A New Wavelet Transform and Its Localization Operators

Abstract

In the present paper we define and study a new wavelet transformation associated to the linear canonical Dunkl transform (LCDT), which has been widely used in signal processing and other related fields. Then we define and study a class of pseudo-differential operators known as time-frequency (or localization) operators and we give criteria for its boundedness and Schatten class properties.

1. Introduction and Preliminaries

The fields of special functions with reflection symmetries and harmonic analysis associated with root systems have seen particularly rapid progress in recent years. The theory of Riemannian symmetric spaces, whose spherical functions may be expressed as multivariate special functions based on certain discrete sets of parameters, provides some motivation for this topic.

Dunkl operators are a crucial tool in the study of special functions with reflection symmetries, which have generated considerable interest in mathematical physics, particularly in conformal field theory. To be more precise, we will fix some notation related to the Dunkl transform. For more details about Dunkl theory, we refer the reader to [1,2,3,4,5].

Let ${\sigma }_{\alpha }$ be the reflection in some hyperplane of ${\mathbb{R}}^N$ orthogonal to $\alpha \in {\mathbb{R}}^N\backslash \left\{0\right\}$ and let $\mathcal{R}$ be a root system such that ${\sigma }_{\alpha }\left(\mathcal{R}\right)=\mathcal{R}$, for every $\alpha \in \mathcal{R}$. Recall that the reflections associated to the root system $\mathcal{R}$ generate a finite group $W\subset O\left(N\right)$, called the reflection group. For a multiplicity function k defined on $\mathcal{R}$, we introduce the index

$${\displaystyle ⟨k⟩=\sum _{\alpha \in {\mathcal{R}}_{+}}k\left(\alpha \right),}$$

(1)

and the weight function

$${\omega }_k\left(t\right)=\prod _{\alpha \in {\mathcal{R}}_{+}}{\left|⟨\alpha ,t⟩\right|}^{2k\left(\alpha \right)},$$

(2)

where ${\mathcal{R}}_{+}$ is a positive root system related to $\mathcal{R}$. Furthermore, we define the Mehta type constant

$${\displaystyle c_k={\int }_{{\mathbb{R}}^N}{e}^{-\frac{{\left|t\right|}^2}{2}}{\omega }_k\left(t\right)dt,}$$

(3)

where $|·|={⟨·,·⟩}^{1/2}$ is the usual norm on ${\mathbb{R}}^N$.

The Dunkl operators ${\left\{T_j\right\}}_{j\in \mathbb{N}}$ are defined by

$${\displaystyle T_{j}u\left(t\right):=\frac{\partial u}{\partial t_j}\left(t\right)+\sum _{\alpha \in {\mathcal{R}}_{+}}k\left(\alpha \right)⟨\alpha ,e_j⟩\frac{u\left(t\right)-u\left({\sigma }_{\alpha }\left(t\right)\right)}{⟨\alpha ,t⟩},t\in {\mathbb{R}}^N,}$$

(4)

for any orthonormal basis ${\left\{e_j\right\}}_{j\in \mathbb{N}}$ of ${\mathbb{R}}^N$.

The Dunkl kernel $E_k(-i·,·)$ is the unique analytic solution on ${\mathbb{R}}^N$ of the system

$$\left\{\begin{array}{ccc}{T}_{j}u(t,z)\hfill & =\hfill & z_{j}u(t,z),j=1,\dots ,d,\hfill \\ u(0,z)\hfill & =\hfill & 1,\hfill \end{array}\right.$$

(5)

where $z\in {\mathbb{R}}^N$. It satisfies

$$|E_k(it,z)|\le 1,\forall t,z\in {\mathbb{R}}^N.$$

(6)

Moreover, it has a unique holomorphic extension to ${\mathbb{C}}^N\times {\mathbb{C}}^N$. In particular

$$E_k(z,0)=1,E_k(t,z)=E_k(z,t),E_k(\omega z,t)=E_k(z,\omega t),$$

for every $z,t\in {\mathbb{C}}^N$ and $\omega \in \mathbb{C}$.

For $1\le p\le \infty$, we define ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ as the space of measurable functions f on ${\mathbb{R}}^N$ such that

$$\begin{array}{ccc}{\parallel u\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\hfill & :=\hfill & {\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|u\left(t\right)\right|}^p{\gamma }_k\left(dt\right)}\right)}^{1/p}<\infty ,p\ge 1,\hfill \\ \\ {\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}\hfill & :=\hfill & \mathrm{ess}\underset{t\in {\mathbb{R}}^N}{sup}\left|u\left(t\right)\right|<\infty ,\hfill \end{array}$$

where ${\gamma }_k\left(dt\right):={c_k}^{-1}{\omega }_k\left(t\right)dt.$ In particular the Hilbert space ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ is equipped with the scalar product

$${\displaystyle {⟨u,v⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}:={\int }_{{\mathbb{R}}^N}u\left(t\right)\overline{v\left(t\right)}{\gamma }_k\left(dt\right).}$$

For a function $u\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$, the Dunkl transform is given by

$${\displaystyle {\mathfrak{F}}_D\left(u\right)\left(z\right)={\int }_{{\mathbb{R}}^N}u\left(t\right)E_k(-it,z){\gamma }_k\left(dt\right),z\in {\mathbb{R}}^N.}$$

(7)

This transformation has recently become an interesting topic in harmonic analysis [6,7,8,9,10,11,12,13,14,15,16,17,18,19].

The Dunkl translation operator [4] is defined on ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ by

$${\mathfrak{F}}_D\left({\tau }_{s}u\right)=E_k(is,·){\mathfrak{F}}_D\left(u\right),s\in {\mathbb{R}}^N.$$

(8)

This operator has an extension on the spaces $C^{\infty }\left({\mathbb{R}}^N\right)$, $C_{b,\mathrm{rad}}\left({\mathbb{R}}^N\right)$ and ${\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, $1\le p\le \infty$. Moreover, if $W={\mathbb{Z}}_2^N$, there is also an extension of the Dunkl translation operator on ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ [5,6,20].

For a matrix $M:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ in $SL(2,\mathbb{R})$, such that $b\ne 0$, we define the linear canonical Dunkl transform (LCDT) of a function $u\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {\mathfrak{F}}_D^M\left(u\right)\left(t\right)=\frac{1}{{\left(ib\right)}^{⟨k⟩+N/2}}{\int }_{{\mathbb{R}}^N}{\mathfrak{J}}_k^M(t,z)u\left(z\right){\gamma }_k\left(dz\right),}$$

(9)

where

$${\mathfrak{J}}_k^M(t,z)=e^{{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{d}{b}}{\left|t\right|}^2+{\displaystyle \frac{a}{b}}{\left|z\right|}^2\right)}{E}_k\left(-i{\displaystyle \frac{t}{b}},z\right).$$

(10)

This transformation is an extension of the classical linear canonical transform (LCT), which was introduced independently by Collins [21] in paraxial optics and Moshinsky-Quesne [22] in quantum mechanics. The LCT is a flexible tool for investigating deep problems in signal processing, optics, and quantum physics [23,24,25,26,27,28] and so on. During the last years, the LCT has attracted a great interest and has been extended to a large class of integral transformations, see for example [29,30,31,32,33,34,35,36,37] and the references therein.

The generalized translation operator in the LCDT setting [38] is given by

$${\mathrm{T}}_y^{M}u\left(z\right)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{(|y|}^2+{|z|}^2)}{\tau }_y\left(e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}u\right)\left(z\right),y,z\in {\mathbb{R}}^N.$$

(11)

Moreover, for every $y,z\in {\mathbb{R}}^N$

$${\displaystyle {\mathrm{T}}_y^{M^{-1}}u\left(z\right)={(-ib)}^{-N/2-⟨k⟩}{e}^{-\frac{i}{2}\frac{a}{b}{\left|z\right|}^2}{\int }_{{\mathbb{R}}^N}{E}_k(it/b,z)\overline{{\mathfrak{J}}_k^M(z,t)}{\mathfrak{F}}_D^M\left(u\right)\left(t\right){\gamma }_k\left(dt\right),}$$

(12)

where $M^{-1}$ is the inverse of the matrix M.

In [39], we have introduced a new Gabor-type transformation associated with the LCDT, in order to concentrate signals in the time-frequency plane, but this concentration is restricted by the uncertainty principles.

It is well-known that wavelet theory is superior to Gabor theory in the localization of signals, because of its ability to measure the time-frequency variations of a signal at different time-frequency resolutions. It is often seen as an alternative to time-frequency analysis, and it has many roots and has become an interdisciplinary field combining applied mathematics [40], harmonic analysis [41,42], and signal and data processing [43].

To overcome the lack of localization in the LCDT setting, we will define in this paper a new wavelet-type transformation (see Section 3 for details), and then investigate its related localization operators (see Section 4 for details). This study extends the results proved in the recent paper [44] to the multidimensional case.

To be more precise, if $\phi \in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ (or $\phi \in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, $p\ne 2$) is an admissible Dunkl linear canonical wavelet, satisfying

$${\displaystyle 0<C_{\phi }^M:={\int }_0^{\infty }|{\mathfrak{F}}_D^M\left(\phi \right)\left(\lambda y\right){|}^2\frac{d\lambda }{\lambda }<\infty ,a.ey\in {\mathbb{R}}^N,}$$

(13)

where ${\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$ is the subspace of radial functions in ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$. Then we define the family ${\phi }_{r,t}^M$ by

$$\forall y\in {\mathbb{R}}^N,{\phi }_{r,t}^M\left(y\right)=r^{⟨k⟩+N/2}\overline{{\mathrm{T}}_t^{M^{-1}}\left({\phi }_r^M\right)(-y)},t\in {\mathbb{R}}^N,r>0,$$

(14)

where

$${\phi }_r^M\left(y\right):={\displaystyle \frac{1}{r^{2⟨k⟩+N}}}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}(1-1/r^2){\left|y\right|}^2}\phi (y/r).$$

(15)

Therefore, the Dunkl linear canonical wavelet transform (DLCWT) is defined on ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {{\rm Y}}_{\phi }^M\left(u\right)(r,t):={\int }_{{\mathbb{R}}^N}u\left(y\right)\overline{{\phi }_{r,t}^M\left(y\right)}{\gamma }_k\left(dy\right).}$$

(16)

For $p\in [1,\infty ]$, the time-frequency localization operator associated to the DLCWT is defined on ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(g\right)\left(x\right)={\int }_{\Omega }\varrho (r,z){{\rm Y}}_u^M\left(g\right)(r,z)v_{r,z}^M\left(x\right)d{\mu }_k(r,z),x\in {\mathbb{R}}^N,}$$

(17)

where $\varrho$ is a measurable function defined on $\Omega =(0,\infty )\times {\mathbb{R}}^N$ (called symbol), $u,v$ are two suitable functions on ${\mathbb{R}}^N$ and ${\mu }_k$ is the weight measure given by $d{\mu }_k(r,t)=r^{-(2⟨k⟩+N+1)}{\gamma }_k\left(dt\right)dr$.

We will prove that the DLCWT satisfies the following orthogonality relation: For all $u_1,u_2\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$,

$${\displaystyle {\int }_{\Omega }{{\rm Y}}_{\phi }^M\left(u_1\right)(r,t)\overline{{{\rm Y}}_{\phi }^M\left(u_2\right)(r,t)}d{\mu }_k(r,t)=C_{\phi }^M{\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{u}_1\left(y\right)\overline{u_2\left(y\right)}{\gamma }_k\left(dy\right).}$$

(18)

In particular we deduce the Plancherel-type formula: For all $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$,

$${\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}|{{\rm Y}}_{\phi }^M{\left(u\right)(r,t)|}^{2}d{\mu }_k(r,t)={\left|b\right|}^{2⟨k⟩+N}{C}_{\phi }^M{\int }_{{\mathbb{R}}^N}{\left|u\left(y\right)\right|}^2{\gamma }_k\left(dy\right).}$$

(19)

More generally we have for all $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and $2\le p\le \infty$,

$${\left({\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{\left|{{\rm Y}}_{\phi }^M\left(u\right)(r,t)\right|}^{p}d{\mu }_k(r,t)}\right)}^{1/p}\le {\left({\left|b\right|}^{2⟨k⟩+N}{C}_{\phi }^M\right)}^{{\displaystyle \frac{1}{p}}}{\parallel \phi \parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^{{\displaystyle \frac{p-2}{p}}}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(20)

Moreover, we will prove the following inversion formula: For every $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$,

$${\displaystyle u\left(y\right)=\frac{1}{C_{\phi }^M{\left|b\right|}^{2⟨k⟩+N}}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M\left(u\right)(r,t){\phi }_{r,t}^M\left(y\right)d{\mu }_k(r,t),a.e.}$$

(21)

Finally, for the DLCWT, we prove the following Calderón-type Reproducing Formula: If $\phi$ is an ${\mathfrak{L}}_k^2$-function, where its LCDT is in ${\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$ and satisfies (13), then for every $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and $0<ϵ<\delta <\infty$, the function

$${\displaystyle u^{ϵ,\delta }\left(y\right)=\frac{1}{C_{\phi }^M{\left(ib\right)}^{2⟨k⟩+N}}{\int }_{ϵ}^{\delta }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M(e^{-i\frac{-a}{b}{|.|}^2}u)(r,t)e^{\frac{ia}{b}{\left|t\right|}^2}{\mathrm{T}}_y^{M^{-1}}\left(\overline{{\theta }_r^M}\right)(-t){\gamma }_k\left(dt\right)\frac{dr}{r^{\frac{2⟨k⟩+N}{2}+1}}}$$

(22)

belongs to ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, such that

$$\underset{ϵ\to 0,\delta \to \infty }{lim}{∥u^{ϵ,\delta }-u∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}=0,$$

(23)

where the function ${\theta }_r^M$ is defined by

$${\mathfrak{F}}_D^M\left(\overline{{\theta }_r^M}\right)\left(\lambda \right)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{\left|\lambda \right|}^2}\overline{{\mathfrak{F}}_D^M\left({\phi }_r^M\right)\left(\lambda \right)}.$$

Daubechies [45] and Ramanathan–Topiwala [46] were the first to introduce and study time-frequency localization operators, then this notion was extended and generalized by many researchers in different settings [47,48,49,50,51,52]. Components of a signal can be located and extracted using these operators from its representation in the time-frequency plane [53]. They have been used in physics as anti-Wick operators, which are tools for quantization processes [54] and in the approximation of pseudo-differential operators [55].

