and let $E\subset {\mathbb{R}}^d$ with finite measure. Then, for all $f\in P{W}_k\left(E\right)$,

$${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2\le \frac{\mathfrak{C}(k,s){\left({\gamma }_k\left(E\right)\right)}^{\frac{2s}{2\gamma +d}}}{C_h}{∥{\parallel x\parallel }^s{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^2$$

(112)

where $P{W}_k\left(E\right):=\left\{f\in L_k^2\left({\mathbb{R}}^d\right):\mathrm{supp}{\mathcal{F}}_D\left(f\right)\subset E\right\}$ is the Paley–Wiener space.

By interchanging the roles of f and ${\mathcal{F}}_D\left(f\right)$ in Proposition 10, we obtain the following:

Corollary 6. 

Let $F\subset {\mathbb{R}}^d$ with finite measure and let $0<t<\gamma +d/2$. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$, we have

$$\parallel f{\parallel }_{L_k^2\left(F\right)}^2\le \mathfrak{C}(k,t){\left({\gamma }_k\left(F\right)\right)}^{\frac{2t}{2\gamma +d}}\parallel {\parallel \xi \parallel }^t{\mathcal{F}}_D\left(f\right){\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

(113)

Using Corollary 6 and by adapting the proof of Theorem 12, we obtain this result.

Corollary 7. 

Let $F\subset {\mathbb{R}}^d$ with finite measure and let $0<t<\gamma +d/2$. Then for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_F|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\le \mathfrak{C}(k,t){\left({\gamma }_k\left(F\right)\right)}^{\frac{2t}{2\gamma +d}}{C}_h{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right).}$$

(114)

Let F be a subset of ${\mathbb{R}}^d$. We define the generalized Paley–Wiener space $GP{W}_k\left(F\right)$:

$$GP{W}_k\left(F\right):=\left\{f\in L_k^2\left({\mathbb{R}}^d\right):\forall a\in {\mathbb{R}}^d,\mathrm{supp}{\mathcal{N}}_h^D\left(f\right)(a,.)\subset F\right\}.$$

Then, by Formula (56), and the previous corollary, we obtain the following result.

Corollary 8. 

Let E and F be two subsets of ${\mathbb{R}}^d$ such that $0<{\gamma }_k\left(E\right),{\gamma }_k\left(F\right)<\infty$, and let $0<s,t<\gamma +d/2$. Then,

(1) 

For any $f\in GP{W}_k\left(F\right)$,

$${\displaystyle {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2\le \mathfrak{C}(k,t){\left({\gamma }_k\left(F\right)\right)}^{\frac{2t}{2\gamma +d}}{\int }_{{\mathbb{R}}^d}{\parallel \xi \parallel }^{2t}|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right).}$$

(115)

(2) 

For any $f\in P{W}_k\left(E\right)\cap GP{W}_k\left(F\right)$,

$$\begin{array}{ccc}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{s+t}\hfill & \le \hfill & {\left(\mathfrak{C}(k,t)\right)}^{\frac{s}{2}}{\left(\mathfrak{C}(k,s)\right)}^{\frac{t}{2}}{\left({\gamma }_k\left(E\right){\gamma }_k\left(F\right)\right)}^{\frac{ts}{2\gamma +d}}\hfill \\ \hfill & \hfill & {\displaystyle {\left({\int }_{{\mathbb{R}}^d}{\parallel \xi \parallel }^{2t}|{\mathcal{F}}_D\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\right)}^{\frac{s}{2}}{\left({\int }_{{\mathbb{R}}^{2d}}{\parallel x\parallel }^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x)\right)}^{\frac{t}{2}}.}\hfill \end{array}$$

(116)

From Theorem 12, we derive the following result.

Theorem 13. 

Let $t>0$ and $0<s<\gamma +d/2$. Then for all $f\in L_k^2\left({\mathbb{R}}^d\right)$,

$${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2\le {\mathcal{C}(k,s,t)\parallel \parallel x\parallel }^s{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{2t}{s+t}}\parallel {\parallel \xi \parallel }^t{\mathcal{F}}_D\left(f\right){\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{2s}{s+t}},$$

(117)

where

$$\mathcal{C}(k,s,t)=\begin{array}{c}{\left(\frac{\mathfrak{C}(k,s){\left(\frac{d_k}{2\gamma +d}\right)}^{\frac{2s}{2\gamma +d}}}{C_h}\right)}^{\frac{t}{s+t}}\left[{\left(\frac{s}{t}\right)}^{\frac{t}{s+t}}+{\left(\frac{t}{s}\right)}^{\frac{s}{s+t}}\right].\end{array}$$

Proof. 

Let $r,t>0$ and $0<s<\gamma +d/2$. Then,

$${\displaystyle {\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2=\left|\right|{\mathcal{F}}_D{\left(f\right)\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2={\int }_{B_d(0,r)}|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)+{\int }_{B_d^c(0,r)}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

(118)

From Theorem 12 and by simple calculation, we have

$${\displaystyle {\int }_{B_d(0,r)}|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\le \frac{\mathfrak{C}(k,s){\left(\frac{d_k}{2\gamma +d}\right)}^{\frac{2s}{2\gamma +d}}}{C_h}{r}^{2s}{\int }_{{\mathbb{R}}^{2d}}{\left|\right|x\left|\right|}^{2s}|{\mathcal{N}}_h^D\left(f\right)(a,x){|}^{2}d{\mu }_k(a,x).}$$

(119)

Moreover it is easy to see that

$${\displaystyle {\int }_{B_d^c(0,r)}|{\mathcal{F}}_D{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\le r^{-2t}{\int }_{{\mathbb{R}}^d}{\left|\right|\xi \left|\right|}^{2t}{\left|{\mathcal{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

