Novel Gabor-Type Transform and Weighted Uncertainty Principles

Abstract

The linear canonical Fourier transform is one of the most celebrated time-frequency tools for analyzing non-transient signals. In this paper, we will introduce and study the deformed Gabor transform associated with the linear canonical Dunkl transform (LCDT). Then, we will formulate several weighted uncertainty principles for the resulting integral transform, called the linear canonical Dunkl-Gabor transform (LCDGT). More precisely, we will prove some variations in Heisenberg’s uncertainty inequality. Then, we will show an analog of Pitt’s inequality for the LCDGT and formulate a Beckner-type uncertainty inequality via two approaches. Finally, we will derive a Benedicks-type uncertainty principle for the LCDGT, which shows the impossibility of a non-trivial function and its LCDGT to both be supported on sets of finite measure. As a side result, we will prove local uncertainty principles for the LCDGT.

1. Introduction

Dunkl operators were introduced and studied by Charles Dunkl [1] as part of the extension of the classical theory of spherical harmonics. Besides their mathematical relevance, these operators are also applied in quantum mechanics in the study of one-dimensional harmonic oscillators governed by Wigner’s commutation rules. The Dunkl transform ${\mathfrak{F}}_D$ associated with a reflection group W and a non-negative multiplicity function k is defined for any function $f\in L_k^1\left({\mathbb{R}}^N\right)\cap L_k^2\left({\mathbb{R}}^N\right)$ by

$$\begin{array}{c}\hfill {\displaystyle {\mathfrak{F}}_D\left(f\right)\left(\xi \right)={\int }_{{\mathbb{R}}^N}f\left(x\right)K(-ix,\xi )d{\gamma }_k\left(x\right),\xi \in {\mathbb{R}}^N,}\end{array}$$

(1)

where $K(-i·,·)$ denotes the Dunkl kernel and $d{\gamma }_k$ is a weighted measure. The Dunkl transform ${\mathfrak{F}}_D$ can be extended uniquely to an isometric isomorphism on $L_k^2\left({\mathbb{R}}^N\right)$ and provides a natural extension of the usual Fourier transform for $k=0$. For a detailed perspective regarding the Dunkl transform, we refer the reader to [2,3,4] and the references therein.

In the context of time-frequency analysis, many important developments have already been reported in the framework of the Dunkl setting. For example, in [5,6,7], the authors have derived several uncertainty inequalities for the Dunkl-Gabor transform. Additionally, several versions of quantitative uncertainty principles associated with the continuous Dunkl wavelet transform have been studied in [8,9].

In view of the contemporary developments of deformed Gabor transforms, we are deeply motivated to study a novel Gabor transform by using the combination of generalized translation and modulation related to the linear canonical Dunkl transform obtained in [10].

In order to investigate the conservation of information and uncertainty under linear mappings of a phase space, Collins [11] in paraxial optics and Moshinsky-Quesne [12] in quantum mechanics have separately invented the classical linear canonical transform (LCT), which is an integral transform associated with a general homogeneous linear lossless mapping in phase space that depends on three parameters independent of the phase space coordinates. The three parameters constitute an augmented matrix consisting of a uni-modular matrix $2\times 2$. It maps the position x and the wave number y into

$$\begin{array}{c}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right),\hfill \end{array}$$

where $ad-bc=1.$ Any convex body can be transformed into another convex body while maintaining the body’s area. These changes make up the homogeneous special group $SL(2,R)$.

The linear canonical transform of any function f with respect to a real augmented matrix

$$\mathbf{m}:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in SL(2,\mathbb{R})$$

is defined by

$${\displaystyle {\mathfrak{F}}^{\mathbf{m}}f\left(x\right)={\displaystyle \frac{1}{{\left(ib\right)}^{\frac{1}{2}}}}{\int }_{\mathbb{R}}{\mathcal{K}}^{\mathbf{m}}(x,y)f\left(y\right)dy,}$$

(2)

where

$${\mathcal{K}}^{\mathbf{m}}(x,y)=e^{{\displaystyle \frac{i}{2}}({\displaystyle \frac{d}{b}}{x}^2+{\displaystyle \frac{a}{b}}{y}^2)}{e}^{{\displaystyle \frac{-ixy}{b}}}.$$

(3)

The LCT is a generalized Fourier transform that is an extension of several well-known integral transformations, such as the Fourier transform [13,14], the Fresnel transform [15], the fractional Fourier transform [16,17], scaling operations, etc. [18,19].

The LCT is more versatile than other transforms because of its additional degrees of freedom and straightforward geometrical representation, making it a useful and effective tool for examining deep issues in signal processing, quantum physics and optics [18,19,20,21,22,23,24,25]. Moreover, the LCT might be a potential tool for solving complex interface problems [26], and it could benefit from the adaptive high-order algorithms for solving large-scale problems [27].