In this paper, we will study the boundedness and compactness of localization operators ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$. In particular, we will prove that for any symbol $\varrho \in {\mathfrak{L}}_{{\mu }_k}^p(\Omega )$, $1⩽p⩽\infty$, ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ are bounded, with

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽{\left({\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}\right)}^{1/p^{\prime }}{\left({\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{\left|\varrho (r,z)\right|}^{p}d{\mu }_k(r,z)}\right)}^{1/p},$$

(24)

and belong to the Schatten class $S_p$, such that

$${\parallel {\mathfrak{T}}_{u,v}^M\left(\varrho \right)\parallel }_{S_p}⩽{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\left({\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{\left|\varrho (r,z)\right|}^{p}d{\mu }_k(r,z)}\right)}^{1/p}.$$

(25)

2. The LCDT and Its Properties

Throughout this paper $M:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is a matrix in $SL(2,\mathbb{R})$, such that $b\ne 0.$ Notice that its inverse $M^{-1}$ is given by $\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$ and belongs to $SL(2,\mathbb{R})$.

2.1. Linear Canonical Dunkl Transform (LCDT)

The LCDT was first defined and studied in [29], then generalized and improved in [38].

Definition 1. 

The LCDT of any function $u\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ is given by

$${\displaystyle {\mathfrak{F}}_D^M\left(u\right)\left(t\right)=\frac{1}{{\left(ib\right)}^{\frac{2⟨k⟩+N}{2}}}{\int }_{{\mathbb{R}}^N}{\mathfrak{J}}_k^M(t,z)u\left(z\right){\gamma }_k\left(dz\right),}$$

(26)

where

$${\mathfrak{J}}_k^M(t,z)=e^{{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{d}{b}}{\left|t\right|}^2+{\displaystyle \frac{a}{b}}{\left|z\right|}^2\right)}{E}_k\left(-i{\displaystyle \frac{t}{b}},z\right).$$

(27)

We denote by ${\Delta }_k^M$ the differential-difference operator given by

$${\Delta }_k^M:={\Delta }_k-i{\displaystyle \frac{d}{b}}\left(\sum _{j=1}^N{M}_j{T}_j-T_j{M}_j\right)-{\displaystyle \frac{d^2}{b^2}}{\left|t\right|}^2,$$

(28)

where $M_j\left(u\left(t\right)\right):=t_{j}u\left(t\right)$ and ${\displaystyle {\Delta }_k=\sum _{j=1}^N{T}_j^2}$ is the Dunkl-Laplacian operator.

Proposition 1. 

Let $u,v\in \mathcal{S}\left({\mathbb{R}}^N\right)$.

1. 

${\Delta }_k^M$ and ${\Delta }_k$ are connected by

$$e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}\circ {\Delta }_k^M\circ e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}={\Delta }_k.$$

(29)

2. 

We have

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\Delta }_k^{M}u\left(t\right)\overline{v\left(t\right)}{\gamma }_k\left(dt\right)={\int }_{{\mathbb{R}}^N}u\left(t\right)\overline{{\Delta }_k^{M}v\left(t\right)}{\gamma }_k\left(dt\right).}$$

(30)

3. 

For every $z\in {\mathbb{R}}^N$, the kernel ${\mathfrak{J}}_k^M(·,z)$ satisfies

$$\left\{\begin{array}{c}{\Delta }_k^M{\mathfrak{J}}_k^M(·,z)=-{\left|{\displaystyle \frac{z}{b}}\right|}^2{\mathfrak{J}}_k^M(·,z),\hfill \\ \\ {\mathfrak{J}}_k^M(0,z)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}{\left|z\right|}^2}.\hfill \end{array}\right.$$

(31)

4. 

For every $t,z\in {\mathbb{R}}^N$

$${\mathfrak{F}}_D^M\left({\left|z\right|}^{2}u\left(z\right)\right)=-b^2{\Delta }_k^M\left({\mathfrak{F}}_D^M\left(u\right)\right),$$

(32)

and

$${\left|t\right|}^2{\mathfrak{F}}_D^M\left(u\right)=-b^2{\mathfrak{F}}_D^M\left({\Delta }_k^{M^{-1}}\left(u\right)\right).$$

(33)

5. 

For every $t,z\in {\mathbb{R}}^N$

$$|{\mathfrak{J}}_k^M(t,z)|\le 1.$$

(34)

2.1.1. Specific Examples [29]

1.

Let $\tau \in \mathbb{R}$. If $M=\left(\begin{array}{cc}1& \tau \\ 0& 1\end{array}\right)$, then ${\mathfrak{F}}_D^M$ is the Fresnel transformation for the following Dunkl-type transformation:

$${\mathcal{W}}_k^{\tau }f\left(t\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{1}{{\left(i\tau \right)}^{\frac{2⟨k⟩+N}{2}}}}{\int }_{{\mathbb{R}}^N}{\mathfrak{E}}_k^{\tau }(t,z)f\left(z\right){\gamma }_k\left(dz\right),}\hfill & \tau \ne 0,\hfill \\ \\ f\left(t\right),\hfill & \tau =0,\hfill \end{array}\right.$$

where ${\mathfrak{E}}_k^{\tau }(t,z)=e^{{\displaystyle \frac{i}{2\tau }}{(|t|}^2+{\left|z\right|}^2)}{E}_k\left(-i{\displaystyle \frac{t}{\tau }},z\right).$

2.

If $M:=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$, then ${\mathfrak{F}}_D^M$ is exactly the Dunkl transformation.

3.

If $M=\left(\begin{array}{cc}cosh\left(s\right)& sinh\left(s\right)\\ sinh\left(s\right)& cosh\left(s\right)\end{array}\right)$, $s\in \mathbb{R},$ then ${\mathfrak{F}}_D^M$ is equal to the following transformation:

$${\mathcal{V}}_k^{s}f\left(t\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{1}{{(isinh\left(s\right))}^{\frac{2⟨k⟩+N}{2}}}}{\int }_{{\mathbb{R}}^N}{R}_k^s(t,z)f\left(z\right){\gamma }_k\left(dz\right),}\hfill & s\ne 0,\hfill \\ \\ f\left(t\right),\hfill & s=0,\hfill \end{array}\right.$$

where $R_k^s(t,z)=e^{{\displaystyle \frac{i}{2}}{coth\left(s\right)(|t|}^2+{\left|z\right|}^2)}{E}_k(-i{\displaystyle \frac{t}{sinh\left(s\right)}},z).$

4.

If $M=\left(\begin{array}{cc}cos\left(s\right)& -sin\left(s\right)\\ sin\left(s\right)& cos\left(s\right)\end{array}\right)$, $s\in \mathbb{R},$ then ${\mathfrak{F}}_D^M$ is the fractional Dunkl-type transform

$${\mathfrak{F}}_D^{s}f\left(t\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{e^{i\frac{2⟨k⟩+N}{2}\left((s-2n\pi )-\widehat{s}\pi /2\right)}}{{|sin\left(s\right)|}^{\frac{2⟨k⟩+N}{2}}}}{\int }_{{\mathbb{R}}^N}{\mathcal{K}}_k^s(t,z)f\left(z\right){\gamma }_k\left(dz\right),}\hfill & (2\mathcal{l}-1)\pi <s<(2\mathcal{l}+1)\pi .\hfill \\ f\left(t\right),\hfill & s=2\mathcal{l}\pi ,\hfill \\ f(-t),\hfill & s=(2\mathcal{l}+1)\pi ,\hfill \end{array}\right.$$

Here $\widehat{s}=\mathrm{sgn}(sin\left(s\right))$, where

$${\mathcal{K}}_k^s(t,z)=e^{-{\displaystyle \frac{i}{2}}{cot\left(s\right)(|t|}^2+{\left|z\right|}^2)}{E}_k\left(-i{\displaystyle \frac{t}{sin\left(s\right)}},z\right).$$

2.1.2. LCDT on ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, $1\le p\le 2$

We define the two operators ${\mathbf{L}}_s$ and $D_s$, $s\in \mathbb{R}$ by

$${\mathbf{L}}_{s}u\left(t\right)=e^{{\displaystyle \frac{is}{2}}{\left|t\right|}^2}u\left(t\right)\mathrm{and}{D}_{s}u\left(t\right)={\left|s\right|}^{-N/2-⟨k⟩}u(t/s),s\ne 0.$$

(35)

Then we have the following properties on ${\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$:

1.

We have

$$D_s\circ {\mathfrak{F}}_D={\mathfrak{F}}_D\circ D_{{\displaystyle \frac{1}{s}}}.$$

(36)

2.

We have

$$e^{i(⟨k⟩+N/2){\displaystyle \frac{\pi }{2}}\mathrm{sgn}\left(b\right)}{\mathfrak{F}}_D^M={\mathbf{L}}_{{\displaystyle \frac{b}{d}}}\circ D_b\circ {\mathfrak{F}}_D\circ {\mathbf{L}}_{{\displaystyle \frac{a}{b}}}.$$

(37)

Theorem 1

(Riemann–Lebesgue-type Inequality). For every $u\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$, its ${\mathfrak{F}}_D^M\left(u\right)$ is in $C_0\left({\mathbb{R}}^N\right)$ such that

$${\left|b\right|}^{⟨k⟩+N/2}\parallel {\mathfrak{F}}_D^M{\left(u\right)\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}\le {\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}.$$

(38)

Proposition 2

(Plancherel-type Formulas).

1. 

For every $u,v\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$, we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(u\right)\left(t\right)\overline{v\left(t\right)}{\gamma }_k\left(dt\right)={\int }_{{\mathbb{R}}^N}u\left(t\right)\overline{{\mathfrak{F}}_D^{M^{-1}}\left(v\right)\left(t\right)}{\gamma }_k\left(dt\right).}$$

(39)

2. 

If u belongs to ${\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, then ${\mathfrak{F}}_D^M\left(u\right)$ belongs to ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ such that

$$\parallel {\mathfrak{F}}_D^M{\left(u\right)\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}={\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(40)

3. 

The LCDT has a unique extension to an isometric isomorphism on ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, which is still denoted by ${\mathfrak{F}}_D^M$.

4. 

For any $u,v\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$,

$${⟨{\mathfrak{F}}_D^M\left(v\right),u⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}={⟨v,{\mathfrak{F}}_D^{M^{-1}}u⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(41)

5. 

For every $u\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ with ${\mathfrak{F}}_D^M\left(u\right)\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right),$

$$\left({\mathfrak{F}}_D^M\circ {\mathfrak{F}}_D^{M^{-1}}\right)\left(u\right)=\left({\mathfrak{F}}_D^{M^{-1}}\circ {\mathfrak{F}}_D^M\right)\left(u\right)=u,a.e.$$

(42)

Definition 2. 

For $1\le p\le 2$, the LCDT is defined on ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ by

$${\mathfrak{F}}_D^M=e^{-i(⟨k⟩+N/2){\displaystyle \frac{\pi }{2}}\mathrm{sgn}\left(b\right)}\left({\mathbf{L}}_{{\displaystyle \frac{d}{b}}}\circ D_b\circ {\mathfrak{F}}_D\circ {\mathbf{L}}_{{\displaystyle \frac{a}{b}}}\right),$$

where ${\mathfrak{F}}_D$ is the Dunkl transform on ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$.

Proposition 3. 

For $1\le p\le 2$, the LCDT extends to a linear bounded operator on ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ withg

$${\left|b\right|}^{\left(1-2/p^{\prime }\right)\left(N/2+⟨k⟩\right)}\parallel {\mathfrak{F}}_D^M\left(u\right){\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}\le {\parallel u\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}.$$

(43)

The last inequality is a Young-type relation for the Dunkl transform.

2.2. Generalized Convolution Product for the LCDT

In this paragraph, we define the convolution product associated with the LCDT and give its properties, see [38].

Definition 3. 

The generalized translation operator associated with the operator ${\Delta }_k^M$ is given by:

$$\begin{array}{c}\hfill {\mathrm{T}}_y^{M}u\left(z\right)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{(|y|}^2+{\left|z\right|}^2)}{\tau }_y\left(e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}u\right)\left(z\right).\end{array}$$

(44)

Then we have the following properties.

Proposition 4. 

Let $x,y,z\in {\mathbb{R}}^N$.

1. 

${\mathrm{T}}_0^M=\mathrm{Id}$ and ${\mathrm{T}}_y^{M}u\left(z\right)={\mathrm{T}}_z^{M}u\left(y\right).$

2. 

The translation product formula is given by:

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{T}}_y^M\left[{\mathfrak{J}}_k^M(·,x)\right]\left(z\right)=e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}{\left|x\right|}^2}{\mathfrak{J}}_k^M(y,x){\mathfrak{J}}_k^M(z,x).\hfill \end{array}\end{array}$$

(45)

3. 

The translation operator ${\mathrm{T}}_y^M$ is continuous from ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ into itself, from $C_{b,\mathrm{rad}}\left({\mathbb{R}}^N\right)$ into itself, and on $L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$. That is, for $u\in L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, we have

$${∥{\mathrm{T}}_y^{M}u∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\le {∥u∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}.$$

(46)

Moreover, if $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, then

$${∥{\mathrm{T}}_y^{M}u∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\le {∥u∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(47)

4. 

If $W={\mathbb{Z}}_2^N$, then for any $u\in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$,

$${∥{\mathrm{T}}_y^{M}u∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\le 2^{{\displaystyle \frac{N}{2p}}|p-2|}{∥u∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}.$$

(48)

5. 

For all $u\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right),$ (resp. ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$), we have

$${\mathfrak{F}}_D^M\left({\mathrm{T}}_y^{M^{-1}}u\right)(·)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}\overline{{\mathfrak{J}}_k^M(y,·)}{\mathfrak{F}}_D^M\left(u\right)(·).$$

(49)

6. 

For every $u\in L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right),p\in (1,2],$

$${\mathfrak{F}}_D^M\left({\mathrm{T}}_y^{M^{-1}}u\right)(·)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}\overline{{\mathfrak{J}}_k^M(y,·)}{\mathfrak{F}}_D^M\left(u\right)(·),a.e.$$

(50)

Corollary 1. 