(120)

Combining the relations (131)–(133), we obtain

$$\begin{array}{ccc}{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2& \le & \frac{\mathfrak{C}(k,s){\left(\frac{d_k}{2\gamma +d}\right)}^{\frac{2s}{2\gamma +d}}}{C_h}{r}^{2s}{\left|\right|\left|\right|x\left|\right|}^s{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^2+r^{-2t}{\left|\right|\left|\right|\xi \left|\right|}^t{\mathcal{F}}_D\left(f\right){\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2.\end{array}$$

We choose

$$r={\left[\frac{t}{s}\frac{C_h}{\mathfrak{C}(k,s){\left(\frac{d_k}{2\gamma +d}\right)}^{\frac{2s}{2\gamma +d}}}\right]}^{\frac{1}{2s+2t}}{\left(\frac{{\left|\right|\left|\right|\xi \left|\right|}^t{\mathcal{F}}_D\left(f\right){\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}}{{\left|\right|\left|\right|x\left|\right|}^s{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}}\right)}^{\frac{1}{s+t}},$$

and we obtain the desired inequality. □

5.3. $L^p$-Local Type Inequality

The aim of this subsection is to derive an $L^p$-local type inequality for the deformed wavelet transform. The following theorem gives the main result.

Theorem 14. 

Let T be any measurable subset in ${\mathbb{R}}^{2d}$ with $0<{\mu }_k\left(T\right)<\infty$. Then, for every $f\in L_k^2\left({\mathbb{R}}^d\right),1<p⩽2$ and $t>0$, we have

$$\begin{array}{c}\hfill {∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\le \left\{\begin{array}{c}{C}_1(t,h){\left({\mu }_k\left(T\right)\right)}^{\frac{2t}{2\gamma +d}}\left[{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}+{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right],t<\frac{d+2\gamma }{2p^{\prime }}\hfill \\ C_2(t,h){\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p^{\prime }}}\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1-\frac{d+2\gamma }{2t{p}^{\prime }}}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2t{p}^{\prime }}},t>\frac{d+2\gamma }{2p^{\prime }}\hfill \\ C_3(t,h){\left({\mu }_k\left(T\right)\right)}^{\frac{1}{2p^{\prime }}}\hfill \\ \left[\parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{1/2}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}^{1/2}+\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1/2}{∥{}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1/2}\right],t=\frac{d+2\gamma }{2p^{\prime }}\hfill \end{array}\right.\end{array}$$

where

$$\begin{array}{cc}\hfill C_1(t,h)& ={\left({\displaystyle \frac{d_k}{(2\gamma +d-2t{p}^{\prime })}}\right)}^{\frac{2t}{2\gamma +d}}{\left(C_h\right)}^{\frac{2\gamma +d-2t{p}^{\prime }}{p^{\prime }(2\gamma +d)}}{\left({\left|\right|h\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}\right)}^{\frac{(2\gamma +d)(p^{\prime }-2)+2t{p}^{\prime }}{p^{\prime }(2\gamma +d)}},\hfill \\ \hfill C_2(t,h)& =\parallel h{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\left({\displaystyle \frac{2t{p}^{\prime }}{2t{p}^{\prime }-2\gamma -d}}\right)}^{\frac{1}{2p}}{\left({\displaystyle \frac{2t{p}^{\prime }}{2\gamma +d}}-1\right)}^{\frac{d+2\gamma }{4tp{p}^{\prime }}}\hfill \\ & {\left(\frac{1}{2pt{d}_k}\frac{\mathrm{\Gamma }\left(\frac{d+2\gamma }{pt}\right)\mathrm{\Gamma }\left(\frac{2p^{\prime }t-2⟨k⟩-d+1}{pt}\right)}{\mathrm{\Gamma }\left(\frac{p^{\prime }}{p}\right)}\right)}^{\frac{1}{2p^{\prime }}}\hfill \\ \hfill C_3(t,h)& =2{\left({\displaystyle \frac{2d_k{C}_h}{2\gamma +d}}\right)}^{\frac{1}{2p^{\prime }}}{\left({\displaystyle \frac{1}{\parallel h{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}}\right)}^{\frac{1}{p}}.\hfill \end{array}$$

Proof. 

(i) For $s>0$, we consider $f_s={\mathbf{1}}_{{}_{B_d(0,s)}}f$ and $f^s=f-f_s$. Using (57) and the fact that $|f^s\left(y\right)|⩽s^{-t}\left|{\left|\right|y\left|\right|}^{t}f\left(y\right)\right|$, we obtain

$$\begin{array}{cc}\hfill {∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left({\mathbf{1}}_{B_d^c(0,s)}f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}& \le {∥{\mathcal{N}}_h^D\left({\mathbf{1}}_{B_d^c(0,s)}f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill \\ & \le {\left(C_h\right)}^{1/p^{\prime }}{\left({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\right)}^{\frac{p^{\prime }-2}{p^{\prime }}}{∥{\mathbf{1}}_{B_d^c(0,s)}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}\hfill \\ \hfill & \le {\left(C_h\right)}^{1/p^{\prime }}{\left({\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\right)}^{\frac{p^{\prime }-2}{p^{\prime }}}{s}^{-t}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}.\hfill \end{array}$$