Recently the LCT has gained considerable attention and has been extended to a wide class of integral transforms; see, for example, refs. [28,29,30,31,32,33,34,35] and the references therein.

The LCT’s application areas have expanded exponentially over the past few decades, making it appropriate for examining deep issues in signal analysis, filter design, phase retrieval problems, pattern recognition, radar analysis, holographic three-dimensional television, quantum physics, and many other fields, such as uncertainty principles [36], Poisson summation formulae [37], convolution theorems [38], and sampling theorems [39]. For further information about the LCT and their applications, see [18,22,24,33,34,40,41].

This paper has two objectives. First by using the main tools related to the linear canonical Dunkl transform (LCDT) [10], we present and study a new Gabor-type transform, which we will call: the linear canonical Dunkl-Gabor transform (LCDGT). Then, we will show the main theorems of harmonic analysis for this transformation.

Our second aim in this paper is to study and prove some uncertainty inequalities for our new Gabor transform, such as local-type, Benedicks-type, Pitt-type and Beckner-type uncertainty principles. Consequently, we will use a variety of methods, such as generalized entropy and the contraction semigroup method of the homogeneous integral transform to derive several types of Heisenberg’s uncertainty inequality.

Notice that uncertainty principles play an important role in both quantum mechanics and harmonic analysis, especially in time-frequency analysis. They have gained considerable attention and have been extended to a wide class of integral transforms; see, for example, reference [42].

The main objectives of this paper are described as follows:

-

To introduce the notion of linear canonical Gabor transform in the Dunkl setting (LCDGT).

-

To derive some fundamental properties of the LCDGT, such as orthogonality relation, Young’s and inversion formula.

-

To investigate for the LCDGT several versions of the Heisenberg-type uncertainty principle via various techniques, including generalized entropy, the semigroup method of contraction and others.

-

To obtain certain concentration uncertainty principles for the LCDGT on sets of finite measure.

This paper’s remaining sections are organized as follows: In Section 2, we will state the necessary materials concerning the harmonic analysis related to the Dunkl transform and the linear canonical Dunkl transform studied in [10]. In Section 3, we will present and study a generalized Gabor-type transform related to the LCDT. More precisely, we will prove a Plancherel-type, a Lieb-type theorem, and an inversion formula. Section 4 is devoted to Heisenberg-type uncertainty principles associated with the LCDGT. The Pitt-type and logarithm uncertainty inequalities for the LCDGT are presented in Section 5. Finally, in Section 6, we will obtain some concentration uncertainty principles, including Benedick-type and local-type uncertainty principles.

2. Linear Canonical Dunkl Transform: Operators and Properties

In this section, we shall present the prerequisites concerning the linear canonical Dunkl transform. First, we will briefly review the conventional Dunkl operators, Dunkl convolutions and the corresponding Dunkl transform. The main references are [1,2,3,43,44].

2.1. Dunkl Transform: Properties, Translation and Convolution

Let $\{e_i,i=1,\dots ,N\}$ be the orthonormal basis of ${\mathbb{R}}^N$ and for $x\in {\mathbb{R}}^N$, we denote by $\parallel x\parallel =\sqrt{⟨x,x⟩}$. The reflection ${\sigma }_{\alpha }$ in the hyperplane $H_{\alpha }\subset {\mathbb{R}}^N$ orthogonal to $\alpha \in {\mathbb{R}}^N\setminus \left\{0\right\}$ is defined by

$${\sigma }_{\alpha }\left(x\right)=x-2{\displaystyle \frac{⟨\alpha ,x⟩}{{\parallel \alpha \parallel }^2}}\alpha .$$

(4)

Let $\mathcal{R}$ be a finite subset of ${\mathbb{R}}^N\setminus \left\{0\right\}$. If $\mathcal{R}\cap \mathbb{R}\alpha =\{\pm \alpha \}$ and if for all $\alpha \in \mathcal{R}$, ${\sigma }_{\alpha }\left(\mathcal{R}\right)=\mathcal{R}$, then $\mathcal{R}$ is a root system.

It is quite obvious that the reflections ${\sigma }_{\alpha },\alpha \in \mathcal{R}$ associated with root system $\mathcal{R}$ generate a finite group $W\subset O\left(N\right)$, called the reflection group.

We fix a positive root system ${\mathcal{R}}_{+}=\left\{\alpha \in \mathcal{R}:⟨\alpha ,\beta ⟩>0\right\}$, for $\beta \in {\mathbb{R}}^N\setminus {\bigcup }_{\alpha \in \mathcal{R}}{H}_{\alpha }$, and we will suppose that for all $\alpha \in {\mathcal{R}}_{+}$, we have $⟨\alpha ,\alpha ⟩=2$.