For every $y,z\in {\mathbb{R}}^N$ and $u\in \mathcal{S}\left({\mathbb{R}}^N\right)$ we have:

$${\displaystyle {\mathrm{T}}_y^{M^{-1}}u\left(z\right)={(-ib)}^{-N/2-⟨k⟩}{e}^{-\frac{i}{2}\frac{a}{b}{\left|z\right|}^2}{\int }_{{\mathbb{R}}^N}{E}_k(it/b,z)\overline{{\mathfrak{J}}_k^M(y,t)}{\mathfrak{F}}_D^M\left(u\right)\left(t\right){\gamma }_k\left(dt\right).}$$

(51)

Definition 4. 

The generalized convolution product associated with ${\mathfrak{F}}_D^M$ of two suitable functions u and v on ${\mathbb{R}}^N$, is the function $u\underset{M}{*}v$ defined by:

$${\displaystyle u\underset{M}{*}v\left(y\right)={\int }_{{\mathbb{R}}^N}{\mathrm{T}}_y^{M}u(-t)e^{-i\frac{d}{b}{\left|t\right|}^2}v\left(t\right){\gamma }_k\left(dt\right).}$$

(52)

Then we have the following properties:

1.

$u\underset{M}{*}v=v\underset{M}{*}u.$

2.

${\mathrm{T}}_y^M\left(u\underset{M}{*}v\right)=\left({\mathrm{T}}_y^{M}u\right)\underset{M}{*}v=u\underset{M}{*}\left({\mathrm{T}}_y^{M}v\right).$

Proposition 5

(Young-type Relation). Let $p,q,r\in [1,\infty ]$ such that $1/p+1/q=1+1/r.$ If $W={\mathbb{Z}}_2^N$, then for every $u\in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ and $v\in {\mathfrak{L}}_k^q\left({\mathbb{R}}^N\right)$, we have $u\underset{M}{*}v\in {\mathfrak{L}}_k^r\left({\mathbb{R}}^N\right)$ and

$${∥u\underset{M}{*}v∥}_{{\mathfrak{L}}_k^r\left({\mathbb{R}}^N\right)}\le 2^{{\displaystyle \frac{N}{2p}}|p-2|}{∥u∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}{∥v∥}_{{\mathfrak{L}}_k^q\left({\mathbb{R}}^N\right)}.$$

(53)

In addition, if $u\in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ and $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^q\left({\mathbb{R}}^N\right)$, then

$${∥u\underset{M}{*}v∥}_{{\mathfrak{L}}_k^r\left({\mathbb{R}}^N\right)}\le {∥u∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}{∥v∥}_{{\mathfrak{L}}_k^q\left({\mathbb{R}}^N\right)}.$$

(54)

Proposition 6.

1. 

For $v\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ and $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$, we have

$${\left(ib\right)}^{-N/2-⟨k⟩}{\mathfrak{F}}_D^M\left(u\underset{M^{-1}}{*}v\right)(·)=e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}{\mathfrak{F}}_D^M\left(u\right)(·){\mathfrak{F}}_D^M\left(v\right)(·).$$

(55)

2. 

If $u\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$ and $v\in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right),p\in (1,2]$, then

$${\left(ib\right)}^{-N/2-⟨k⟩}{\mathfrak{F}}_D^M\left(u\underset{M^{-1}}{*}v\right)(·)=e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}{\mathfrak{F}}_D^M\left(u\right)(·){\mathfrak{F}}_D^M\left(v\right)(·),a.e.$$

(56)

3. 

For $u,v,w\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$, we have

$$\left(u\underset{M}{*}v\right)\underset{M}{*}w=u\underset{M}{*}\left(v\underset{M}{*}w\right).$$

(57)

4. 

It is not necessary for the functions to be radials for the previous three results to hold if $W={\mathbb{Z}}_2^N$.

5. 

For all $u,v\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, we have

$${\int }_{{\mathbb{R}}^N}{\left|\left(u\underset{M^{-1}}{*}v\right)\left(t\right)\right|}^2{\gamma }_k\left(dt\right)={\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^M{\left(u\right)\left(\lambda \right)|}^2{|{\mathfrak{F}}_D^M\left(v\right)\left(\lambda \right)|}^2{\gamma }_k\left(d\lambda \right).$$

(58)

3. The DLCWT

We denote by $\Omega :=\{(r,t)\in {\mathbb{R}}^{N+1}:r>0\}$ and ${\mathfrak{L}}_{{\mu }_k}^p(\Omega ),$$1⩽p⩽\infty$, denotes the space of functions on $\Omega$ such that

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )}& :=& {\left({\displaystyle {\int }_{\Omega }{\left|f(r,t)\right|}^{p}d{\mu }_k(r,t)}\right)}^{1/p}<\infty ,\hfill \\ \hfill {\parallel f\parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}& :=& \begin{array}{c}esssup\\ (r,t)\in \Omega \end{array}\left|f\right(r,t\left)\right|<\infty ,\hfill \end{array}$$

where $d{\mu }_k(r,t)={\displaystyle \frac{{\gamma }_k\left(dt\right)dr}{r^{2⟨k⟩+N+1}}}$.

In this section, we will introduce the generalized linear canonical wavelet transform associated with the operator ${\Delta }_k^M$, and we give some of its properties.

Definition 5. 

Let $\phi \in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ or $\phi \in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, $p\ne 2$. We say that φ is an admissible Dunkl linear canonical wavelet if for almost $y\in {\mathbb{R}}^N$

$${\displaystyle 0<C_{\phi }^M:={\int }_0^{\infty }|{\mathfrak{F}}_D^M\left(\phi \right)\left(\lambda y\right){|}^2\frac{d\lambda }{\lambda }<\infty .}$$

(59)

Notice that, in the case of the reflection group $W={\mathbb{Z}}_2^N$, we can take in the previous definition $\phi \in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, $p\in [1,\infty ]$.

Example 1. 

The function $g_s$, $s>0$ defined on ${\mathbb{R}}^N$ by

$$g_s\left(y\right)={\displaystyle \frac{1}{{\left(2t\right)}^{N/2+⟨k⟩}}}{e}^{-{\displaystyle \frac{ia}{2b}}{\left|y\right|}^2-{\displaystyle \frac{{\left|y\right|}^2}{4s}}}$$

(60)

satisfies

$${\mathfrak{F}}_D^M\left(g_s\right)\left(\lambda \right)=L_{{\displaystyle \frac{d}{b}}}\left(\lambda \right)e^{{\displaystyle \frac{id}{2b}}{\left|\lambda \right|}^2-{\displaystyle \frac{s}{b^2}}{\left|\lambda \right|}^2}.$$

The function $\phi \left(y\right)=-{\displaystyle \frac{d}{ds}}{g}_s\left(y\right)$ is an admissible Dunkl linear canonical wavelet on ${\mathbb{R}}^N$ and we have $C_{\phi }^M={\displaystyle \frac{1}{8s}}$.

For $r>0$, $t\in {\mathbb{R}}^N$ and $\phi$ a suitable function, we define the family ${\phi }_{r,t}^M$ by

$$\forall y\in {\mathbb{R}}^N,{\phi }_{r,t}^M\left(y\right)=r^{{\displaystyle \frac{2⟨k⟩+N}{2}}}\overline{{\mathrm{T}}_t^{M^{-1}}\left({\phi }_r^M\right)(-y)},$$

(61)

where

$${\phi }_r^M\left(y\right):={\displaystyle \frac{1}{r^{2⟨k⟩+N}}}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}(1-1/r^2){\left|y\right|}^2}\phi (y/r).$$

(62)

Proposition 7. 

Let $(r,t)\in \Omega$ and $\lambda \in {\mathbb{R}}^N$.

1. 

For all $\phi \in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$,

$${∥{\phi }_{r,t}^M∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\le {\parallel \phi \parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(63)

2. 

For every $\phi \in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\cup {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$,

$${\mathfrak{F}}_D^M\left({\phi }_r^M\right)\left(\lambda \right)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}(1-1/r^2){\left|\lambda \right|}^2}{\mathfrak{F}}_D^M\left(\phi \right)\left(r\lambda \right).$$

(64)

3. 

For all $\phi \in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\cup {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$

$${\mathfrak{F}}_D^M\left(\overline{{\phi }_{r,t}^M}\right)\left(\lambda \right)=r^{⟨k⟩+N/2}{e}^{{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{d}{b}}(1-1/r^2){\left|\lambda \right|}^2-{\displaystyle \frac{a}{b}}{\left|t\right|}^2\right)}{E}_k(-t,\lambda /b){\mathfrak{F}}_D^M\left(\phi \right)\left(r\lambda \right).$$

(65)

4. 

For all $\phi \in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, $p\in [1,\infty ]$

$${∥{\phi }_{r,t}^M∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\le r^{(2⟨k⟩+N)({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}})}{\parallel \phi \parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}.$$

(66)

5. 

If $W={\mathbb{Z}}_2^N$, then for all $\phi \in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, $p\in [1,\infty ]$

$${∥{\phi }_{r,t}^M∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\le 2^{{\displaystyle \frac{N}{2p}}|p-2|}{r}^{(2⟨k⟩+N)({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}})}{\parallel \phi \parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}.$$

(67)

Proof. 

The first assertion is proved by using (47), (61) and (62). Involving the relations (26) and (62), we infer (64). Using (49), (61) and (64) we derive (65). The relation (66) is proved by using (46), (61) and (62). Similarly, involving (48), (61) and (62), we derive (67). □

Definition 6. 

Let φ be an admissible Dunkl linear canonical wavelet. Then the Dunkl linear canonical wavelet transform (DLCWT) of any regular function $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ is denoted by ${{\rm Y}}_{\phi }^M\left(u\right)$ and is defined as

$${\displaystyle {{\rm Y}}_{\phi }^M\left(u\right)(r,t):={\int }_{{\mathbb{R}}^N}u\left(y\right)\overline{{\phi }_{r,t}^M\left(y\right)}{\gamma }_k\left(dy\right),(r,t)\in \Omega ,}$$

(68)

where ${\phi }_{r,t}^M$ is given by (61).

The last formula can also be written as

$${{\rm Y}}_{\phi }^M\left(u\right)(r,t)=r^{{\displaystyle \frac{2⟨k⟩+N}{2}}}{\mathbf{L}}_{-{\displaystyle \frac{2a}{b}}}u\underset{M^{-1}}{*}{\phi }_r^M\left(t\right).$$

(69)

The DLCWT satisfies the following properties:

1.

By (63), (68) and Cauchy–Schwartz’s inequality, we have for any $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$

$$\parallel {{\rm Y}}_{\phi }^M{\left(u\right)\parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}\le {\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(70)

2.

If $\phi \in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right),1\le p\le \infty ,$ then by (66), (68) and Hölder’s inequality, we have for all $u\in {\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)$

$$\left|{{\rm Y}}_{\phi }^M\left(u\right)(r,t)\right|\le r^{(2⟨k⟩+N)({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}})}{\parallel u\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)},\forall (r,t)\in \Omega .$$

(71)

3.

If $W={\mathbb{Z}}_2^N$, as above for any $\phi \in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right),1\le p\le \infty ,$ and for all $u\in {\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)$

$$\left|{{\rm Y}}_{\phi }^M\left(u\right)(r,t)\right|\le 2^{{\displaystyle \frac{N}{2p}}|p-2|}{r}^{(2⟨k⟩+N)({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2}})}{\parallel u\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}{∥\phi ∥}_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)},\forall (r,t)\in \Omega .$$

(72)

4.

Let $\phi \in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$. Then by (56) and (69), we have

$${\mathfrak{F}}_D^M\left({{\rm Y}}_{\phi }^M\left(u\right)(r,·)\right)\left(\lambda \right)={\left(irb\right)}^{⟨k⟩+N/2}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{\left|\lambda \right|}^2}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}u\right)\left(\lambda \right){\mathfrak{F}}_D^M\left({\phi }_r^M\right)\left(\lambda \right).$$

(73)

In the remainder of this section, $\phi \in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ will be an admissible linear canonical Dunkl wavelet.

Theorem 2

(Plancherel-type formula). For all $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, we have

$${\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}|{{\rm Y}}_{\phi }^M{\left(u\right)(r,t)|}^{2}d{\mu }_k(r,t)={\left|b\right|}^{2⟨k⟩+N}{C}_{\phi }^M{\int }_{{\mathbb{R}}^N}{\left|u\left(y\right)\right|}^2{\gamma }_k\left(dy\right).}$$

(74)

Proof. 

From (56), (58), (64), (73) and Fubini’s theorem we have

$$\begin{array}{ccc}& & {\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{\left|{{\rm Y}}_{\phi }^M\left(u\right)(r,t)\right|}^{2}d{\mu }_k(r,t)}\hfill \\ & & {\displaystyle ={\left|b\right|}^{2⟨k⟩+N}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}u\right)\left(\lambda \right){|}^2{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(\lambda r\right)\right|}^2{\gamma }_k\left(d\lambda \right)\frac{dr}{r}}\hfill \\ & & {\displaystyle ={\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}u\right)\left(\lambda \right)\right|}^2\left({\int }_0^{\infty }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(\lambda r\right)\right|}^2\frac{dr}{r}\right){\gamma }_k\left(d\lambda \right).}\hfill \end{array}$$

Using the identity (59) and Parseval’s formula (40), we derive

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}|{{\rm Y}}_{\phi }^M{\left(u\right)(r,t)|}^{2}d{\mu }_k(r,t)={\left|b\right|}^{2⟨k⟩+N}{C}_{\phi }^M{∥{\mathbf{L}}_{\frac{-2a}{b}}u∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}=C_{\phi }^M{\left|b\right|}^{2⟨k⟩+N}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.}\end{array}$$

As desired. □

Involving (70), (74), and the Riesz–Thorin interpolation theorem, we derive the following proposition.

Proposition 8. 