On the other hand, relation (52) and Hölder’s inequality imply that

$$\begin{array}{cc}\hfill {∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left({\mathbf{1}}_{{}_{B_d(0,s)}}f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}& \le {∥{\mathbf{1}}_T∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}{∥{\mathcal{N}}_h^D\left({\mathbf{1}}_{{}_{B_d(0,s)}}f\right)∥}_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2d}\right)}\hfill \\ \hfill & \le \parallel h{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\left({\mu }_k\left(T\right)\right)}^{1/p^{\prime }}{∥{\mathbf{1}}_{{}_{B_d(0,s)}}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}\hfill \\ \hfill & \le \parallel h{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\left({\mu }_k\left(T\right)\right)}^{1/p^{\prime }}{∥{\left|\right|y\left|\right|}^{-t}{\mathbf{1}}_{{}_{B_d(0,s)}}∥}_{L_k^{2p^{\prime }}\left({\mathbb{R}}^d\right)}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}.\hfill \end{array}$$

Moreover, a simple calculation yields

$$\begin{array}{c}\hfill {∥{\left|\right|y\left|\right|}^{-t}{\mathbf{1}}_{{}_{B_d(0,s)}}∥}_{L_k^{2p^{\prime }}\left({\mathbb{R}}^d\right)}={\left({\displaystyle \frac{d_k}{(2\gamma +d-2t{p}^{\prime })}}\right)}^{1/2p^{\prime }}{s}^{-t+\frac{d+2\gamma }{2p^{\prime }}}.\end{array}$$

Consequently, we have

$$\begin{array}{cc}\hfill {∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}& \le {∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left(f_s\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}+{∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left(f^s\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill \\ \hfill & \le s^{-t}\parallel h{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\left[{\left(C_h\right)}^{1/p^{\prime }}{\left(\parallel h{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\right)}^{-2/p^{\prime }}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}\right.\hfill \\ \hfill & \left.+{\left(\frac{d_k}{(2\gamma +d-2t{p}^{\prime })}\right)}^{\frac{1}{2p^{\prime }}}{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p^{\prime }}}{s}^{\frac{d+2\gamma }{2p^{\prime }}}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right].\hfill \end{array}$$

With the choice

$$\begin{array}{c}\hfill s={\left(\frac{C_h}{{\left|\right|h\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^2}\right)}^{\frac{2}{2\gamma +d}}{\left({\displaystyle \frac{d_k}{(2\gamma +d-2t{p}^{\prime })}}\right)}^{\frac{-1}{2\gamma +d}}{\left({\mu }_k\left(T\right)\right)}^{\frac{-2}{2\gamma +d}},\end{array}$$

we obtain the first inequality.

(ii) Assume that ${\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}<\infty .$ Again applying Hölder’s inequality and (52), we obtain

$$\begin{array}{cc}\hfill {∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}& \le {∥{\mathbf{1}}_T∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}{∥{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2d}\right)}\hfill \\ \hfill & \le \parallel h{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\left({\mu }_k\left(T\right)\right)}^{1/p^{\prime }}\parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.\hfill \end{array}$$

(121)

Using Hölder’s inequality together with the fact that $t>\frac{d+2\gamma }{2p^{\prime }}$, we obtain

$$\begin{array}{cc}\hfill \parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{2p}& ={\left({\displaystyle {\int }_{{\mathbb{R}}^d}{\left(1+|y{|}^{2pt}\right)}^{1/p}|f\left(y\right){|}^2\frac{d{\gamma }_k\left(y\right)}{{(1+|\left|y\right||}^{2pt}{)}^{\frac{1}{p}}}}\right)}^p\hfill \\ \hfill & \le {\left({\displaystyle {\int }_{{\mathbb{R}}^d}\frac{d{\gamma }_k\left(y\right)}{{(1+|\left|y\right||}^{2pt}{)}^{p^{\prime }/p}}}\right)}^{p/p^{\prime }}\left(\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{2p}+{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{2p}\right)\hfill \\ \hfill & \le {\left({\displaystyle \frac{d_k}{2pt}}{\displaystyle \frac{\mathrm{\Gamma }\left(\frac{d+2\gamma }{2pt}\right)\mathrm{\Gamma }\left(\frac{2p^{\prime }t-2\gamma ⟩-d}{2pt}\right)}{\mathrm{\Gamma }\left(\frac{p^{\prime }}{p}\right)}}\right)}^{p/p^{\prime }}\left(\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{2p}+{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{2p}\right)<\infty .\hfill \end{array}$$

Upon replacing $f\left(y\right)$ by ${\delta }_{\lambda }f\left(y\right):=f\left(\lambda y\right),\lambda >0$, in the above inequality, we obtain

$$\begin{array}{cc}\hfill \parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{2p}& \le {\left({\displaystyle \frac{d_k}{2pt}}{\displaystyle \frac{\mathrm{\Gamma }\left(\frac{d+2\gamma }{2pt}\right)\mathrm{\Gamma }\left(\frac{2p^{\prime }t-2\gamma ⟩-d}{2pt}\right)}{\mathrm{\Gamma }\left(\frac{p^{\prime }}{p}\right)}}\right)}^{p/p^{\prime }}\hfill \\ \hfill & \times \left({\lambda }^{(2\gamma +d)(p-1)}\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{2p}+{\lambda }^{(2\gamma +d)(p-1)-2pt}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{2p}\right).\hfill \end{array}$$

In particular, the inequality also holds at

$$\begin{array}{c}\hfill \lambda ={\left(\frac{2p^{\prime }t}{2\gamma +d}-1\right)}^{1/2pt}{\left({\displaystyle \frac{{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}}{\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}}}\right)}^{1/t},\end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\le C\left(a\right)\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1-\frac{d+2\gamma }{2t{p}^{\prime }}}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2t{p}^{\prime }}},\end{array}$$

(122)

with

$$\begin{array}{cc}\hfill C\left(a\right)& ={\left(\frac{2p^{\prime }t}{2t{p}^{\prime }-(2\gamma +d)}\right)}^{1/2p}{\left(\frac{2t{p}^{\prime }}{2\gamma +d}-1\right)}^{\frac{d+2\gamma }{4tp{p}^{\prime }}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{d_k}{2pt}}{\displaystyle \frac{\mathrm{\Gamma }\left(\frac{d+2\gamma }{2pt}\right)\mathrm{\Gamma }\left(\frac{2p^{\prime }t-2\gamma ⟩-d}{2pt}\right)}{\mathrm{\Gamma }\left(\frac{p^{\prime }}{p}\right)}}\right)}^{1/2p^{\prime }}.\hfill \end{array}$$

(123)

Therefore, the desired result follows immediately from (121)–(123).