If a function $k:\mathcal{R}⟶[0,\infty )$ remains invariant under the action of the reflection group W, then it is considered as a multiplicity function.

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

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

(5)

and we introduce the index

$${\displaystyle \gamma =\gamma \left(k\right)=\sum _{\alpha \in {\mathcal{R}}_{+}}k\left(\alpha \right).}$$

(6)

It is evident that ${\omega }_k$ is homogeneous of degree $2\gamma$ and W-invariant.

Finally, we introduce the well-known Mehta-type constant:

$${\displaystyle c_k={\int }_{{\mathbb{R}}^N}{e}^{-\frac{{\parallel x\parallel }^2}{2}}{\omega }_k\left(x\right)dx.}$$

(7)

For any function $f\in C^1\left({\mathbb{R}}^N\right)$, the Dunkl operators $T_j,j=1,\dots ,N$ associated with the finite reflection group W and multiplicity function k are defined by

$${\displaystyle T_{j}f\left(x\right):=\frac{\partial f}{\partial x_j}\left(x\right)+\sum _{\alpha \in {\mathcal{R}}_{+}}k\left(\alpha \right){\alpha }_j\frac{f\left(x\right)-f\left({\sigma }_{\alpha }\left(x\right)\right)}{⟨\alpha ,x⟩},x\in {\mathbb{R}}_{\mathrm{reg}}^N={\mathbb{R}}^N\setminus {\cup }_{\alpha \in \mathcal{R}}{H}_{\alpha },}$$

(8)

where ${\alpha }_j=⟨\alpha ,e_j⟩$, and $C^p\left({\mathbb{R}}^N\right)$ is the space of functions of class $C^p$ on ${\mathbb{R}}^N$.

The Dunkl operators form a commutative system of differential-difference operators. Similarly, we can define the Dunkl-Laplacian operator ${\mathsf{\Delta }}_k$, for any $f\in C^2\left({\mathbb{R}}^N\right)$, as

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{k}f\left(x\right)& {\displaystyle :=\sum _{j=1}^N{T}_j^{2}f\left(x\right)=\mathsf{\Delta }f\left(x\right)+2\sum _{\alpha \in {\mathcal{R}}^{+}}k\left(\alpha \right)\left({\displaystyle \frac{⟨\nabla f\left(x\right),\alpha ⟩}{⟨\alpha ,x⟩}}-{\displaystyle \frac{f\left(x\right)-f\left({\sigma }_{\alpha }\left(x\right)\right)}{{⟨\alpha ,x⟩}^2}}\right),x\in {\mathbb{R}}_{\mathrm{reg}}^N,}\hfill \end{array}$$

where $\mathsf{\Delta }$ and ∇ denote the Euclidean Laplacian and the gradient operators on ${\mathbb{R}}^N$, respectively.

For $y\in {\mathbb{R}}^N$, the system

$$\left\{\begin{array}{ccc}{T}_{j}u(x,y)\hfill & =\hfill & y_{j}u(x,y),j=1,\dots ,N,\hfill \\ u(0,y)\hfill & =\hfill & 1,\hfill \end{array}\right.$$

(9)

admits a unique analytic solution on ${\mathbb{R}}^N$ called the Dunkl kernel. It has the following properties:

• $K(-i·,·)$ has a unique holomorphic extension to ${\mathbb{C}}^N\times {\mathbb{C}}^N$.
• For all $z,t,\in {\mathbb{C}}^N$ and $\lambda \in \mathbb{C}$, we have $K(z,t)=K(t,z)$, $K(z,0)=1$ and $K(\lambda z,t)=K(z,\lambda t)$.
• For all $\nu \in {\mathbb{N}}^N,x\in {\mathbb{R}}^N$ and $z\in {\mathbb{C}}^N$, we have

  $$\left|D_z^{\nu }K(-ix,z)\right|\le {\parallel x\parallel }^{\left|\nu \right|}exp\left(\parallel x\parallel \parallel \mathrm{Re}z\parallel \right),$$

  (10)

  where $D_z^{\nu }={\displaystyle \frac{{\partial }^{\left|\nu \right|}}{\partial z_1^{{\nu }_1}\cdots \partial z_N^{{\nu }_N}}}$ and $\left|\nu \right|={\nu }_1+\cdots +{\nu }_N$. In particular, $\left|K\right(ix,y\left)\right|\le 1$, for all $x,y\in {\mathbb{R}}^N$.
• $K(-i·,·)$ admits the following Laplace-type integral representation:

  $${\displaystyle K(-ix,z)={\int }_{{\mathbb{R}}^N}{e}^{⟨y,z⟩}d{\nu }_x\left(y\right),}$$

  (11)

  where ${\nu }_x$ denotes the positive probability measure on ${\mathbb{R}}^N$, with support in $B_N(0,\parallel x\parallel )$ of center 0 and radius $\parallel x\parallel$ (see [45]).