For $u\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and $2\le p\le \infty$, we have

$${∥{{\rm Y}}_{\phi }^M\left(u\right)∥}_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )}\le {\left({\left|b\right|}^{2⟨k⟩+N}{C}_{\phi }^M\right)}^{{\displaystyle \frac{1}{p}}}{\parallel \phi \parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^{1-2/p}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(75)

Theorem 3

(Orthogonality Property). For all $f_1,f_2\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}{f}_1\left(y\right)\overline{f_2\left(y\right)}{\gamma }_k\left(dy\right)=\frac{1}{C_{\phi }^M{\left|b\right|}^{2⟨k⟩+N}}{\int }_{\Omega }{{\rm Y}}_{\phi }^M\left(f_1\right)(r,t)\overline{{{\rm Y}}_{\phi }^M\left(f_2\right)(r,t)}d{\mu }_k(r,t).}$$

(76)

Proof. 

Plancherel-type formula for DLCWT implies that for all $f\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, the function

$$I\left(r\right):={\left({\int }_{{\mathbb{R}}^N}{\left|{{\rm Y}}_{\phi }^M\left(f\right)(r,t)\right|}^2{\gamma }_k\left(dt\right)\right)}^{1/2}$$

is in ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$. Consequently, ${{\rm Y}}_{\phi }^M\left(f\right)(r,·)\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ for almost all $r\in {\mathbb{R}}_{+}$. In regard to convolution and Parseval’s formula applied to the variable t, we obtain

$$\begin{array}{ccc}& & {\int }_{\Omega }{{\rm Y}}_{\phi }^M\left(f_1\right)(r,t)\overline{{{\rm Y}}_{\phi }^M\left(f_2\right)(r,t)}d{\mu }_k(r,t)\hfill \\ & & ={\int }_{\Omega }{r}^{2⟨k⟩+N}\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\underset{M^{-1}}{*}{\phi }_r^M\right)\left(t\right)\overline{\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\underset{M^{-1}}{*}{\phi }_r^M\right)\left(t\right)}d{\mu }_k(r,t)\hfill \\ & & ={\left|b\right|}^{2⟨k⟩+N}{\int }_{\Omega }{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\right)\left(\lambda \right)}{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\lambda \right)\right|}^2{\gamma }_k\left(d\lambda \right){\displaystyle \frac{dr}{r}}.\hfill \end{array}$$

In view of Fubini’s theorem, we can justify the interchange of the integral

$$\begin{array}{ccc}& & {\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\right)\left(\lambda \right)}\left({\displaystyle {\int }_0^{\infty }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\lambda \right)\right|}^2\frac{dr}{r}}\right){\gamma }_k\left(d\lambda \right)\hfill \\ & & ={\left|b\right|}^{2⟨k⟩+N}{C}_{\phi }^M{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_1\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{f}_2\right)\left(\lambda \right)}{\gamma }_k\left(d\lambda \right).\hfill \end{array}$$

The conclusion follows due to Parseval’s formula for LCDT. □

Now will provide a weak inversion formula for the DLCWT.

Theorem 4

(Inversion Formula). The inversion formula states

$${\displaystyle \forall f\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right),f\left(y\right)=\frac{1}{C_{\phi }^M{\left|b\right|}^{N+2⟨k⟩}}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M\left(f\right)(r,t){\phi }_{r,t}^M\left(y\right)d{\mu }_k(r,t).}$$

(77)

Proof. 

Let $\phi \in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ be an admissible linear canonical Dunkl wavelet and $f\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$. We have

$$\begin{array}{cc}\hfill {\displaystyle {\int }_0^{\infty }({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}}& {{\rm Y}}_{\phi }^M\left(f\right)(r,t){\phi }_{r,t}^M\left(y\right)d{\mu }_k(r,t))\overline{g\left(y\right)}{\gamma }_k\left(dy\right)\hfill \\ & {\displaystyle ={\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M\left(f\right)(r,t)\left({\displaystyle {\int }_{{\mathbb{R}}^N}\overline{g\left(y\right)\overline{{\phi }_{r,t}^M\left(y\right)}}{\gamma }_k\left(dy\right)}\right)d{\mu }_k(r,t)}\hfill \\ & {\displaystyle ={\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M\left(f\right)(r,t)\overline{{{\rm Y}}_{\phi }^M\left(g\right)(r,t)}d{\mu }_k(r,t)}\hfill \\ & {\displaystyle =C_{\phi }^M{\left|b\right|}^{N+2⟨k⟩}{\int }_{{\mathbb{R}}^N}f\left(y\right)\overline{g\left(y\right)}{\gamma }_k\left(dy\right).}\hfill \end{array}$$

Then the result follows. □

Equation (59) can be generalized as follows.

Definition 7. 

Let u and v be in ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$. We say that the pair $(u,v)$ is an admissible Dunkl linear canonical two-wavelets on ${\mathbb{R}}^N$ if

$${\displaystyle C_{u,v}^M:={\int }_0^{\infty }{\mathfrak{F}}_D^M\left(u\right)\left(\lambda y\right)\overline{{\mathfrak{F}}_D^M\left(v\right)\left(\lambda y\right)}\frac{d\lambda }{\lambda }}$$

(78)

is constant for almost every $y\in {\mathbb{R}}^N$.

Theorem 5. 

Let $(u,v)$ be an admissible Dunkl linear canonical two-wavelets. Then, for any $f,g\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, we have

$${\displaystyle {\int }_{\Omega }{{\rm Y}}_u^M\left(f\right)(r,t)\overline{{{\rm Y}}_v^M\left(g\right)(r,t)}d{\mu }_k(r,t)=C_{u,v}^M{\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}f\left(y\right)\overline{g\left(y\right)}{\gamma }_k\left(dy\right).}$$

(79)

Proof. 

By using Parseval’s formula (41), Relations (56) and (69), we have

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_{\Omega }{{\rm Y}}_u^M\left(f\right)(r,t)\overline{{{\rm Y}}_v^M\left(g\right)(r,t)}d{\mu }_k(r,t)}\hfill \\ \hfill & {\displaystyle ={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{r}^{2⟨k⟩+N}\left({\mathbf{L}}_{\frac{-2a}{b}}f\underset{M^{-1}}{*}{u}_r^M\right)\left(t\right)\overline{\left({\mathbf{L}}_{\frac{-2a}{b}}g\underset{M^{-1}}{*}{v}_r^M\right)}\left(t\right)d{\mu }_k(r,t)}\hfill \\ \hfill & {\displaystyle ={\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}f\right)\left(\xi \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}g\right)}\left(\xi \right){\mathfrak{F}}_D^M\left(u\right)\left(r\lambda \right)\overline{{\mathfrak{F}}_D^M\left(v\right)\left(r\lambda \right)}{\gamma }_k\left(d\lambda \right)\frac{dr}{r}.}\hfill \end{array}$$

By Fubini’s theorem and Relation (78), we observe that

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_{\Omega }{{\rm Y}}_u^M\left(f\right)(r,t)\overline{{{\rm Y}}_v^M\left(g\right)(r,t)}d{\mu }_k(r,t)}\hfill \\ \hfill & {\displaystyle ={\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}f\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}g\right)\left(\lambda \right)}\left({\displaystyle {\int }_0^{\infty }{\mathfrak{F}}_D^M\left(u\right)\left(r\lambda \right)\overline{{\mathfrak{F}}_D^M\left(v\right)\left(r\lambda \right)}\frac{dr}{r}}\right){\gamma }_k\left(d\lambda \right)}\hfill \\ \hfill & {\displaystyle =C_{u,v}^M{\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}f\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}g\right)\left(\lambda \right)}{\gamma }_k\left(d\lambda \right).}\hfill \end{array}$$

Finally, Parseval’s formula (41) yields the required result. □

Theorem 6

(Calderón’s Reproducing Formula). Let $\phi \in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ be an admissible Dunkl linear canonical wavelet such that ${\mathfrak{F}}_D^M\left(\phi \right)$ belongs to ${\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$. Then, for any f in ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and $0<ϵ<\delta <\infty$, the function

$${\displaystyle f^{ϵ,\delta }\left(y\right)=\frac{1}{C_{\phi }^M{\left(ib\right)}^{2⟨k⟩+N}}{\int }_{ϵ}^{\delta }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M\left({\mathbf{L}}_{\frac{2a}{b}}f\right)(r,t)e^{\frac{ia}{b}{\left|t\right|}^2}{T}_y^{M^{-1}}\left(\overline{{\theta }_r^M}\right)(-t){\gamma }_k\left(dt\right)\frac{dr}{r^{\frac{2⟨k⟩+N}{2}+1}},y\in {\mathbb{R}}^N,}$$

(80)

belongs to ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, and satisfies

$$\underset{ϵ\to 0,\delta \to \infty }{lim}{∥f^{ϵ,\delta }-f∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}=0,$$

(81)

where the function ${\theta }_r^M$ is defined by

$${\mathfrak{F}}_D^M\left(\overline{{\theta }_r^M}\right)\left(\lambda \right)=e^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{\left|\lambda \right|}^2}\overline{{\mathfrak{F}}_D^M\left({\phi }_r^M\right)\left(\lambda \right)}.$$

The following lemmas are required in order to prove Theorem 6.

Lemma 1. 

Let $K_{ϵ,\delta }$ be the function defined by

$${\displaystyle K_{ϵ,\delta }\left(\xi \right)=\frac{1}{C_{\phi }^M}{\int }_{ϵ}^{\delta }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\xi \right)\right|}^2\frac{dr}{r},\xi \in {\mathbb{R}}^N,}$$

(82)

then under the assumptions of Theorem 6, we have

$$0<K_{ϵ,\delta }\left(\xi \right)\le 1,a.e.\xi \in {\mathbb{R}}^N,$$

(83)

and

$$\underset{ϵ\to 0,\delta \to \infty }{lim}{K}_{ϵ,\delta }\left(\xi \right)=1.$$

(84)

Proof. 

As $\phi$ is an admissible Dunkl linear canonical wavelet, then from (59)

$${\displaystyle {\int }_{ϵ}^{\delta }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\xi \right)\right|}^2\frac{dr}{r}\le C_{\phi }^M,a.e.\xi \in {\mathbb{R}}^N.}$$

Therefore

$$0<K_{ϵ,\delta }\left(\xi \right)\le 1,a.e.\xi \in {\mathbb{R}}^N.$$

(85)

Finally, using the fact that

$${\displaystyle C_{\phi }^M=\underset{ϵ\to 0,\delta \to \infty }{lim}{\int }_{ϵ}^{\delta }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\xi \right)\right|}^2\frac{dr}{r}}$$

(86)

we derive the result. □

Lemma 2. 

The function $f^{ϵ,\delta }$ defined by (80) belongs to ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and satisfies

$${\mathfrak{F}}_D^M\left(f^{ϵ,\delta }\right)\left(\xi \right)={\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)K_{ϵ,\delta }\left(\xi \right),\xi \in {\mathbb{R}}^N.$$

(87)

Proof. 

According to (52) and (69), the function $f^{ϵ,\delta }$ can also be stated as

$${\displaystyle f^{ϵ,\delta }\left(y\right)=\frac{1}{C_{\phi }^M{\left(ib\right)}^{2⟨k⟩+N}}{\int }_{ϵ}^{\delta }\left(f\underset{M^{-1}}{*}{\phi }_r^M\right)\underset{M^{-1}}{*}\overline{{\theta }_r^M}\left(y\right)\frac{dr}{r}.}$$

(88)

Then

$${\displaystyle {\left|f^{ϵ,\delta }\left(y\right)\right|}^2\le {\displaystyle \frac{1}{{\left(C_{\phi }^M\right)}^2{\left|b\right|}^{4⟨k⟩+2N}}}\left({\displaystyle {\int }_{ϵ}^{\delta }\frac{dr}{r}}\right){\int }_{ϵ}^{\delta }{\left|\left(f\underset{M^{-1}}{*}{\phi }_r^M\right)\underset{M^{-1}}{*}\overline{{\theta }_r^M}\left(y\right)\right|}^2\frac{dr}{r}.}$$

Therefore

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\left|f^{ϵ,\delta }\left(y\right)\right|}^2{\gamma }_k\left(dy\right)\le {\displaystyle \frac{1}{{\left(C_{\phi }^M\right)}^2{\left|b\right|}^{4⟨k⟩+2N}}}\left({\displaystyle {\int }_{ϵ}^{\delta }\frac{dr}{r}}\right){\int }_{ϵ}^{\delta }\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|\left(f\underset{M^{-1}}{*}{\phi }_r^M\right)\underset{M^{-1}}{*}\overline{{\theta }_r^M}\left(y\right)\right|}^2{\gamma }_k\left(dy\right)}\right)\frac{dr}{r}.}$$

From Plancherel’s formula (40), (56) and (58), we deduce that

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\left|f^{ϵ,\delta }\left(y\right)\right|}^2{\gamma }_k\left(dy\right)\le {\displaystyle \frac{1}{{\left(C_{\phi }^M\right)}^2}}\left({\displaystyle {\int }_{ϵ}^{\delta }\frac{dr}{r}}\right){\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\right|}^2\left({\displaystyle {\int }_{ϵ}^{\delta }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\xi \right)\right|}^4\frac{dr}{r}}\right){\gamma }_k\left(d\xi \right).}$$

Using (59) and (38) we have

$${\displaystyle {\int }_{ϵ}^{\delta }|{\mathfrak{F}}_D^M{\left(\phi \right)\left(r\xi \right)|}^4\frac{dr}{r}\le C_{\phi }^M{\parallel {\mathfrak{F}}_D^M\left(\phi \right)\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}^2.}$$

Therefore

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\left|f^{ϵ,\delta }\left(y\right)\right|}^2{\gamma }_k\left(dy\right)\le {\displaystyle \frac{1}{C_{\phi }^M}}\left({\displaystyle {\int }_{ϵ}^{\delta }\frac{dr}{r}}\right){∥{\mathfrak{F}}_D^M\left(\phi \right)∥}_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}^2{\parallel {\mathfrak{F}}_D^M\left(f\right)\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^2.}$$

Additionally (40) gives

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\left|f^{ϵ,\delta }\left(y\right)\right|}^2{\gamma }_k\left(dy\right)\le {\displaystyle \frac{1}{C_{\phi }^M}}\left({\displaystyle {\int }_{ϵ}^{\delta }\frac{dr}{r}}\right)\parallel {\mathfrak{F}}_D^M{\left(\phi \right)\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^2<\infty .}$$

Thus, we conclude that $f^{ϵ,\delta }\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right).$

On the other hand, let $\psi$ be in $\mathcal{S}\left({\mathbb{R}}^N\right)$. Then by Equation (88),