(iii) Using the inequality

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\left|\right|y\left|\right|}{r}}\right)}^{\frac{d+2\gamma }{4p^{\prime }}}\le 1+{\left(\frac{\left|\right|y\left|\right|}{r}\right)}^{\frac{d+2\gamma }{2p^{\prime }}},r>0,\end{array}$$

we obtain

$$\begin{array}{c}\hfill {\parallel \left|\right|y\left|\right|}^{\frac{d+2\gamma }{4p^{\prime }}}f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}\le r^{\frac{d+2\gamma }{4p^{\prime }}}\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}+r^{-\frac{d+2\gamma }{4p^{\prime }}}{∥{\left|\right|y\left|\right|}^{\frac{d+2\gamma }{2p^{\prime }}}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}.\end{array}$$

By optimizing over r, we obtain

$$\begin{array}{c}\hfill {∥{\left|\right|y\left|\right|}^{\frac{d+2\gamma }{4p^{\prime }}}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\le 2\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1/2}{∥{\left|\right|y\left|\right|}^{\frac{d+2\gamma }{2p^{\prime }}}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1/2}.\end{array}$$

Similarly, we can prove that

$$\begin{array}{c}\hfill {∥{\left|\right|y\left|\right|}^{\frac{d+2\gamma }{4p^{\prime }}}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}\le 2\parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{1/2}{\left.∥{\left|\right|y\left|\right|}^{\frac{d+2\gamma }{2p^{\prime }}}f\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{1/2}.\end{array}$$

Thus, we conclude that

$$\begin{array}{cc}\hfill & {∥{\mathbf{1}}_T{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}\hfill \\ \hfill & \le C_1\left(\frac{d+2\gamma }{4p^{\prime }},h\right){\left({\mu }_k\left(T\right)\right)}^{1/2p^{\prime }}\left[{∥{\left|\right|y\left|\right|}^{\frac{d+2\gamma }{4p^{\prime }}}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}+{∥{\left|\right|y\left|\right|}^{\frac{d+2\gamma }{4p^{\prime }}}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right]\hfill \\ & \le 2C_1\left(\frac{d+2\gamma }{4p^{\prime }},h\right){\left({\mu }_k\left(T\right)\right)}^{1/2p^{\prime }}\hfill \\ \hfill & \times \left[\parallel f{\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^{1/2}{∥{\left|\right|y\left|\right|}^{t}f∥}_{L_k^2\left({\mathbb{R}}^d\right)}^{1/2}+\parallel f{\parallel }_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1/2}{∥{}^{t}f∥}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1/2}\right].\hfill \end{array}$$

This completes the proof of Theorem 14. □

Remark 6. 

We note that when $t=\frac{d+2\gamma }{2p^{\prime }}$, we can obtain a family of inequalities and improve the inequality given in Theorem 14. Indeed, if we apply the first inequality with

$$s=(1-\epsilon )\frac{d+2\gamma }{2p^{\prime }},\epsilon \in (0,1),$$

and then apply the classical inequality

$${\parallel \left|\right|y\left|\right|}^{t-t\epsilon }{f\parallel }_{L_k^r\left({\mathbb{R}}^d\right)}⩽{C\left|\right|f\left|\right|}_{L_k^r\left({\mathbb{R}}^d\right)}^{\epsilon }\left|\right|{\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^r\left({\mathbb{R}}^d\right)}^{1-\epsilon },$$

(124)

we obtain, for all $\epsilon \in (0,1)$,

$$\left|\right|{\mathbf{1}}_T{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}⩽C_4(\epsilon ,t,h){\left({\mu }_k\left(T\right)\right)}^{\frac{1-\epsilon }{p^{\prime }}}\left[{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\epsilon }{\left|\right|\left|\right|y\left|\right|}^t{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{1-\epsilon }+{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\epsilon }{\left|\right|\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1-\epsilon }\right].$$

(125)

Definition 8. 

Let $0\le \epsilon <1$, $S\subset {\mathbb{R}}^d$ and $T\subset {\mathbb{R}}^{2d}$.

(1) 

A nonzero $f\in L_k^2\left({\mathbb{R}}^d\right)$ is called an ε-concentrated function on S in $L_k^2\left({\mathbb{R}}^d\right)$-norm if

$$\parallel {\mathbf{1}}_{S^c}{f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\le \epsilon {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.$$

(2) 

A nonzero $f\in L_k^2\left({\mathbb{R}}^d\right)$ is called an ε-bandlimited function on T in $L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)$-norm if

$$\parallel {\mathbf{1}}_{T^c}{\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}\le \epsilon {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}.$$

Here, $A^c$ is the complement of A.

Corollary 9. 

Let $h\in L_k^2\left({\mathbb{R}}^d\right)$ such that $C_h=1$.