Let $L_k^p\left({\mathbb{R}}^N\right),1\le p<\infty ,$ the space of measurable functions f on ${\mathbb{R}}^N$ satisfying

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

where $d{\gamma }_k\left(x\right):={c_k}^{-1}{\omega }_k\left(x\right)dx.$ In particular, $L_k^2\left({\mathbb{R}}^N\right)$ constitutes a Hilbert space equipped with the scalar product

$${\displaystyle {⟨f,g⟩}_{L_k^2\left({\mathbb{R}}^N\right)}:={\int }_{{\mathbb{R}}^N}f\left(x\right)\overline{g\left(x\right)}d{\gamma }_k\left(x\right).}$$

If $\mathcal{F}$ denotes the space of all $\mathbb{C}$-valued functions on ${\mathbb{R}}^N$, then we set

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

as the subspace of radial functions in $\mathcal{F}$. Therefore, for any $f\in {\mathcal{F}}_{\mathrm{rad}}$, there exists a unique function $F:{\mathbb{R}}_{+}\to \mathbb{C}$ such that $f\left(x\right)=F(\parallel x\parallel )$, for all $x\in {\mathbb{R}}^N$.

Remark 1.

Using the homogeneity of ${\omega }_k$, it is proved in [43] that for a radial function $f\in L_k^1\left({\mathbb{R}}^N\right)$, the function F defined by $f(·)=F(\parallel ·\parallel )$ is integrable with respect to the measure $r^{2\gamma +N-1}dr$. More precisely,

$${\displaystyle {\int }_{{\mathbb{R}}^N}f\left(x\right)d{\gamma }_k\left(x\right)=d_k{\int }_{{\mathbb{R}}^N}F\left(r\right)r^{2\gamma +N-1}dr,}$$

(12)

where

$$d_k:={\displaystyle \frac{1}{2^{\gamma +\frac{N}{2}-1}\mathsf{\Gamma }\left(\gamma +{\displaystyle \frac{N}{2}}\right)}}.$$

(13)

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

$${\displaystyle {\mathfrak{F}}_D\left(f\right)\left(\xi \right)={\int }_{{\mathbb{R}}^N}f\left(x\right)K(-ix,\xi )d{\gamma }_k\left(x\right),\xi \in {\mathbb{R}}^N.}$$

(14)

In the following, we state some properties of the Dunkl transform (see [2,3]).

• For any $f\in L_k^1\left({\mathbb{R}}^N\right)$, we have

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

  (15)
• Inversion Formula: if $f\in L_k^1\left({\mathbb{R}}^N\right)$ is a function such that its Dunkl transform ${\mathfrak{F}}_D\left(f\right)$ is in $L_k^1\left({\mathbb{R}}^N\right)$, then

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

  (16)
• The Dunkl transform ${\mathfrak{F}}_D$ is a topological isomorphism from the Schwartz space $\mathcal{S}\left({\mathbb{R}}^N\right)$ onto itself. If we take $f\in \mathcal{S}\left({\mathbb{R}}^N\right)$ such that

  $$\overline{{\mathfrak{F}}_D}\left(f\right)\left(\xi \right)={\mathfrak{F}}_D\left(f\right)(-\xi ),\xi \in {\mathbb{R}}^N,$$

  (17)

  then,

  $${\mathfrak{F}}_D\overline{{\mathfrak{F}}_D}=\overline{{\mathfrak{F}}_D}{\mathfrak{F}}_D=Id.$$
• Plancherel’s Formula: for any $f\in \mathcal{S}\left({\mathbb{R}}^N\right)$,

  $${\displaystyle {\int }_{{\mathbb{R}}^N}{\left|f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right)={\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

  (18)
• Parseval’s Formula: for any $f,g\in \mathcal{S}\left({\mathbb{R}}^N\right)$, we have

  $${\displaystyle {\int }_{{\mathbb{R}}^N}f\overline{g}d{\gamma }_k={\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D\left(f\right)\overline{{\mathfrak{F}}_D\left(g\right)}d{\gamma }_k.}$$

  (19)

Proposition 1.