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{f}^{ϵ,\delta }\left(y\right)\overline{{\left({\mathfrak{F}}_D^M\right)}^{-1}\left(\psi \right)\left(y\right)}{\gamma }_k\left(dy\right)}\hfill \\ \hfill & {\displaystyle ={\int }_{{\mathbb{R}}^N}\left({\displaystyle \frac{1}{C_{\phi }^M{\left(ib\right)}^{2⟨k⟩+N}}{\int }_{ϵ}^{\delta }\left(f\underset{M^{-1}}{*}{\phi }_r^M\right)\underset{M^{-1}}{*}\overline{{\theta }_r^M}\left(y\right)\frac{dr}{r}}\right)\overline{{\left({\mathfrak{F}}_D^M\right)}^{-1}\left(\psi \right)\left(y\right)}{\gamma }_k\left(dy\right).}\hfill \end{array}$$

(89)

The second member of the relation (89) can also be written in the form

$${\displaystyle \frac{1}{C_{\phi }^M{\left(ib\right)}^{2⟨k⟩+N}}{\int }_{ϵ}^{\delta }\left({\displaystyle {\int }_{{\mathbb{R}}^N}\left(f\underset{M^{-1}}{*}{\phi }_r^M\right)\underset{M^{-1}}{*}\overline{{\theta }_r^M}\left(y\right)\overline{{\left({\mathfrak{F}}_D^M\right)}^{-1}\left(\psi \right)\left(y\right)}{\gamma }_k\left(dy\right)}\right)\frac{dr}{r}.}$$

(90)

By (41) and (56), Relation (90) becomes

$${\displaystyle {\displaystyle \frac{1}{C_{\phi }^M}}{\int }_{ϵ}^{\delta }\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right){\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\xi \right)\right|}^2\overline{\psi \left(\xi \right)}{\gamma }_k\left(d\xi \right)}\right)\frac{dr}{r}.}$$

Then

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\left({\displaystyle {\displaystyle \frac{1}{C_{\phi }^M}}{\int }_{ϵ}^{\delta }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\xi \right)\right|}^2\frac{dr}{r}}\right)\overline{\psi \left(\xi \right)}{\gamma }_k\left(d\xi \right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)K_{ϵ,\delta }\left(\xi \right)\overline{\psi \left(\xi \right)}{\gamma }_k\left(d\xi \right).}\hfill \end{array}$$

(91)

However, by (41), the first member of (89) is equal to

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left(f^{ϵ,\delta }\right)\left(\xi \right)\overline{\psi \left(\xi \right)}{\gamma }_k\left(d\xi \right).}$$

(92)

By (91) and (92), we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}\left({\mathfrak{F}}_D^M\left(f^{ϵ,\delta }\right)\left(\xi \right)-{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)K_{ϵ,\delta }\left(\xi \right)\right)\overline{\psi \left(\xi \right)}{\gamma }_k\left(d\xi \right)=0.}$$

Then

$${\mathfrak{F}}_D^M\left(f^{ϵ,\delta }\right)\left(\xi \right)={\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)K_{ϵ,\delta }\left(\xi \right),\xi \in {\mathbb{R}}^N.$$

 □

Proof of Theorem 6. 

By Lemma 2 and Equation (40),

$$\begin{array}{ccc}\hfill {∥f^{ϵ,\delta }-f∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^2& =& {\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M(f^{ϵ,\delta }-f)\left(\xi \right)\right|}^2{\gamma }_k\left(d\xi \right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\left(K_{ϵ,\delta }\left(\xi \right)-1\right)\right|}^2{\gamma }_k\left(d\xi \right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\right|}^2{\left|1-K_{ϵ,\delta }\left(\xi \right)\right|}^2{\gamma }_k\left(d\xi \right).}\hfill \end{array}$$

Furthermore, Lemma 6 implies that

$$\underset{ϵ\to 0,\delta \to \infty }{lim}{\left|{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\right|}^2{\left|1-K_{ϵ,\delta }\left(\xi \right)\right|}^2=0,$$

and

$${\left|{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\right|}^2{\left|1-K_{ϵ,\delta }\left(\xi \right)\right|}^2\le C{\left|{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\right|}^2,$$

with ${\left|{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\right|}^2$ in ${\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$. □

Proposition 9

(An Inversion Formula for ${{\rm Y}}_{\phi }^M$). For all $f\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ (resp. $f\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ ) such that ${\mathfrak{F}}_D^M\left(f\right)\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ (resp. ${\mathfrak{F}}_D^M\left(f\right)\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$),

$${\displaystyle f\left(y\right)=\frac{1}{C_{\phi }^M{\left|b\right|}^{2⟨k⟩+N}}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M\left({\mathbf{L}}_{\frac{2a}{b}}f\right)(r,t)\overline{{\psi }_{r,y}^M\left(t\right)}d{\mu }_k(r,t),a.e,}$$

(93)

where the function ${\psi }_{r,y}^M$ is defined by

$$\overline{{\mathfrak{F}}_D^M\left({\psi }_{r,y}^M\right)\left(\lambda \right)}=r^{{\displaystyle \frac{2⟨k⟩+N}{2}}}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{\left|\lambda \right|}^2}\overline{{\mathfrak{J}}_k^M(y,\lambda )}\overline{{\mathfrak{F}}_D^M\left({\phi }_r^M\right)\left(\lambda \right)}.$$

Proof. 

Let

$${\displaystyle \mathfrak{I}(r,y)={\int }_{{\mathbb{R}}^N}{{\rm Y}}_{\phi }^M\left({\mathbf{L}}_{\frac{2a}{b}}f\right)(r,t)\overline{{\psi }_{r,y}^M\left(t\right)}{\gamma }_k\left(dt\right)}$$

(94)

and let

$${\displaystyle \mathfrak{J}\left(y\right)=\frac{1}{C_{\phi }^M{\left|b\right|}^{2⟨k⟩+N}}{\int }_0^{\infty }\mathfrak{I}(r,y)\frac{dr}{r^{2⟨k⟩+N+1}}.}$$

(95)

We will prove (93), in the case of $f\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ and ${\mathfrak{F}}_D^M\left(f\right)\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$. Using Parseval’s formula, Equations (39), (50) and (56), we have

$${\displaystyle \mathfrak{I}(r,y)=r^{2⟨k⟩+N}{\left(ib\right)}^{\frac{2⟨k⟩+N}{2}}{\int }_{{\mathbb{R}}^N}\overline{{\mathfrak{J}}_k^M(y,\lambda )}{\mathfrak{F}}_D^M\left(f\right)\left(\lambda \right){\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\lambda \right)\right|}^2{\gamma }_k\left(d\lambda \right).}$$

(96)

Then for all $y\in {\mathbb{R}}^N,$

$${\displaystyle \mathfrak{J}\left(y\right)=\frac{1}{C_{\phi }^M{(-ib)}^{\frac{2⟨k⟩+N}{2}}}{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}\overline{{\mathfrak{J}}_k^M(y,\lambda )}{\mathfrak{F}}_D^M\left(f\right)\left(\lambda \right){\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\lambda \right)\right|}^2{\gamma }_k\left(d\lambda \right)\frac{dr}{r}.}$$

(97)

Thus, by Fubini’s theorem, we obtain

$$\begin{array}{ccc}\hfill \mathfrak{J}\left(y\right)& =& {\displaystyle \frac{1}{C_{\phi }^M{(-ib)}^{\frac{2⟨k⟩+N}{2}}}{\int }_{{\mathbb{R}}^N}\overline{{\mathfrak{J}}_k^M(y,\lambda )}{\mathfrak{F}}_D^M\left(f\right)\left(\lambda \right)\left({\displaystyle {\int }_0^{\infty }{\left|{\mathfrak{F}}_D^M\left(\phi \right)\left(r\lambda \right)\right|}^2\frac{dr}{r}}\right){\gamma }_k\left(d\lambda \right)}\hfill \\ & =& {\displaystyle \frac{1}{{(-ib)}^{\frac{2⟨k⟩+N}{2}}}{\int }_{{\mathbb{R}}^N}\overline{{\mathfrak{J}}_k^M(y,\lambda )}{\mathfrak{F}}_D^M\left(f\right)\left(\lambda \right){\gamma }_k\left(d\lambda \right).}\hfill \end{array}$$

Involving the inversion Formula (42), we obtain the desired relation.

On the other hand. if $f\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and ${\mathfrak{F}}_D^M\left(f\right)\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$, we have $f\underset{M^{-1}}{*}{\phi }_r^M$ belongs to ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and

$${\mathfrak{F}}_D^M\left(f\underset{M^{-1}}{*}{\phi }_r^M\right)(·)={\left(ib\right)}^{{\displaystyle \frac{2⟨k⟩+N}{2}}}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{d}{b}}{|·|}^2}{\mathfrak{F}}_D^M\left(f\right)(·){\mathfrak{F}}_D^M\left({\phi }_r^M\right)(·).$$

Next, using a similar argument as in the first case, we obtain the result. □

4. Localization Operators (LO) for the DLCWT

Let $K\in B\left({\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\right)$ be a compact operator. Then its singular values ${\left\{s_j\left(K\right)\right\}}_{j\in \mathbb{N}}$ are the eigenvalues of the positive self-adjoint operator $\left|K\right|=\sqrt{K^{*}K}$.

The set of all compact operators whose singular values are in ${\mathfrak{L}}^p\left(\mathbb{N}\right)$ is known as the Schatten class $S_p$, $1\le p<\infty$, and are equipped with the norm

$${\parallel K\parallel }_{S_p}:={\left(\sum _{j=1}^{\infty }{\left(s_j\left(K\right)\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

(98)

We define $S_{\infty }:=B\left({\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\right)$, with its norm,

$${\parallel K\parallel }_{S_{\infty }}:=\underset{{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}=1}{sup}{\parallel Kv\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(99)

If K is an operator in $S_1$, then its trace is defined by

$$tr\left(K\right)=\sum _{j=1}^{\infty }{⟨K{u}_j,u_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}$$

(100)

for any orthonormal basis ${\left(u_j\right)}_j$ of ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and if K is nonnegative then

$$tr\left(K\right)={\parallel K\parallel }_{S_1}.$$

(101)

Furthermore, if the positive operator $K^{*}K$ is in $S_1$, for a compact operator $K\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, then K is Hilbert–Schmidt, such that

$${\parallel K\parallel }_{HS}^2:={\parallel K\parallel }_{S_2}^2=\parallel K^{*}{K\parallel }_{S_1}=tr\left(K^{*}K\right)=\sum _{j=1}^{\infty }{\parallel K{u}_j\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^2$$

(102)

for any orthonormal basis $\left(u_j\right)$ of ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$.

Definition 8. 

Let ϱ be a measurable function on Ω and let $u,v$ be two admissible Dunkl linear canonical wavelets. Then for $p\in [1,\infty ]$, the (two-wavelet) LO associated to the DLCWT is defined on ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ by

$${\displaystyle {\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f\right)\left(x\right)={\int }_{\Omega }\varrho (r,z){{\rm Y}}_u^M\left(f\right)(r,z)v_{r,z}^M\left(x\right)d{\mu }_k(r,z),x\in {\mathbb{R}}^N.}$$

(103)

The definition of (103) is frequently easier to understand in a weak sense: More precisely for $1\le p\le \infty$, if $f_1\in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ and $f_2\in {\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)$,

$$\begin{array}{c}\hfill {\displaystyle {⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_1\right),f_2⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}={\int }_{\Omega }\varrho (r,z){{\rm Y}}_u^M\left(f_1\right)(r,z)\overline{{{\rm Y}}_v^M\left(f_2\right)(r,z)}d{\mu }_k(r,z).}\end{array}$$

(104)

For $1\le p<\infty$, the adjoint of ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ is ${\mathfrak{T}}_{v,u}^M\left(\overline{\varrho }\right):{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)$, that is,

$${\left({\mathfrak{T}}_{u,v}^M\left(\varrho \right)\right)}^{*}={\mathfrak{T}}_{v,u}^M\left(\overline{\varrho }\right).$$

(105)

In this section, u and v are two admissible Dunkl linear canonical wavelets in ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ such that ${\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}={\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}=1.$

4.1. Boundedness of LO

In this subsection we will prove that, for all $\varrho \in {\mathfrak{L}}_{{\mu }_k}^p(\Omega )$, $1\le p\le \infty$, the LO ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ is bounded. To do this, we first consider the problem for $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$ and then for $\varrho \in {\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )$, and the conclusion follows by interpolation theory.

Proposition 10. 

For $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, the LO ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)\in S_{\infty }$, such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

(106)

Proof. 

If $f_1,f_2\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, then by (70) and (104),

$$\begin{array}{cc}\hfill \left|{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_1\right),f_2⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|& {\displaystyle ⩽{\int }_{\Omega }\left|\varrho (r,z)\right|\left|{{\rm Y}}_u^M\left(f_1\right)(r,z)\right|\left|\overline{{{\rm Y}}_v^M\left(f_2\right)(r,z)}\right|d{\mu }_k(r,z)}\hfill \\ \hfill & ⩽{∥{{\rm Y}}_u^M\left(f_1\right)∥}_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}{∥{{\rm Y}}_v^M\left(f_2\right)∥}_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}{∥\varrho ∥}_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}\hfill \\ \hfill & ⩽\parallel f_1{\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\parallel f_2{\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.\hfill \end{array}$$

Thus,

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

 □

Proposition 11. 

For $\varrho \in {\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )$, the LO ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)\in S_{\infty }$, such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽{\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}.$$

Proof. 

If $f_1,f_2\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, then by Hölder’s inequality

$$\begin{array}{cc}\hfill \left|{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_1\right),f_2⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|& {\displaystyle ⩽{\int }_{\Omega }\left|\varrho (r,z)\right|\left|{{\rm Y}}_u^M\left(f_1\right)(r,z)\right|\left|\overline{{{\rm Y}}_v^M\left(f_2\right)(r,z)}\right|d{\mu }_k(r,z)}\hfill \\ \hfill & ⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}{∥{{\rm Y}}_u^M\left(f_1\right)∥}_{{\mathfrak{L}}_{{\mu }_k}^2(\Omega )}{∥{{\rm Y}}_v^M\left(f_2\right)∥}_{{\mathfrak{L}}_{{\mu }_k}^2(\Omega )}.\hfill \end{array}$$

From (74), we obtain

$$\begin{array}{c}\hfill \left|{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_1\right),f_2⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|⩽\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\parallel f_1{\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\parallel f_2{\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}.\end{array}$$

Thus,

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽{\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}.$$

 □

Consequently for each $\varrho \in {\mathfrak{L}}_{{\mu }_k}^p(\Omega )$, $1\le p\le \infty$, the LO ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ is well defined and belongs to $S_{\infty }$.