(1) 

If $0<t<\frac{d+2\gamma }{4}$, then there exists a positive constant C such that for every function f that is ε-bandlimited on T,

$${\left({\mu }_k\left(T\right)\right)}^{\frac{4t}{2\gamma +2d}}{\left[{\left|\right|\left|\right|y\left|\right|}^t{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}\right]}^2\ge C\left(1-{\epsilon }^2\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

(126)

(2) 

If $t>\frac{d+2\gamma }{4}$, then there exists a positive constant C such that for every function f that is ε-bandlimited on T,

$${\mu }_k{\left(T\right)\left|\right|f\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}^{2-\frac{d+2\gamma }{2t}}{\left|\right|\left|\right|y\left|\right|}^t{f\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2t}}\ge C\left(1-{\epsilon }^2\right){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

(127)

(3) 

For all $s\in (0,1)$, there exists a positive constant C such that for every function f that is ε-bandlimited on T,

$${\left({\mu }_k\left(T\right)\right)}^{1-s}{\left[{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^s{\left|\right|\left|\right|y\left|\right|}^t{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{1-s}+{\left|\right|f\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}^s{\left|\right|\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}^{1-s}\right]}^2\ge C(1-{\epsilon }^2){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

(128)

Proof. 

Since $f\in L_k^2\left({\mathbb{R}}^d\right)$ is $\epsilon$-bandlimited on T, it follows that

$$\parallel {\mathbf{1}}_T{\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^2=C_h{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2-\parallel {\mathbf{1}}_{T^c}{\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^2\ge (1-{\epsilon }^2){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}^2.$$

(129)

In view of (129), the inequalities (126) and (127) follow from the first and the second local inequalities in Theorem 14, respectively, while (128) follows from (125). □

Corollary 10. 

Let $1<p⩽2$ and $t>0$. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$, we have

$$\begin{array}{cc}\left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{\frac{(2\gamma +d)p^{\prime }}{2\gamma +d-2p^{\prime }t},p^{\prime }}\left({\mathbb{R}}^{2d}\right)}⩽C_1(t,h)\left[{\left|\right|\left|\right|y\left|\right|}^t{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right],\hfill & \mathrm{for}t<\frac{d+2\gamma }{2p^{\prime }}\hfill \\ \left|\right|{\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{\infty ,p^{\prime }}\left({\mathbb{R}}^{2d}\right)}⩽C_2{(t,h)\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1-\frac{d+2\gamma }{2t{p}^{\prime }}}{\left|\right|\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2t{p}^{\prime }}},\hfill & \mathrm{for}t>\frac{d+2\gamma }{p^{\prime }}\hfill \\ \left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{2p^{\prime },p^{\prime }}\left({\mathbb{R}}^{2d}\right)}⩽C_3(t,h)\left[{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^t{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}+{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^{t}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}\right],\hfill & \mathrm{for}t=\frac{d+2\gamma }{2p^{\prime }}\hfill \end{array}$$

where $L_{{\mu }_k}^{p,q}\left({\mathbb{R}}^{2d}\right)$ denotes the Lorentz space corresponding to the norm

$$\begin{array}{c}\hfill {\left|\right|g\left|\right|}_{L_{{\mu }_k}^{p,q}\left({\mathbb{R}}^{2d}\right)}:=\underset{T\subset {\mathbb{R}}^{2d},0<{\mu }_k\left(T\right)<\infty }{sup}{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p}-\frac{1}{q}}\left|\right|{\mathbf{1}}_{T}g{\left|\right|}_{L_{{\mu }_k}^q\left({\mathbb{R}}^{2d}\right)},\end{array}$$

(130)

and $C_j(t,h)$, for $j=1,2,3,$ are the constants given in Theorem 14.

Theorem 15. 

Let $s,t>0$ and $1<p⩽2$. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$, we have

$$\left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}⩽\left\{\begin{array}{cc}{C}_1(s,t,h){\left[{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right]}^{\frac{t}{2s+t}}\hfill & \\ {∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{\frac{2s}{2s+t}},\hfill & \mathrm{for}s<\frac{d+2\gamma }{2p^{\prime }},\hfill \\ C_2(s,t,h){\left[{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{1-\frac{d+2\gamma }{2s{p}^{\prime }}}{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2s{p}^{\prime }}}\right]}^{\frac{t{p}^{\prime }}{2\gamma +d+t{p}^{\prime }}}\hfill \\ {∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{\frac{2\gamma +d}{2\gamma +d+t{p}^{\prime }}},\hfill & \mathrm{for}s>\frac{d+2\gamma }{2p^{\prime }},\hfill \\ C_3{(s,t,h)\left[\right|\left|f\right||}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^{t}f|{|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}+\hfill \\ {\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{]}^{\frac{2t}{2\gamma +d+2t{p}^{\prime }}}{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{\frac{2\gamma +d}{2\gamma +d+2t{p}^{\prime }}},\hfill & \mathrm{for}s=\frac{d+2\gamma }{2p^{\prime }},\hfill \end{array}\right.$$