For all f in $D\left({\mathbb{R}}^N\right)$ (resp. $\mathcal{S}\left({\mathbb{R}}^N\right))$, we have the following relations:

$${\mathfrak{F}}_D\left(\overline{f}\right)\left(\lambda \right)=\overline{{\mathfrak{F}}_D\left(\breve{f}\right)\left(\lambda \right)},\forall \lambda \in {\mathbb{R}}^N,$$

(20)

and

$${\mathfrak{F}}_D\left(f\right)\left(\lambda \right)={\mathfrak{F}}_D\left(\breve{f}\right)(-\lambda ),\forall \lambda \in {\mathbb{R}}^N,$$

(21)

where $\breve{f}$ is the function defined by $\breve{f}\left(x\right)=f(-x),$ and $D\left({\mathbb{R}}^N\right)$ is the space of $C^{\infty }$-functions on ${\mathbb{R}}^N$, which are of compact support.

Definition 1.

The Dunkl translation operator $f\mapsto {\tau }_{x}f$ is defined on $L_k^2\left({\mathbb{R}}^N\right)$ by (see [43])

$${\mathfrak{F}}_D\left({\tau }_{x}f\right)=K(ix,.){\mathfrak{F}}_D\left(f\right),x\in {\mathbb{R}}^N.$$

(22)

Let ${\mathcal{W}}_k\left({\mathbb{R}}^N\right)$ be the Wigner-type space of all functions, which satisfies (22) point-wise. That is

$${\mathcal{W}}_k\left({\mathbb{R}}^N\right):=\left\{f\in L_k^1\left({\mathbb{R}}^N\right):{\mathfrak{F}}_D\left(f\right)\in L_k^1\left({\mathbb{R}}^N\right)\right\}.$$

The Dunkl translation operator satisfies the following properties (see [43,46]):

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

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

  (23)
• For any $f\in {\mathcal{W}}_k\left({\mathbb{R}}^N\right)$, we have

  $${\displaystyle {\tau }_{x}f\left(y\right)={\int }_{{\mathbb{R}}^N}K(ix,\xi )K(iy,\xi ){\mathfrak{F}}_D\left(f\right)\left(\xi \right)d{\gamma }_k\left(\xi \right),y\in {\mathbb{R}}^N.}$$
• For all f in ${\mathcal{W}}_k\left({\mathbb{R}}^N\right)$, we have

  $${\tau }_{x}f\left(y\right)={\tau }_y\left(f\right)\left(x\right).$$

  (24)

Remark 2.

We note that, in [4] Trimèche has defined the Dunkl translation on the space of functions of classes $C^{\infty }\left({\mathbb{R}}^N\right)$ as

$${\tau }_{x}f\left(y\right)={\left(V_k\right)}_x{\left(V_k\right)}_y\left[{\left(V_k\right)}^{-1}f(x+y)\right],x,y\in {\mathbb{R}}^N,$$

(25)

where $V_k$ is the Dunkl transmutation operator.

Until now, an explicit formula for the Dunkl translation operator is known only in the following cases: If $N=1$ and $W={\mathbb{Z}}_2$, then for all $x\in \mathbb{R}$ and $f\in C\left(\mathbb{R}\right)$, we have

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

where

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

From this, one can also give an explicit formula for the translation operator in the case of $W={\mathbb{Z}}_2^N$. Moreover, we have the following boundedness result (see [46]).

Proposition 2.

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

$${\parallel {\tau }_{y}f\parallel }_{L_k^p\left({\mathbb{R}}^N\right)}\le 2^{{\displaystyle \frac{N}{2p}}|p-2|}{\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^N\right)}.$$

(26)

Moreover, if $f\in {\mathcal{W}}_k\left({\mathbb{R}}^N\right)$ is radial, then

$${\displaystyle {\tau }_{y}f\left(x\right)={\int }_{B_N(0,\parallel x\parallel )}{f}_0\left(\sqrt{{\parallel x\parallel }^2+{\parallel y\parallel }^2+2⟨x,z⟩}\right)d{\nu }_x\left(z\right),\forall x\in {\mathbb{R}}^N,}$$

where $f_0$ is the function given by $f\left(x\right)=f_0(\parallel x\parallel )$ and ${\nu }_x$ is the measure given by (11).

Let $L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$ be the subspace of radial functions in $L_k^p\left({\mathbb{R}}^N\right)$. Then, we have the following proposition (see [46]).

Proposition 3.

1. 

If $f\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$ is non-negative, then we have

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

and

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

(27)

2. 

For all $f\in L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, $1\le p\le \infty$, we have

$${\parallel {\tau }_{y}f\parallel }_{L_k^p\left({\mathbb{R}}^N\right)}\le {\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^N\right)},\forall y\in {\mathbb{R}}^N.$$

(28)

The Dunkl convolution product may be defined by using the Dunkl translation operator (see [4,46]).

Definition 2.