Theorem 7. 

Let $1\le p\le \infty$ and let $\varrho \in {\mathfrak{L}}_{{\mu }_k}^p(\Omega )$. Then there is a unique operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ in $S_{\infty }$, satisfying

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽{\left({\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}\right)}^{1/p^{\prime }}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )}.$$

(107)

Proof. 

For $f\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ let ${\mathcal{T}}_M:{\mathfrak{L}}_{{\mu }_k}^1(\Omega )\cap {\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ the function defined by ${\mathcal{T}}_M\left(\varrho \right):={\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f\right).$ Then, from the previous two propositions, we have

$$\parallel {\mathcal{T}}_M{\left(\varrho \right)\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}$$

(108)

and

$$\parallel {\mathcal{T}}_M{\left(\varrho \right)\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\le \sqrt{{\left|b\right|}^{2⟨k⟩+N}{C}_u^M{C}_v^M}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(109)

Thus, using interpolation theory (greensee ([56], Theorem 2.11), ${\mathcal{T}}_M$ may be uniquely extended to an operator on ${\mathfrak{L}}_{{\mu }_k}^p(\Omega )$, with

$$\parallel {\mathcal{T}}_M{\left(\varrho \right)\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\le {\left({\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}\right)}^{1/p^{\prime }}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )}{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(110)

As desired. □

4.2. Schatten Class Properties of LO

In this subsection we will show that the LO

$${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$$

belongs to the Schatten class $S_p$.

Proposition 12. 

Let ϱ be in ${\mathfrak{L}}_{{\mu }_k}^1(\Omega )$. Then the localization operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is in $S_2$ and we have

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_2}⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

(111)

Proof. 

Given that ${\left\{h_j\right\}}_{j\in \mathbb{N}}$ an orthonormal basis of ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$. Next, using Parseval’s identity, (68), (104) and (105), we obtain

$$\begin{array}{ccc}{\displaystyle \sum _{j=1}^{\infty }{\parallel {\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right)\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^2}\hfill & =\hfill & {\displaystyle \sum _{j=1}^{\infty }{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right),{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right)⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}}\hfill \\ & =\hfill & {\displaystyle \sum _{j=1}^{\infty }{\int }_{\Omega }\varrho (r,z){⟨h_j,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\overline{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right),v_{r,z}^M⟩}}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z)}\hfill \\ & =\hfill & {\displaystyle {\int }_{\Omega }\varrho (r,z)\sum _{j=1}^{\infty }{⟨h_j,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{⟨{\left({\mathfrak{T}}_{u,v}^M\left(\varrho \right)\right)}^{*}\left(v_{r,z}^M\right),h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z)}\hfill \\ & =\hfill & {\displaystyle {\int }_{\Omega }\varrho (r,z){⟨{\left({\mathfrak{T}}_{u,v}^M\left(\varrho \right)\right)}^{*}{v}_{r,z}^M,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z).}\hfill \end{array}$$

Thus, from (63), (105) and (106) we obtain

$${\displaystyle \sum _{j=1}^{\infty }{∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right)∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^2\le {\int }_{\Omega }\left|\varrho (r,z)\right|{∥{\left({\mathfrak{T}}_{u,v}^M\left(\varrho \right)\right)}^{*}∥}_{S_{\infty }}d{\mu }_k(r,z)\le {\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}^2<\infty .}$$

(112)

Hence by (112) and by Proposition 2.8 in the book [56], the operator

$${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$$

is in $S_2$ and hence compact. □

Proposition 13. 

Let $1⩽p<\infty$. If ϱ is in ${\mathfrak{L}}_{{\mu }_k}^p(\Omega ),$ then ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is compact.

Proof. 

Assume that $\varrho \in {\mathfrak{L}}_{{\mu }_k}^p(\Omega )$. Let ${\left({\varrho }_j\right)}_{j\in \mathbb{N}}$ be a sequence in ${\mathfrak{L}}_{{\mu }_k}^1(\Omega )\cap {\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )$ such that ${\varrho }_j\to \varrho$ in ${\mathfrak{L}}_{{\mu }_k}^p(\Omega )$. Then from Theorem 7

$${∥{\mathfrak{T}}_{u,v}^M\left({\varrho }_j\right)-{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}\le {\left({\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{∥{\varrho }_j-\varrho ∥}_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )}.$$

Therefore ${\mathfrak{T}}_{u,v}^M\left({\varrho }_j\right)\to {\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ in $S_{\infty }$ as $j\to \infty$. Moreover, since ${\mathfrak{T}}_{u,v}^M\left({\varrho }_j\right)$ is in $S_2$ hence compact, it follows that ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is compact. □

Theorem 8. 

If $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, then ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ belongs to $S_1$, such that

$$\parallel \tilde{\varrho }{\parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}⩽\parallel {\mathfrak{T}}_{u,v}^M{\left(\varrho \right)\parallel }_{S_1}⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )},$$

(113)

where $\tilde{\varrho }$ is given by

$$\forall (r,z)\in \Omega ,\tilde{\varrho }(r,z)={⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)u_{r,z}^M,v_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(114)

Proof. 

Since $\varrho$ is in ${\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, by Proposition 12, ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is in $S_2$, then from the canonical form for compact operators given in ([56], Theorem 2.2), there is an orthonormal basis ${\left\{h_j\right\}}_{j\in \mathbb{N}}$ for the orthogonal complement of the kernel of the operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$, consisting of eigenvectors of $|{\mathfrak{T}}_{u,v}^M\left(\varrho \right)|$ and ${\left\{{\psi }_j\right\}}_{j\in \mathbb{N}}$ an orthonormal set in ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, satisfying

$$\begin{array}{c}\hfill {\displaystyle {\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f\right)=\sum _{j=1}^{\infty }{\alpha }_j{⟨f,h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\psi }_j,}\end{array}$$

(115)

where ${\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ are the nonnegative singular values of ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ related to $h_j$. Then

$$\begin{array}{c}\hfill {\displaystyle {∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_1}=\sum _{j=1}^{\infty }{\alpha }_j=\sum _{j=1}^{\infty }{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right),{\psi }_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.}\end{array}$$

Therefore from Bessel inequality and Equations (63) and (68), we have

$$\begin{array}{ccc}\hfill {∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_1}& =& {\displaystyle \sum _{j=1}^{\infty }{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right),{\psi }_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}}\hfill \\ & =& {\displaystyle \sum _{j=1}^{\infty }{\int }_{\Omega }\varrho (r,z){{\rm Y}}_u^M\left(h_j\right)(r,z)\overline{{{\rm Y}}_v^M\left({\psi }_j\right)(r,z)}d{\mu }_k(r,z)}\hfill \\ & ⩽& {\displaystyle {\int }_{\Omega }\left|\varrho (r,z)\right|{\left(\sum _{j=1}^{\infty }{\left|{{\rm Y}}_u^M\left(h_j\right)(r,z)\right|}^2\right)}^{\frac{1}{2}}{\left(\sum _{j=1}^{\infty }{\left|{{\rm Y}}_v^M\left({\psi }_j\right)(r,z)\right|}^2\right)}^{\frac{1}{2}}d{\mu }_k(r,z)}\hfill \\ & ⩽& {\displaystyle {\int }_{\Omega }\left|\varrho (r,z)\right|\parallel u_{r,z}^M{\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v_{r,z}^M\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z)}\hfill \\ & ⩽& {\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.\hfill \end{array}$$

Thus

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_1}⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

Since $\tilde{\varrho }\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$, then by (115), we have

$$\begin{array}{ccc}\hfill |\tilde{\varrho }(r,z)|& =& \left|{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(u_{r,z}^M\right),v_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|\hfill \\ & =& \left|\sum _{j=1}^{\infty }{\alpha }_j{⟨u_{r,z}^M,h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{⟨h_j,v_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|\hfill \\ & ⩽& {\displaystyle \frac{1}{2}}\sum _{j=1}^{\infty }{\alpha }_j\left({\left|{⟨u_{r,z}^M,h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|}^2+{\left|{⟨v_{r,z}^M,h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|}^2\right).\hfill \end{array}$$

Then, from Fubini’s theorem, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle {\int }_{\Omega }\left|\tilde{\varrho }(r,z)\right|d{\mu }_k(r,z)}& \le & {\displaystyle \frac{1}{2}\sum _{j=1}^{\infty }{\alpha }_j({\int }_{\Omega }{\left|{⟨u_{r,z}^M,h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|}^{2}d{\mu }_k(r,z)}\hfill \\ & +& {\displaystyle {\int }_{\Omega }{\left|{⟨v_{r,z}^M,h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|}^{2}d{\mu }_k(r,z)).}\hfill \end{array}$$

Thus ${\displaystyle {\int }_{\Omega }\left|\tilde{\varrho }(r,z)\right|d{\mu }_k(r,z)\le \sum _{j=1}^{\infty }{\alpha }_j={∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_1}.}$ □

Corollary 2. 

For ϱ in ${\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, we have the following trace formula

$${\displaystyle tr\left({\mathfrak{T}}_{u,v}^M\left(\varrho \right)\right)={\int }_{\Omega }\varrho (r,z){⟨v_{r,z}^M,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z).}$$

(116)

Proof. 

Given ${\left\{h_j\right\}}_{j\in \mathbb{N}}$ an orthonormal basis of ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$. We have from Theorem 8, that ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ belongs to $S_1$. Therefore by (100) and Parseval’s relation,

$$\begin{array}{ccc}\hfill tr\left({\mathfrak{T}}_{u,v}^M\left(\varrho \right)\right)& =& {\displaystyle \sum _{j=1}^{\infty }{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(h_j\right),h_j⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}}\hfill \\ & =& {\displaystyle \sum _{j=1}^{\infty }{\int }_{\Omega }\varrho (r,z){⟨h_j,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\overline{⟨h_j,v_{r,z}^M⟩}}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z)}\hfill \\ & =& {\displaystyle {\int }_{\Omega }\varrho (r,z)\sum _{j=1}^{\infty }{⟨h_j,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\overline{⟨h_j,v_{r,z}^M⟩}}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z)}\hfill \\ & =& {\displaystyle {\int }_{\Omega }\varrho (r,z){⟨v_{r,z}^M,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z)}\hfill \\ & =& {\displaystyle {\int }_{\Omega }\varrho (r,z){⟨v_{r,z}^M,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}d{\mu }_k(r,z).}\hfill \end{array}$$

 □

Corollary 3. 

Let $1⩽p⩽\infty$. If $\varrho \in {\mathfrak{L}}_{{\mu }_k}^p(\Omega )$, then ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ belongs to $S_p$, such that

$$\parallel {\mathfrak{T}}_{u,v}^M{\left(\varrho \right)\parallel }_{S_p}⩽{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )}.$$

(117)

Proof. 

This is the consequence of interpolating (see ([56], Theorem 2.10 and Theorem 2.11)), Proposition 11, and Theorem 8. □

Remark 1. 

If $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$ is positive and real valued and if $u=v$, then

$${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$$

is nonnegative. Thus from Corollary 2 and Equation (101),

$${\displaystyle {∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_1}={\int }_{\Omega }\varrho (r,z){∥u_{r,z}^M∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}^{2}d{\mu }_k(r,z).}$$

(118)

Now we state a result concerning the trace of products of localization operators.

Corollary 4. 

Let ${\varrho }_1$ and ${\varrho }_2$ be any real-valued and non-negative functions in ${\mathfrak{L}}_{{\mu }_k}^1(\Omega )$. We assume that $u=v$ such that ${\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}=1$. Then the localization operators ${\mathfrak{T}}_{u,v}^M\left({\varrho }_1\right)$ and ${\mathfrak{T}}_{u,v}^M\left({\varrho }_2\right)$ are positive and trace class operators, such that

$$\begin{array}{ccc}\hfill {∥{\left({\mathfrak{T}}_{u,v}^M\left({\varrho }_1\right){\mathfrak{T}}_{u,v}^M\left({\varrho }_2\right)\right)}^n∥}_{S_1}& =& tr{\left({\mathfrak{T}}_{u,v}^M\left({\varrho }_1\right){\mathfrak{T}}_{u,v}^M\left({\varrho }_2\right)\right)}^n\hfill \\ & \le & {\left(tr\left({\mathfrak{T}}_{u,v}^M\left({\varrho }_1\right)\right)\right)}^n{\left(tr\left({\mathfrak{T}}_{u,v}^M\left({\varrho }_2\right)\right)\right)}^n\hfill \\ & =& {∥{\mathfrak{T}}_{u,v}^M\left({\varrho }_1\right)∥}_{S_1}^n{∥{\mathfrak{T}}_{u,v}^M\left({\varrho }_2\right)∥}_{S_1}^n,\hfill \end{array}$$

for any natural number n.

Proof. 

By Theorem 1 in the paper [57], we know that if the operators A and B are in the trace class $S_1$ and are positive, then

$$\forall n\in \mathbb{N},tr{\left(AB\right)}^n\le {\left(tr\left(A\right)\right)}^n{\left(tr\left(B\right)\right)}^n.$$

(119)

Hence, if we take $A={\mathfrak{T}}_{u,v}^M\left({\varrho }_1\right)$ and $B={\mathfrak{T}}_{u,v}^M\left({\varrho }_2\right)$, then the result follows by Remark 1. □

5. ${\mathfrak{L}}_{\mathbf{k}}^{\mathbf{p}}$-Boundedness and ${\mathfrak{L}}_{\mathbf{k}}^{\mathbf{p}}$-Compactness of LO

For $1\le p\le \infty$, let $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^{p^{\prime }}\left({\mathbb{R}}^N\right)$ and $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$ be two admissible Dunkl linear canonical wavelets.

5.1. ${\mathfrak{L}}_k^p$ Boundedness

In this subsection we will prove that ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is bounded from ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ to itself.

Proposition 14. 