where

$$\begin{array}{ccc}{C}_1(s,t,h)\hfill & =\hfill & {\left[{\left(\frac{t}{2s}\right)}^{\frac{2s}{2s+t}}+{\left(\frac{2s}{t}\right)}^{\frac{t}{2s+t}}\right]}^{\frac{1}{p^{\prime }}}{\left(C_1(s,h)\right)}^{\frac{t}{2s+t}}{\left(\frac{d_k{\left(\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)\right)}^2}{4(2\gamma +d)\mathrm{\Gamma }\left(2\gamma +d\right)}\right)}^{\frac{st}{(2s+t)(2\gamma +d)}},\hfill \\ \\ C_2(s,t,h)\hfill & =\hfill & {\left[{\left(\frac{t{p}^{\prime }}{2\gamma +d}\right)}^{\frac{2\gamma +d}{2\gamma +d+t{p}^{\prime }}}+{\left(\frac{2\gamma +d}{t{p}^{\prime }}\right)}^{\frac{t{p}^{\prime }}{2\gamma +d+t{p}^{\prime }}}\right]}^{\frac{1}{p^{\prime }}}{\left(\frac{\sqrt{d_k}\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right){\left(C_2(s,h)\right)}^{p^{\prime }}}{2\sqrt{2\gamma +d\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{t}{2\gamma +d+t{p}^{\prime }}},\hfill \\ \\ C_3(s,t,h)\hfill & =\hfill & {\left[{\left(\frac{2t{p}^{\prime }}{2\gamma +d}\right)}^{\frac{2\gamma +d}{2\gamma +d+t{p}^{\prime }}}+{\left(\frac{2\gamma +d}{2t{p}^{\prime }}\right)}^{\frac{2t{p}^{\prime }}{2\gamma +d+t{p}^{\prime }}}\right]}^{\frac{1}{p^{\prime }}}{\left(\frac{\sqrt{d_k}\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right){\left(C_3(s,h)\right)}^{2p^{\prime }}}{2\sqrt{(2\gamma +d)\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{t}{2\gamma +d+2t{p}^{\prime }}},\hfill \end{array}$$

and $C_j(s,h)$, for $j=1,2,3,$ are the constants given in Theorem 14.

Proof. 

(i) Let $0<s<\frac{d+2\gamma }{2p^{\prime }}$, $t>0$ and $r>0$. Then,

$$\left|\right|{\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}⩽\left|\right|{\mathbf{1}}_{V_r}{\mathcal{N}}_h^D{\left(f\right)\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}+\left|\right|{\mathbf{1}}_{V_r^c}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}.$$

(131)

From Theorem 14, we have

$$\begin{array}{ccc}\left|\right|{\mathbf{1}}_{V_r}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}\hfill & ⩽\hfill & {\left(C_1(s,h)\right)}^{p^{\prime }}{\left(\frac{d_k{\left(\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)\right)}^2}{4(2\gamma +d)\mathrm{\Gamma }\left(2\gamma +d\right)}\right)}^{\frac{s{p}^{\prime }}{2\gamma +d}}\hfill \\ & \hfill & r^{2s{p}^{\prime }}{\left[{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right]}^{p^{\prime }}.\hfill \end{array}$$

(132)

Moreover, it is easy to see that

$${∥{\mathbf{1}}_{V_r^c}{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}⩽r^{-t{p}^{\prime }}{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}.$$

(133)

Combining the relations (131)–(133), we obtain

$$\begin{array}{c}\left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}⩽{\left(C_1(s,h)\right)}^{p^{\prime }}{\left(\frac{d_k{\left(\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)\right)}^2}{4(2\gamma +d)\mathrm{\Gamma }\left(2\gamma +d\right)}\right)}^{\frac{s{p}^{\prime }}{2\gamma +d}}\hfill \\ \hfill r^{2s{p}^{\prime }}{\left[{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}\right]}^{p^{\prime }}+r^{-t{p}^{\prime }}{\parallel {\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)\parallel }_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}.\end{array}$$

We choose

$$r={\left(\frac{t}{2s{\left(C_1(s,h)\right)}^{p^{\prime }}{\left(\frac{d_k{\left(\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)\right)}^2}{4(2\gamma +d)\mathrm{\Gamma }\left(2\gamma +d\right)}\right)}^{\frac{s{p}^{\prime }}{2\gamma +d}}}\right)}^{\frac{1}{(2s+t)p^{\prime }}}{\left(\frac{{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}}{{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}}\right)}^{\frac{1}{2s+t}}$$

to obtain the first inequality.

(ii) Let $s>\frac{d+2\gamma }{2p^{\prime }}$, $t>0$ and $r>0$. From Theorem 14, we have

$${∥{\mathbf{1}}_{V_r}{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}⩽\frac{\sqrt{d_k}\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right){\left(C_2(s,h)\right)}^{p^{\prime }}}{2\sqrt{(2\gamma +d)\mathrm{\Gamma }(2\gamma +d)}}{r}^{2\gamma +d}{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{p^{\prime }-\frac{d+2\gamma }{2s}}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2s}}.$$

(134)

Combining the relations (131), (133), and (134), we obtain

$$\begin{array}{ccc}\left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}\hfill & ⩽\hfill & \frac{\sqrt{d_k}\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right){\left(C_2(s,h)\right)}^{p^{\prime }}}{2\sqrt{(2\gamma +d)\mathrm{\Gamma }(2\gamma +d)}}{r}^{2\gamma +d}{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{p^{\prime }-\frac{d+2\gamma }{2s}}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2s}}\hfill \\ & +\hfill & r^{-t{p}^{\prime }}\left|\right|{∥{(b,y)\left|\right|}^s{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}.\hfill \end{array}$$

We choose

$$r={\left(\frac{t{p}^{\prime }\mathrm{\Gamma }(2\gamma +d)}{(2\gamma +d)\frac{\sqrt{d_k}\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right){\left(C_2(s,h)\right)}^{p^{\prime }}}{2\sqrt{2\gamma +d\mathrm{\Gamma }(2\gamma +d)}}}\right)}^{\frac{1}{2\gamma +d+t{p}^{\prime }}}{\left(\frac{{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}}{{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{p^{\prime }-\frac{d+2\gamma }{2s}}{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{2s}}}\right)}^{\frac{1}{2\gamma +d+t{p}^{\prime }}}$$

to obtain the second inequality.