The Dunkl convolution product of any $f,g\in D\left({\mathbb{R}}^N\right)$ is defined by

$${\displaystyle f{\ast }_{D}g\left(x\right)={\int }_{{\mathbb{R}}^N}{\tau }_{x}f(-y)g\left(y\right)d{\gamma }_k\left(y\right),\forall x\in {\mathbb{R}}^N.}$$

(29)

The Dunkl convolution is both associative and commutative and satisfies the following properties (see [4,46]).

Proposition 4.

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

1. 

If $f\in L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$ and $g\in L_k^q\left({\mathbb{R}}^N\right),$ then $f{\ast }_{D}g$ belongs to $L_k^r\left({\mathbb{R}}^N\right)$ and

$${∥f{\ast }_{D}g∥}_{L_k^r\left({\mathbb{R}}^N\right)}\le {∥f∥}_{L_k^p\left({\mathbb{R}}^N\right)}{∥g∥}_{L_k^q\left({\mathbb{R}}^N\right)}.$$

(30)

2. 

If $W={\mathbb{Z}}_2^N$, then for all $f\in L_k^p\left({\mathbb{R}}^N\right)$ and $g\in L_k^q\left({\mathbb{R}}^N\right)$, the function $f{\ast }_{D}g\in L_k^r\left({\mathbb{R}}^N\right)$ and

$${∥f{\ast }_{D}g∥}_{L_k^r\left({\mathbb{R}}^N\right)}\le 2^{\frac{N}{2p}|p-2|}{∥f∥}_{L_k^p\left({\mathbb{R}}^N\right)}{∥g∥}_{L_k^q\left({\mathbb{R}}^N\right)}.$$

(31)

3. 

If $f,g\in L_k^2\left({\mathbb{R}}^N\right)$, then $f{\ast }_{D}g\in L_k^2\left({\mathbb{R}}^N\right)$ if and only if ${\mathfrak{F}}_D\left(f\right){\mathfrak{F}}_D\left(g\right)\in L_k^2\left({\mathbb{R}}^N\right)$, and in this case, we have

$${\mathfrak{F}}_D\left(f{\ast }_{D}g\right)={\mathfrak{F}}_D\left(f\right){\mathfrak{F}}_D\left(g\right).$$

(32)

4. 

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

$${\displaystyle {\int }_{{\mathbb{R}}^N}|f{\ast }_D{g\left(x\right)|}^{2}d{\gamma }_k\left(x\right)={\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D{\left(f\right)\left(\xi \right)|}^2{\left|{\mathfrak{F}}_D\left(g\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right),}$$

(33)

where both sides are finite or infinite.

2.2. Linear Canonical Dunkl Transform

In this article, $\mathbf{m}:=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ denotes a matrix in $SL(2,\mathbb{R})$ with $b\ne 0.$ In this subsection, we recall some results proved in [10,47].

Definition 3.

The linear canonical Dunkl transform of a function $f\in L_k^1\left({\mathbb{R}}^N\right)$ is defined by

$${\displaystyle {\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(x\right)=\frac{1}{{\left(ib\right)}^{\frac{2\gamma +N}{2}}}{\int }_{{\mathbb{R}}^N}{\mathfrak{B}}_k^{\mathbf{m}}(x,y)f\left(y\right)d{\gamma }_k\left(y\right),}$$

(34)

where

$${\mathfrak{B}}_k^{\mathbf{m}}(x,y)=e^{\frac{i}{2}\left(\frac{d}{b}{\parallel x\parallel }^2+\frac{a}{b}{\parallel y\parallel }^2\right)}K\left(-i{\displaystyle \frac{x}{b}},y\right).$$

(35)

Proposition 5.

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

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

(36)

where $M_j\left(u\left(x\right)\right):=x_{j}u\left(x\right)$.

1. 

The operator ${\mathsf{\Delta }}_k^{\mathbf{m}}$ is related to the deformed Laplace operator ${\mathsf{\Delta }}_k$ by

$$e^{-\frac{i}{2}\frac{d}{b}{\parallel x\parallel }^2}\circ {\mathsf{\Delta }}_k^{\mathbf{m}}\circ e^{\frac{i}{2}\frac{d}{b}{\parallel x\parallel }^2}={\mathsf{\Delta }}_k.$$

(37)

2. 

For every $f,g\in \mathcal{S}\left({\mathbb{R}}^N\right)$

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\mathsf{\Delta }}_k^{\mathbf{m}}f\left(x\right)\overline{g\left(x\right)}d{\gamma }_k\left(x\right)={\int }_{{\mathbb{R}}^N}f\left(x\right)\overline{{\mathsf{\Delta }}_k^{\mathbf{m}}g\left(x\right)}d{\gamma }_k\left(x\right).}$$

(38)

3. 