Let $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$ and $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^{\infty }\left({\mathbb{R}}^N\right)$. For $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, the localization operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ is bounded, with

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\right)}⩽{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}{\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}.$$

(120)

Proof. 

Let $f\in {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$. Then by (66), (71) and (103),

$$\begin{array}{ccc}\hfill \parallel {\mathfrak{T}}_{u,v}^M{\left(\varrho \right)\left(f\right)\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}& ⩽& {\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_{\Omega }\left|\varrho (r,z)\right|\left|{{\rm Y}}_u^M\left(f\right)(r,z)\right|\left|v_{r,z}^M\left(y\right)\right|d{\mu }_k(r,z){\gamma }_k\left(dy\right)}\hfill \\ & ⩽& {\parallel f\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.\hfill \end{array}$$

Thus,

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\right)}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

 □

Proposition 15. 

Let $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^{\infty }\left({\mathbb{R}}^N\right)$ and $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$. Then for $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, the localization operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$ is bounded and we have

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)\right)}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

(121)

Proof. 

From (66), (71) and (103), we have for every $f\in {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$

$$\begin{array}{ccc}\hfill \forall y\in {\mathbb{R}}^N,\left|{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f\right)\left(y\right)\right|& \le & {\displaystyle {\int }_{\Omega }\left|\varrho (r,z)\right|\left|{{\rm Y}}_u^M\left(f\right)(r,z)\right|\left|v_{r,z}^M\left(y\right)\right|d{\mu }_k(r,z)}\hfill \\ & \le & {\parallel f\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.\hfill \end{array}$$

Thus,

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)\right)}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

 □

Remark 2. 

Proposition 15 is also a corollary of Proposition 14, since ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$ is the adjoint of ${\mathfrak{T}}_{v,u}^M\left(\overline{\varrho }\right):{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$.

Using an interpolation of Propositions 14 and 15, we obtain the following result.

Theorem 9. 

Let u and v be functions in ${\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$. Then for all ϱ in ${\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, there exists a unique linear bounded operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, $1\le p\le \infty ,$ satisfying

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}\le \parallel u{\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}\parallel v{\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}^{{\displaystyle \frac{1}{p}}}\parallel u{\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}^{{\displaystyle \frac{1}{p}}}\parallel v{\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}\parallel \varrho {\parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

(122)

Another variant of the ${\mathfrak{L}}^p$-boundedness can be provided. To do this, we start by improving Proposition 15.

Proposition 16. 

Let $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^{p^{\prime }}\left({\mathbb{R}}^N\right)$ and $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, $1<p\le \infty$. Then for $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, the localization operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ is bounded such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

(123)

Proof. 

For $f_1\in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, let ${\mathcal{I}}_{f_1}^M:{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)\to \mathbb{C}$ the function defined by ${\mathcal{I}}_{f_1}^M\left(f_2\right)={⟨f_2,{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_1\right)⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}$. Then from (71) and (104)

$$\begin{array}{ccc}\hfill \left|{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_1\right),f_2⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|& \le & {\displaystyle {\int }_{\Omega }\left|\varrho (r,z)\right|\left|{{\rm Y}}_u^M\left(f_1\right)(r,z)\right|\left|\overline{{{\rm Y}}_v^M\left(f_2\right)(r,z)}\right|d{\mu }_k(r,z)}\hfill \\ & \le & {\parallel u\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\parallel f_1{\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\parallel f_2{\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.\hfill \end{array}$$

Then ${\mathcal{I}}_{f_1}^M$ is continuous on ${\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)$, and we have

$${∥{\mathcal{I}}_{f_1}^M∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}\parallel f_1{\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

Since ${\mathcal{I}}_{f_1}^M\left(f_2\right)={⟨f_2,{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_1\right)⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}$, then by Riesz’s representation theorem, we conclude the desired result. □

The following result is obtained by using the Propositions 14 and 16.

Theorem 10. 

Let $1\le p\le \infty$ and let $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^{p^{\prime }}\left({\mathbb{R}}^N\right)$, $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$. Then for $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, the LO ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ is bounded, such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

(124)

With Schur’s technique, we can obtain an ${\mathfrak{L}}_k^p$-boundedness result as in the previous Theorem, but the estimate for the norm ${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}$ is weaker than that of (124).

Proposition 17. 

Let ϱ be in ${\mathfrak{L}}_{{\mu }_k}^1(\Omega )$ and let $u,v\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$. Then there exists a unique linear bounded operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, $1\le p\le \infty$ such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽max\left({\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{,\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\right){\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

Proof. 

Define ${\mathcal{S}}_k^M$ on ${\mathbb{R}}^N\times {\mathbb{R}}^N$ by

$${\displaystyle {\mathcal{S}}_k^M(y,w)={\int }_{\Omega }\varrho (r,z)\overline{u_{r,z}^M\left(w\right)}{v}_{r,z}^M\left(y\right)d{\mu }_k(r,z).}$$

(125)

Then we have

$${\displaystyle {\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f\right)\left(y\right)={\int }_{{\mathbb{R}}^N}{\mathcal{S}}_k^M(y,w)f\left(w\right){\gamma }_k\left(dw\right).}$$

By simple calculations, it is easy to see that

$${\displaystyle \forall y\in {\mathbb{R}}^N,{\int }_{{\mathbb{R}}^N}|{\mathcal{S}}_k^M(y,·)|{\gamma }_k\left(dy\right)\le {\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )},}$$

and

$${\displaystyle \forall w\in {\mathbb{R}}^N,{\int }_{{\mathbb{R}}^N}|{\mathcal{S}}_k^M(·,w)|{\gamma }_k\left(dw\right)\le {\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.}$$

Thus by Schur’s Lemma [58], the linear operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ is bounded for any $1\le p\le \infty$, and we have

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽max\left({\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)},{\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\right){\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}.$$

 □

Theorem 11. 

Let ϱ be in ${\mathfrak{L}}_{{\mu }_k}^s(\Omega )$, $s\in [1,2]$, and let $u,v\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$ be two admissible linear canonical wavelets. Then for all $p\in [s,s^{\prime }]$, there is a unique linear bounded operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right),$ satisfying

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\mathfrak{C}}_1^t{\mathfrak{C}}_2^{1-t}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )},$$

(126)

where

$$\begin{array}{ccc}{\mathfrak{C}}_1\hfill & =\hfill & {\left({\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{2}{s}}-1}{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}},\hfill \\ {\mathfrak{C}}_2\hfill & =\hfill & {\left({\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{2}{s}}-1}{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}},\hfill \end{array}$$

and

$${\displaystyle \frac{t}{s}}+{\displaystyle \frac{1-t}{s^{\prime }}}={\displaystyle \frac{1}{p}}.$$

Proof. 

Let ${\mathcal{I}}^M:\left({\mathfrak{L}}_{{\mu }_k}^1(\Omega )\cap {\mathfrak{L}}_{{\mu }_k}^2(\Omega )\right)\times \left({\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\right)$ defined by ${\mathcal{I}}^M(\varrho ,f)={\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f\right)$. Then from Theorem 7 and Proposition 14, we obtain

$${∥{\mathcal{I}}^M(\varrho ,f)∥}_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel f\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}$$

(127)

and

$${∥{\mathcal{I}}^M(\varrho ,f)∥}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\le {\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{2}}}{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^2(\Omega )}.$$

(128)

Therefore, by (127), (128) and the the multi-linear interpolation theory (see Section 10.1 in [59]), we obtain a unique linear bounded operator

$${\mathcal{I}}^M(\varrho ,f):{\mathfrak{L}}_{{\mu }_k}^s(\Omega )\times {\mathfrak{L}}_k^s\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^s\left({\mathbb{R}}^N\right)$$

that satisfies

$${∥{\mathcal{I}}^M(\varrho ,f)∥}_{{\mathfrak{L}}_k^s\left({\mathbb{R}}^N\right)}\le {\mathfrak{C}}_1{\parallel f\parallel }_{{\mathfrak{L}}_k^s\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )},$$

(129)

where

$${\mathfrak{C}}_1={\left({\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\right)}^{\theta }{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1-\theta }{2}}}$$

and

$${\displaystyle \frac{\theta }{1}}+{\displaystyle \frac{1-\theta }{2}}={\displaystyle \frac{1}{s}}.$$

By the definition of ${\mathcal{I}}^M$, we have

$$\begin{array}{ccc}\hfill & & {∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^s\left({\mathbb{R}}^N\right)\right)}\le {\left({\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{2}{s}}-1}\hfill \\ & \times & {\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )}.\hfill \end{array}$$

(130)

Since ${\mathfrak{T}}_{v,u}^M\left(\overline{\varrho }\right)$ is the adjoint of ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$, then the linear operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is bounded on ${\mathfrak{L}}_k^{s^{\prime }}\left({\mathbb{R}}^N\right)$ such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^{s^{\prime }}\left({\mathbb{R}}^N\right)\right)}={∥{\mathfrak{T}}_{\overline{v},\overline{u}}^M\left(\overline{\varrho }\right)∥}_{B\left({\mathfrak{L}}_k^s\left({\mathbb{R}}^N\right)\right)}\le {\mathfrak{C}}_2{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )},$$

(131)

where

$${\mathfrak{C}}_2={\left({\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{2}{s}}-1}{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}}.$$

Using an interpolation of (130) and (131), we have for any $p\in [s,s^{\prime }]$,

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\mathfrak{C}}_1^t{\mathfrak{C}}_2^{1-t}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^p(\Omega )},$$

with ${\displaystyle \frac{t}{s}}+{\displaystyle \frac{1-t}{s^{\prime }}}={\displaystyle \frac{1}{p}}.$ □

Theorem 12. 

Let $\varrho \in {\mathfrak{L}}_{{\mu }_k}^s(\Omega )$, $s\in [1,\infty ]$, and let $u,v\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$. Then there exists a unique linear bounded operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, $p\in \left[{\displaystyle \frac{2s}{s+1}},{\displaystyle \frac{2s}{s-1}}\right]$, such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\mathfrak{C}}_3^{{\displaystyle \frac{t}{s}}}{\mathfrak{C}}_4^{{\displaystyle \frac{1-t}{s}}}{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )},$$

(132)

where

$${\mathfrak{C}}_3={\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)},{\mathfrak{C}}_4={\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)},$$

and $t={\displaystyle \frac{s+1}{2}}-{\displaystyle \frac{s}{p}}.$

To prove this theorem, we need the following lemmas.

Lemma 3. 

Let $\varrho \in {\mathfrak{L}}_{{\mu }_k}^s(\Omega )$, $s\in [1,\infty ]$ and let $u\in {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$, $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$. Then there is a unique linear bounded operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s+1}}}\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s+1}}}\left({\mathbb{R}}^N\right)$ satisfying

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s+1}}}\left({\mathbb{R}}^N\right)\right)}⩽{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}}{\left({\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{s}}}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )}.$$

(133)

Proof. 

Consider ${\mathcal{I}}_M:{\mathfrak{L}}_{{\mu }_k}^1(\Omega )\cap {\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )\to B\left({\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\right)\cap B\left({\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\right)$ be the function defined by ${\mathcal{I}}_M\left(\varrho \right)={\mathfrak{T}}_{u,v}^M\left(\varrho \right)$. Then by Proposition 14 and Theorem 7

$$\parallel {\mathcal{I}}_M{\parallel }_{B({\mathfrak{L}}_{{\mu }_k}^1(\Omega ),B\left({\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)\right))}⩽{\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}$$

(134)

and

$$\parallel {\mathcal{I}}_M{\parallel }_{B\left({\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega ),B\left({\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\right)\right)}⩽\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}.$$

(135)

Interpolating (134) with (135) we obtain the desired result. □

Lemma 4. 

Let $\varrho \in {\mathfrak{L}}_{{\mu }_k}^s(\Omega )$, $s\in [1,\infty ]$ and let $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, $v\in {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$. Then there is a unique operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s-1}}}\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s-1}}}\left({\mathbb{R}}^N\right)$ satisfying

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s-1}}}\left({\mathbb{R}}^N\right)\right)}⩽{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}}{\left({\parallel u\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{s}}}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )}.$$

(136)

Proof. 

Since ${\mathfrak{T}}_{v,u}^M\left(\overline{\varrho }\right):{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s+1}}}\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s+1}}}\left({\mathbb{R}}^N\right)$ is the adjoint of ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{2-1}}}\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^{{\displaystyle \frac{2s}{s-1}}}\left({\mathbb{R}}^N\right)$, then we have the desired result by using Lemma 3 and a duality argument. □

Prof of Theorem 12. 

Interpolating Equations (133) and (136), we obtain for every $p\in \left[{\displaystyle \frac{2s}{s+1}},{\displaystyle \frac{2s}{s-1}}\right]$,

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\mathfrak{C}}_3^{{\displaystyle \frac{t}{s}}}{\mathfrak{C}}_4^{{\displaystyle \frac{1-t}{s}}}{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )},$$

with

$$t={\displaystyle \frac{s+1}{2}}-{\displaystyle \frac{s}{p}}.$$

 □

Proposition 18. 

Let $p,s\in [1,\infty ]$ be such that $p\in \left[{\displaystyle \frac{2s}{s+1}},2\right]$. Let $\varrho \in {\mathfrak{L}}_{{\mu }_k}^s(\Omega )$ and let $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$, $v\in {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)\cap {\mathfrak{L}}_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$. Then there exists a unique operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right),$ with

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{B\left({\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)\right)}⩽{\left(\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}\right)}^{{\displaystyle \frac{1}{s^{\prime }}}}{\mathfrak{C}}_5^t{\mathfrak{C}}_6^{1-t}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^s(\Omega )},$$

(137)

where

$${\mathfrak{C}}_5={\left({\parallel u\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right)}^{{\displaystyle \frac{1}{q}}},{\mathfrak{C}}_6={\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}$$

and

$$t={\displaystyle \frac{(s-1)q}{(q-1)s}},q={\displaystyle \frac{(2p-2)s}{p-(2-p)s}}.$$

Proof. 

The proof follows from Theorem 10 and Theorem 7 with $p=1$, q instead of p, and interpolation theory. □

5.2. Compactness of ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ for Symbols in ${\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$

Proposition 19. 

Under hypothesis of the Theorem 11, the LO ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ is compact.

Proof. 