(iii) Let $s=\frac{d+2\gamma }{2p^{\prime }}$, $s>0$ and $r>0$. From Theorem 14, we have

$$\begin{array}{c}\left|\right|{\mathbf{1}}_{V_r}{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}⩽{\left(\frac{\sqrt{d_k}\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right){\left(C_3(s,h)\right)}^{2p^{\prime }}}{2\sqrt{(2\gamma +d)\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{1}{2}}{r}^{\frac{2\gamma +d}{2}}\hfill \\ \hfill {\left[{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}+{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}\right]}^{p^{\prime }}.\end{array}$$

(135)

Combining the relations (131), (133), and (135), we obtain

$$\begin{array}{c}\left|\right|{\mathcal{N}}_h^D\left(f\right){\left|\right|}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}⩽{\left(\frac{\sqrt{d_k}\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right){\left(C_3(s,h)\right)}^{2p^{\prime }}}{2\sqrt{(2\gamma +d)\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{1}{2}}{r}^{\frac{2\gamma +d}{2}}\hfill \\ \hfill {\left[{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}+{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}\right]}^{p^{\prime }}+r^{-t{p}^{\prime }}{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}^{p^{\prime }}.\end{array}$$

We choose

$$r=C(d,s,t,a,p^{\prime },k){\left(\frac{{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2d}\right)}}{{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^s{f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}+{\left|\right|f\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^{s}f{\left|\right|}_{L_k^{2p}\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}}\right)}^{\frac{2p^{\prime }}{2\gamma +d+2t{p}^{\prime }}},$$

where

$$C(d,s,t,a,p^{\prime },k)={\left(\frac{4t{p}^{\prime }\sqrt{\mathrm{\Gamma }(2\gamma +d)}}{{d_k}^{\frac{1}{4}}\sqrt{2\gamma +d}\sqrt{\mathrm{\Gamma }\left(\frac{2\gamma +d}{2}\right)}{\left(C_3(s,h)\right)}^{p^{\prime }}}\right)}^{\frac{2}{2\gamma +d+2t{p}^{\prime }}}$$

to obtain the third inequality. □

Corollary 11. 

Let $s,t>0$. Then, for all $f\in L_k^2\left({\mathbb{R}}^d\right)$, we have

$${\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}⩽\left\{\begin{array}{cc}\frac{C_1(s,t,h)}{\sqrt{C_h}}{\left[{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^2\left({\mathbb{R}}^d\right)}+{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^4\left({\mathbb{R}}^d\right)}\right]}^{\frac{t}{2s+t}}{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{2s}{2s+t}},\hfill & \mathrm{for}0<s<\frac{d+2\gamma }{2}\hfill \\ \frac{C_2(s,t,h)}{\sqrt{C_h}}{\left({\left|\right|f\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}^{1-\frac{d+2\gamma }{4s}}{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^4\left({\mathbb{R}}^d\right)}^{\frac{d+2\gamma }{4s}}\right)}^{\frac{t}{2\gamma +d+t}}{∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{t}{2\gamma +d+t}},\hfill & \mathrm{for}s>\frac{d+2\gamma }{2}\hfill \\ \frac{C_3(s,t,h)}{\sqrt{C_h}}{\left[{\left|\right|f\left|\right|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{\left|\right|\left|\right|y\left|\right|}^{s}f|{|}_{L_k^2\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}+{\left|\right|f\left|\right|}_{L_k^4\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}{∥{\left|\right|y\left|\right|}^{s}f∥}_{L_k^4\left({\mathbb{R}}^d\right)}^{\frac{1}{2}}\right]}^{\frac{t}{2s+t}}\hfill \\ {∥{\left|\right|(b,y)\left|\right|}^t{\mathcal{N}}_h^D\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{2s}{2s+t}},\hfill & \mathrm{for}s=\frac{d+2\gamma }{2}.\hfill \end{array}\right.$$

We close this subsection with the following local uncertainty principle version:

Theorem 16 (Faris–Price’s uncertainty principle for ${\mathcal{N}}_h^D$).

Let $\eta ,p$ be two real numbers such that $0<\eta <2\gamma +d$ and $p⩾1$. Then, there is a positive constant $C_k(\eta ,p)$ such that for all $f\in L_k^2\left({\mathbb{R}}^d\right)$ and any $T\subset {\mathbb{R}}^{2d}$ with $0<{\mu }_k\left(T\right)<\infty$,

$$\begin{array}{ccc}\hfill {\left({\displaystyle {\int }_T{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^{p}d{\mu }_k(a,x)}\right)}^{\frac{1}{p}}& ⩽\hfill & \hfill C_k(\eta ,p){\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}\parallel {\parallel (a,x)\parallel }^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{4\gamma +2d}{(2\gamma +d+\eta )(p+1)}}\\ & & \hfill {\left({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}\right)}^{\frac{(2\gamma +d+\eta )(p+1)-(4\gamma +2d)}{(2\gamma +d+\eta )(p+1)}}.\end{array}$$

Proof. 

Assume that ${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}={\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}=1$. Then, for every $r>1$,

$$\begin{array}{c}\hfill \parallel {\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^p\left(T\right)}⩽\parallel {\mathcal{N}}_h^D\left(f\right){\mathbf{1}}_{V_r}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}+{\parallel {\mathcal{N}}_h^D\left(f\right){\mathbf{1}}_{V_r^c}\parallel }_{L_{{\mu }_k}^p\left(T\right)},\end{array}$$

where $V_r:=\left\{(a,x)\in {\mathbb{R}}^{2d}:\parallel (a,x)\parallel \le r\right\}.$ However, by Hölder’s inequality and relation (52) we obtain, for every $\eta \in (0,2\gamma +d)$,