For each $y\in {\mathbb{R}}^N$, the kernel ${\mathfrak{B}}_k^{\mathbf{m}}(·,y)$ of the linear canonical Dunkl transform ${\mathfrak{F}}_D^{\mathbf{m}}$ satisfies the following:

$$\left\{\begin{array}{c}{\mathsf{\Delta }}_k^{\mathbf{m}}{\mathfrak{B}}_k^{\mathbf{m}}(·,y)=-{\parallel \frac{y}{b}\parallel }^2{\mathfrak{B}}_k^{\mathbf{m}}(·,y),\hfill \\ \\ {\mathfrak{B}}_k^{\mathbf{m}}(0,y)=e^{\frac{i}{2}\frac{a}{b}{\parallel y\parallel }^2}.\hfill \end{array}\right.$$

(39)

4. 

For every $f\in \mathcal{S}\left({\mathbb{R}}^N\right)$, we have

$${\mathfrak{F}}_D^{\mathbf{m}}\left({\parallel y\parallel }^{2}f\left(y\right)\right)=-b^2{\mathsf{\Delta }}_k^{\mathbf{m}}\left({\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\right),$$

(40)

and

$${\parallel x\parallel }^2{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)=-b^2{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathsf{\Delta }}_k^{{\mathbf{m}}^{-1}}\left(f\right)\right).$$

(41)

5. 

For each $x,y\in {\mathbb{R}}^N$,

$$|{\mathfrak{B}}_k^{\mathbf{m}}(x,y)|\le 1.$$

(42)

2.2.1. Particular Cases

• In the case $\mathbf{m}:=\mathbf{m}\left(\tau \right)=\left(\begin{array}{cc}1& \tau \\ 0& 1\end{array}\right)$, $\tau \in \mathbb{R}$, ${\mathfrak{F}}_D^{\mathbf{m}}$ coincides with the Fresnel transform associated with the Dunkl transform (see [10]):

  $${\mathcal{W}}_k^{\tau }f\left(x\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{1}{{\left(i\tau \right)}^{\frac{2\gamma +N}{2}}}}{\int }_{{\mathbb{R}}^N}{E}_k^{\tau }(x,y)f\left(y\right)d{\gamma }_k\left(y\right),}\hfill & \tau \ne 0,\hfill \\ \\ f\left(x\right),\hfill & \tau =0,\hfill \end{array}\right.$$

  where $E_k^{\tau }(x,y)=e^{\frac{i}{2\tau }{(\parallel x\parallel }^2+{\parallel y\parallel }^2)}K\left(-i\frac{x}{\tau },y\right).$
• In the case $\mathbf{m}:=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$, ${\mathsf{\Delta }}_k^{\mathbf{m}}$ is reduced to the deformed Laplace operator ${\mathsf{\Delta }}_k$ and ${\mathfrak{F}}_D^{\mathbf{m}}$ coincides with the Dunkl transform ${\mathfrak{F}}_D$ (except for a constant unimodular factor ${(e^{i\frac{\pi }{2}})}^{\frac{2\gamma +N}{2}}$) (see [10]).
• When $\mathbf{m}:=\mathbf{m}\left(\theta \right)=\left(\begin{array}{cc}cosh\left(\theta \right)& sinh\left(\theta \right)\\ sinh\left(\theta \right)& cosh\left(\theta \right)\end{array}\right)$, $\theta \in \mathbb{R},{\mathfrak{F}}_D^{\mathbf{m}}$ coincides with the following integral transform (see [10]):

  $${\mathcal{V}}_k^{\theta }f\left(x\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{1}{{(isinh\left(\theta \right))}^{\frac{2\gamma +N}{2}}}}{\int }_{{\mathbb{R}}^N}{R}_k^{\theta }(x,y)f\left(y\right)d{\gamma }_k\left(y\right),}\hfill & \theta \ne 0,\hfill \\ \\ f\left(x\right),\hfill & \theta =0,\hfill \end{array}\right.$$

  where $R_k^{\theta }(x,y)=e^{\frac{i}{2}{coth\left(\theta \right)(\parallel x\parallel }^2+{\parallel y\parallel }^2)}K(-i\frac{x}{sinh\left(\theta \right)},y).$
• In the case $\mathbf{m}:=\mathbf{m}\left(\theta \right)=\left(\begin{array}{cc}cos\left(\theta \right)& -sin\left(\theta \right)\\ sin\left(\theta \right)& cos\left(\theta \right)\end{array}\right)$, $\theta \in \mathbb{R},{\mathfrak{F}}_D^{\mathbf{m}}$ coincides with the fractional Dunkl transform ${\mathfrak{F}}_D^{\theta }$ (see [10]):