Given ${\left(f_n\right)}_{n\in \mathbb{N}}$ a sequence that satisfies $f_n\rightharpoonup 0$ weakly in ${\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ as $n\to \infty$. It is enough to show that ${\displaystyle \underset{n\to \infty }{lim}{\parallel {\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_n\right)\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}=0.}$ Since

$${\displaystyle {∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_n\right)∥}_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}⩽{\int }_{{\mathbb{R}}^N}{\int }_{\Omega }\left|\varrho (r,z)\right|\left|{⟨f_n,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|\left|v_{r,z}^M\left(y\right)\right|d{\mu }_k(r,z){\gamma }_k\left(dy\right),}$$

(138)

then using the fact that $f_n\rightharpoonup 0$ weakly in ${\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$, we deduce that

$$\forall r>0,\forall y,z\in {\mathbb{R}}^N,\underset{n\to \infty }{lim}\left|\varrho (r,z)\right|\left|{⟨f_n,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|\left|v_{r,z}^M\left(y\right)\right|=0.$$

(139)

However, since $f_n\rightharpoonup 0$ weakly in ${\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$, there is $C>0$ such that $\parallel f_n{\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\le C$. Therefore, using (66) and a basic computations, we obtain for all $r>0$ and $y,z\in {\mathbb{R}}^N$,

$$\left|\varrho (r,z)\right|\left|{⟨f_n,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|\left|v_{r,z}^M\left(y\right)\right|\le {C\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}\left|\varrho (r,z)\right|\left|T_x^{M^{-1}}{v}_r^M(-y)\right|.$$

(140)

Moreover, from Fubini’s theorem and (66), we have

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_{\Omega }\left|\varrho (r,z)\right|\left|{⟨f_n,u_{r,z}^M⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\right|\left|v_{r,z}^M\left(y\right)\right|d{\mu }_k(r,z){\gamma }_k\left(dy\right)}\hfill \\ & \le & {\displaystyle {C\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\int }_{\Omega }\left|\varrho (r,z)\right|{\int }_{{\mathbb{R}}^N}\left|T_x^{M^{-1}}{v}_r^M(-y)\right|{\gamma }_k\left(dy\right)d{\mu }_k(r,z)}\hfill \\ & \le & {C\parallel u\parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}{\parallel v\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^1(\Omega )}<\infty .\hfill \end{array}$$

(141)

Thus, from the (138)–(141) we deduce that ${\displaystyle \underset{n\to \infty }{lim}{∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f_n\right)∥}_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}=0}$. □

Theorem 13. 

Let $u\in {\mathfrak{L}}_{k,\mathrm{rad}}^{p^{\prime }}\left({\mathbb{R}}^N\right)$ and $v\in {\mathfrak{L}}_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, $1\le p\le \infty$. Then for $\varrho \in {\mathfrak{L}}_{{\mu }_k}^1(\Omega )$, the operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$ is compact.

Proof. 

From the previous proposition, we only need to show that the conclusion holds for $p=\infty$. In fact, the operator ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$ is the adjoint of ${\mathfrak{T}}_{v,u}^M\left(\overline{\varrho }\right):{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$, which is compact by the previous proposition. Then by the duality property ${\mathfrak{T}}_{u,v}^M\left(\varrho \right):{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)⟶{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$ is compact. Finally, by an interpolation of the compactness on ${\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)$ and on ${\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$, such as the one given on pages 202 and 203 of the book [60], we obtain the result. □

Theorem 14. 

In accordance with the assumptions of Theorem 11, the LO is compact for every $p\in [s,s^{\prime }]$.

Proof. 

This is a direct result by interpolating Proposition 19 and Corollary 3, (see ([60], pages 202 and 203)). □

Remark 3. 

Notice that, in the case of the reflection group $W={\mathbb{Z}}_2^N$, all results of this section remain true for the admissible Dunkl linear canonical wavelets $u,v\in {\mathfrak{L}}_k^p\left({\mathbb{R}}^N\right)$, $p\in [1,\infty ]$.

5.3. Examples

As per Wong’s perspective in his work [61], we will give some examples of localization operators.

1.

Linear canonical Dunkl multiplier (LCDM): For $m\in {\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$, we define the linear operator

$${\mathcal{T}}_m^M:{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)\to {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$$

by

$${\mathcal{T}}_m^{M}f:={\mathfrak{F}}_D^{M^{-1}}\left(m{\mathfrak{F}}_D^M\left(f\right)\right),f\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right).$$

(142)

It is called the linear canonical Dunkl multiplier with symbol m. We will show that if the symbol $\varrho$ is a function that depends only on the second variable, then ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is an LCDM.

Proposition 20. 

Suppose that ϱ is a function defined on Ω by

$$\forall (r,z)\in \Omega ,\varrho (r,z)=\sigma \left(r\right),$$

where σ is a function defined on $(0,\infty )$. Then ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)={\mathcal{T}}_m^M,$ where ${\mathcal{T}}_m^M$ is the LCDM associated to the symbol

$${\displaystyle m\left(\xi \right)={\left|b\right|}^{2⟨k⟩+N}{\int }_0^{\infty }\sigma \left(r\right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{v}_r^M\right)\left(\xi \right)}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{u}_r^M\right)\left(\xi \right)\frac{dr}{r},\xi \in {\mathbb{R}}^N.}$$

Proof. 

For all $j\in \mathbb{N}$, let $I_j$ defined by

$$\begin{array}{ccc}\hfill I_j& =& {\displaystyle {\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }\sigma \left(r\right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{v}_r^M\right)\left(\xi \right)}\left({\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{u}_r^M\right)\overline{{\mathfrak{F}}_D^M\left(g\right)}\underset{M^{-1}}{*}{h}_j\right)\left(\xi \right)}\hfill \\ & \times & {\mathfrak{F}}_D^M\left(f\right)\left(\xi \right){\displaystyle \frac{dr}{r}}{\gamma }_k\left(d\xi \right)\hfill \end{array}$$

where

$$h_j\left(t\right):=j^{{\displaystyle \frac{2⟨k⟩+N}{2}}}{e}^{-j{\displaystyle \frac{{\left|t\right|}^2}{2}}}.$$

On the other hand, by simple calculations, we obtain

$${\mathfrak{F}}_D^M({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{u}_r^M)\overline{{\mathfrak{F}}_D^M\left(g\right)}\underset{M^{-1}}{*}{h}_j⟶{\mathfrak{F}}_D^M({\mathbf{L}}_{{\displaystyle \frac{-2a}{b}}}{u}_r^M)\overline{{\mathfrak{F}}_D^M\left(g\right)}$$

(143)

in ${\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ and almost everywhere on ${\mathbb{R}}^N$ as $j\to \infty$. Then by (143) we obtain

$${\displaystyle I_j⟶{\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }\sigma \left(r\right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{v}_r^M\right)\left(\xi \right)}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{u}_r^M\right)\left(\xi \right)\overline{{\mathfrak{F}}_D^M\left(g\right)\left(\xi \right)}{\mathfrak{F}}_D^M\left(f\right)\left(\xi \right)\frac{dr}{r}{\gamma }_k\left(d\xi \right),}$$

as $j\to \infty$. Therefore for every $f,g\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$,

$$I_j⟶{⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)f,g⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

Thus

$${⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)f,g⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}={⟨{\mathcal{T}}_m^{M}f,g⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)},$$

where ${\mathcal{T}}_m^M$ is the LCDM, such that m is defined by

$${\displaystyle m\left(\xi \right)={\left|b\right|}^{2⟨k⟩+N}{\int }_0^{\infty }\sigma \left(r\right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{v}_r^M\right)\left(\xi \right)}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}{u}_r^M\right)\left(\xi \right)\frac{dr}{r},\xi \in {\mathbb{R}}^N.}$$

 □

2.

Paraproduct: Now we will assume that $\varrho$ depends only on the second variable, that is,

$$\varrho (r,z)=\sigma \left(z\right),(r,z)\in \Omega ,$$

where $\sigma$ is suitable function on ${\mathbb{R}}^N$. Using Parseval’s formula (41) and Fubini’s theorem, we obtain for all $f,g\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$

$${\displaystyle {⟨{\mathfrak{T}}_{u,v}^M\left(\varrho \right)\left(f\right),g⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}={\int }_{\Omega }\varrho (r,z){{\rm Y}}_u^M\left(f\right)(r,z)\overline{{{\rm Y}}_v^M\left(g\right)(r,z)}d{\mu }_k(r,z).}$$

(144)

Moreover, Fubini’s theorem gives

$$\begin{array}{ccc}{⟨{\mathfrak{T}}_{u,u}^M\left(\varrho \right)\left(f\right),g⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}\hfill & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}\sigma \left(z\right)p_{u,v}^M(f,g)\left(z\right){\gamma }_k\left(dz\right),}\hfill \end{array}$$

(145)

where

$${\displaystyle \forall z\in {\mathbb{R}}^N,p_{u,v}^M(f,g)\left(z\right)={\int }_0^{\infty }\left(u_r^M\underset{M^{-1}}{*}{\mathbf{L}}_{\frac{-2a}{b}}f\right)\left(z\right)\overline{\left(v_r^M\underset{M^{-1}}{*}{\mathbf{L}}_{\frac{-2a}{b}}g\right)\left(z\right)}\frac{dr}{r}.}$$

From (145) and since $\varrho$ is in ${\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )$, then ${\mathfrak{T}}_{u,v}^M\left(\varrho \right)$ is bounded such that

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽\sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel \varrho \parallel }_{{\mathfrak{L}}_{{\mu }_k}^{\infty }(\Omega )}.$$

We can therefore provide an ${\mathfrak{L}}_k^1$-estimate on the paraproduct $p_{u,v}^M(·,·)$.

Lemma 5. 

For all $f,g\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$, we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}{p}_{u,v}^M(f,g)\left(z\right){\gamma }_k\left(dz\right)={\left|b\right|}^{2⟨k⟩+N}{C}_{u,v}^M{⟨f,g⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.}$$

(146)

Proof. 

By Equations (41) and (69),

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^N}{p}_{u,v}^M(f,g)\left(z\right){\gamma }_k\left(dz\right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_0^{\infty }\left(u_r^M\underset{M^{-1}}{*}{\mathbf{L}}_{\frac{-2a}{b}}f\right)\left(z\right)\overline{\left(v_r^M\underset{M^{-1}}{*}{\mathbf{L}}_{\frac{-2a}{b}}g\right)\left(z\right)}\frac{dr}{r}{\gamma }_k\left(dz\right)}\hfill \\ & =& {\displaystyle {\left|b\right|}^{2⟨k⟩+N}{\int }_0^{\infty }\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}f\right)\left(\xi \right){\mathfrak{F}}_D^M\left(u\right)\left(r\xi \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}g\right)\left(\xi \right)}\overline{{\mathfrak{F}}_D^M\left(v\right)\left(r\xi \right)}{\gamma }_k\left(d\xi \right)}\right)\frac{dr}{r}}\hfill \\ & =& {\displaystyle {\left|b\right|}^{2⟨k⟩+N}{\int }_{{\mathbb{R}}^N}\left({\displaystyle {\int }_0^{\infty }{\mathfrak{F}}_D^M\left(u\right)\left(r\xi \right)\overline{{\mathfrak{F}}_D^M\left(v\right)\left(r\xi \right)}\frac{dr}{r}}\right){\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}f\right)\left(\xi \right)\overline{{\mathfrak{F}}_D^M\left({\mathbf{L}}_{\frac{-2a}{b}}g\right)\left(\xi \right)}{\gamma }_k\left(d\xi \right)}\hfill \\ & =& C_{u,v}^M{\left|b\right|}^{2⟨k⟩+N}{⟨f,g⟩}_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

 □

Proposition 21. 

For every $f,g\in {\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)$ we have

$$\parallel p_{u,v}^M{(f,g)\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\le {\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel g\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

(147)

Proof. 

From Proposition 11,

$${∥{\mathfrak{T}}_{u,v}^M\left(\varrho \right)∥}_{S_{\infty }}⩽{\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}{\parallel \varrho \parallel }_{L_{{\mu }_k}^{\infty }(\Omega )}={\left|b\right|}^{2⟨k⟩+N}\sqrt{C_u^M{C}_v^M}{\parallel \sigma \parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}.$$

Therefore using (145) we deduce that

$${\left|b\right|}^{-N-2⟨k⟩}\left|{\displaystyle {\int }_{{\mathbb{R}}^N}\sigma \left(z\right)p_{u,v}^M(f,g)\left(z\right){\gamma }_k\left(dz\right)}\right|\le \sqrt{C_u^M{C}_v^M}{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel g\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel \sigma \parallel }_{{\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)}.$$

It follows from the Hahn–Banach theorem that $p_{u,v}^M(f,g)$ is in the dual of ${\mathfrak{L}}_k^{\infty }\left({\mathbb{R}}^N\right)$ and

$$\parallel p_{u,v}^M{(f,g)\parallel }_{{\mathfrak{L}}_k^1\left({\mathbb{R}}^N\right)}\le \sqrt{C_u^M{C}_v^M}{\left|b\right|}^{2⟨k⟩+N}{\parallel f\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}{\parallel g\parallel }_{{\mathfrak{L}}_k^2\left({\mathbb{R}}^N\right)}.$$

The proof is complete. □

6. Conclusions and Perspectives

Using the harmonic analysis of the LCDT, we have introduced and studied a new type of wavelet transformation (called DLCWT), by proving all of its known properties, such that a Plancherel-type formula, an orthogonality relation, an inversion and a Calderón-type Reproducing formulas. Then we define the notion of localization operators associated with the DLCWT. For these operators, we studied the boundedness, compactness, and Schatten–von Neumann class properties.

It is well-known that the uncertainty principles set a limit to the simultaneous localization of a signal and its LCDT. In future work, we will study concentrations operators, which are a special case of localization operators presented in this paper, to measure the time-frequency content of the ${\mathfrak{L}}_k^2$-functions on some subset of finite measure. In particular, we will define and study the scalogram associated with the DCLWT, and show that the signal space spanned by the first eigenfunctions of the concentration operator has the maximum localized scalogram in the region of interest in the time-frequency plane. Then we will show that any essentially concentrated function (functions that are almost concentrated on a subset of finite measure) can be approximated by a linear combination of such eigenfunctions, and corresponding error estimates can be given.