$$\begin{array}{cc}\hfill \parallel {\mathcal{N}}_h^D\left(f\right){\mathbf{1}}_{V_r}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}& ={\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^p{\mathbf{1}}_{V_r}(a,x){\mathbf{1}}_T(a,x)d{\mu }_k(a,x)}\right)}^{\frac{1}{p}}\hfill \\ & ⩽\parallel {\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2d}\right)}^{\frac{p}{p+1}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^{\frac{p}{p+1}}{\mathbf{1}}_{V_r}(a,x){\mathbf{1}}_T(a,x)d{\mu }_k(a,x)}\right)}^{\frac{1}{p}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\parallel {\mathcal{N}}_h^D\left(f\right){\mathbf{1}}_{V_r}\parallel }_{L_{{\mu }_k}^1\left({\mathbb{R}}^{2d}\right)}^{\frac{1}{p+1}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\parallel \left|\right|(a,x)\left|\right|}^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{1}{p+1}}{\parallel \left|\right|(a,x)\left|\right|}^{-\eta }{\mathbf{1}}_{V_r}{\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{1}{p+1}}.\hfill \end{array}$$

On the other hand by simple calculations, we see that

$${\parallel \left|\right|(a,x)\left|\right|}^{-\eta }{\mathbf{1}}_{V_r}{\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}\le \left(\frac{d_k\mathrm{\Gamma }(\gamma +\frac{d}{2})}{2\sqrt{(2\gamma +d-\eta )\mathrm{\Gamma }(2\gamma +d)}}\right)r^{2\gamma +d-\eta }.$$

Thus, we obtain

$$\begin{array}{ccc}\parallel {\mathcal{N}}_h^D\left(f\right){\mathbf{1}}_{V_r}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}& ⩽& {\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\left(\frac{d_k\mathrm{\Gamma }(\gamma +\frac{d}{2})}{2\sqrt{(2\gamma +d-\eta )\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{1}{p+1}}\\ & & r^{\frac{2\gamma +d-\eta }{p+1}}{\parallel \left|\right|(a,x)\left|\right|}^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{1}{p+1}}.\end{array}$$

On the other hand, and again by Hölder’s inequality and Relation (52), we deduce that

$$\begin{array}{cc}\hfill \parallel {\mathcal{N}}_h^D\left(f\right){\mathbf{1}}_{V_r^c}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}& ⩽\parallel {\mathcal{N}}_h^D{\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2d}\right)}^{\frac{p-1}{p+1}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^{\frac{2p}{p+1}}{\mathbf{1}}_{V_r^c}(a,x){\mathbf{1}}_T(a,x)d{\mu }_k(a,x)}\right)}^{\frac{1}{p}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2d}}{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^2{\mathbf{1}}_{V_r^c}(a,x)d{\mu }_k(a,x)}\right)}^{\frac{1}{p+1}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\parallel \left|\right|(a,x)\left|\right|}^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{2}{p+1}}{r}^{-\frac{2\eta }{p+1}}.\hfill \end{array}$$

Hence, for every $\eta \in (0,2\gamma +d)$,

$$\begin{array}{cc}& {\left({\displaystyle {\int }_T{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^{p}d{\mu }_k(a,x)}\right)}^{\frac{1}{p}}⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\parallel \left|\right|(a,x)\left|\right|}^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{1}{p+1}}\hfill \\ & \left({\left(\frac{d_k\mathrm{\Gamma }(\gamma +\frac{d}{2})}{2\sqrt{(2\gamma +d-\eta )\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{1}{p+1}}{r}^{\frac{2\gamma +d-\eta }{p+1}}+{\parallel \left|\right|(a,x)\left|\right|}^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{1}{p+1}}{r}^{-\frac{2\eta }{p+1}}\right).\hfill \end{array}$$

In particular, the inequality holds for

$$r_0=\frac{{\left(\frac{2\eta }{2\gamma +d-\eta }\right)}^{\frac{p+1}{2\gamma +d+\eta }}}{{\left(\frac{d_k\mathrm{\Gamma }(\gamma +\frac{d}{2})}{2\sqrt{(2\gamma +d-\eta )\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{1}{2\gamma +d+\eta }}}{\parallel \left|\right|(a,x)\left|\right|}^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{1}{2\gamma +d+\eta }}$$

and therefore

$$\begin{array}{cc}& {\left({\displaystyle {\int }_T{\left|{\mathcal{N}}_h^D\left(f\right)(a,x)\right|}^{p}d{\mu }_k(a,x)}\right)}^{\frac{1}{p}}⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\left(\frac{d_k\mathrm{\Gamma }(\gamma +\frac{d}{2})}{2\sqrt{(2\gamma +d-\eta )\mathrm{\Gamma }(2\gamma +d)}}\right)}^{\frac{2\eta }{(2\gamma +d+\eta )(p+1)}}\hfill \\ & {\parallel \left|\right|(a,x)\left|\right|}^{\eta }{\mathcal{N}}_h^D\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2d}\right)}^{\frac{4\gamma +2d}{(2\gamma +d+\eta )(p+1)}}{\left(\frac{2\eta }{2\gamma +d-\eta }\right)}^{\frac{-2\eta }{2\gamma +d+\eta }}\left(\frac{2\gamma +d+\eta }{2\gamma +d-\eta }\right).\hfill \end{array}$$

Finally, the general result follows from the above inequality by replacing f with $\frac{f}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}$ and h with $\frac{h}{{\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^d\right)}}$. □

6. Conclusions

In this paper, we have established some new uncertainty principles for the deformed wavelet transform in which localization is measured by generalized dispersions such as in the Heisenberg-type uncertainty inequalities, and other principles in which localization is measured by smallness of the supports such as in Benedicks-type and local uncertainty inequalities. We have provided detailed proofs for these uncertainty principles. Future studies will focus on the explicit expression of this transformation in some special cases and then give some applications of the proposed transform in image processing.