  $${\mathfrak{F}}_D^{\theta }f\left(x\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{e^{i\left(\left(\frac{2\gamma +N}{2}\right)\right)((\theta -2n\pi )-\widehat{\theta }\pi /2)}}{{|sin\left(\theta \right)|}^{\frac{2\gamma +N}{2}}}}{\int }_{{\mathbb{R}}^N}{\mathcal{K}}_k^{\theta }(x,y)f\left(y\right)d{\gamma }_k\left(y\right),}\hfill & (2j-1)\pi <\theta <(2j+1)\pi ,\hfill \\ f\left(x\right),\hfill & \theta =2j\pi ,\hfill \\ f(-x),\hfill & \theta =(2j+1)\pi ,\hfill \end{array}\right.$$

  where $\widehat{\theta }=\mathrm{sgn}(sin\left(\theta \right))$ and

  $${\mathcal{K}}_k^{\theta }(x,y)=e^{-\frac{i}{2}{cot\left(\theta \right)(\parallel x\parallel }^2+{\parallel y\parallel }^2)}K\left(-i{\displaystyle \frac{x}{sin\left(\theta \right)}},y\right).$$

2.2.2. Linear Canonical Dunkl Transform on $L_k^p\left({\mathbb{R}}^N\right)$, $1\le p\le 2$

Definition 4.

The chirp multiplication operator ${\mathbf{L}}_s$, $s\in \mathbb{R}$ and the dilation operator ${\mathsf{\Delta }}_s$ are defined, respectively, by

$${\mathbf{L}}_{s}f\left(x\right)=e^{\frac{is}{2}{\parallel x\parallel }^2}f\left(x\right)\mathrm{and}{\mathsf{\Delta }}_{s}f\left(x\right)={\displaystyle \frac{1}{{\left|s\right|}^{\frac{2\gamma +N}{2}}}}f(x/s),s\ne 0.$$

(43)

We recall in the following proposition some useful properties of ${\mathbf{L}}_s$ and ${\mathsf{\Delta }}_s$.

Proposition 6.

The following equalities hold on $L_k^1\left({\mathbb{R}}^N\right)$.

1. 

For all $s\in {\mathbb{R}}^{\ast }$, we have

$${\mathsf{\Delta }}_s\circ {\mathfrak{F}}_D={\mathfrak{F}}_D\circ {\mathsf{\Delta }}_{\frac{1}{s}}.$$

(44)

2. 

For $\mathbf{m}\in SL(2,\mathbb{R})$, we have

$$e^{i\left(\frac{2\gamma +N}{2}\right)\frac{\pi }{2}sgn\left(b\right)}{\mathfrak{F}}_D^{\mathbf{m}}={\mathbf{L}}_{\frac{b}{d}}\circ {\mathsf{\Delta }}_b\circ {\mathfrak{F}}_D\circ {\mathbf{L}}_{\frac{a}{b}}.$$

(45)

Theorem 1

(Riemann–Lebesgue’s lemma). For all $f\in L_k^1\left({\mathbb{R}}^N\right)$, the linear canonical Dunkl transform ${\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)$ belongs to $C_0\left({\mathbb{R}}^N\right)$ and satisfies

$$\parallel {\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\parallel }_{L_k^{\infty }\left({\mathbb{R}}^N\right)}\le {\left|b\right|}^{-\frac{2\gamma +N}{2}}{\parallel f\parallel }_{L_k^1\left({\mathbb{R}}^N\right)}.$$

(46)

Theorem 2

(Plancherel’s Theorem). 

1. 

For every $f,g\in L_k^1\left({\mathbb{R}}^N\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(x\right)\overline{g\left(x\right)}d{\gamma }_k\left(x\right)={\int }_{{\mathbb{R}}^N}f\left(x\right)\overline{{\mathfrak{F}}_D^{{\mathbf{m}}^{-1}}\left(g\right)\left(x\right)}d{\gamma }_k\left(x\right).}$$

(47)

2. 

If $f\in L_k^1\left({\mathbb{R}}^N\right)\cap L_k^2\left({\mathbb{R}}^N\right)$, then ${\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\in L_k^2\left({\mathbb{R}}^N\right)$, and we have

$$\parallel {\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(48)

3. 

The linear canonical Dunkl transform has a unique extension to an isometric isomorphism of $L_k^2\left({\mathbb{R}}^N\right)$. The extension is also denoted by ${\mathfrak{F}}_D^{\mathbf{m}}\left(f\right):L_k^2\left({\mathbb{R}}^N\right)\to L_k^2\left({\mathbb{R}}^N\right)$.

4. 

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

$${⟨{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right),g⟩}_{L_k^2\left({\mathbb{R}}^N\right)}={⟨f,{\mathfrak{F}}_D^{{\mathbf{m}}^{-1}}g⟩}_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(49)

5. 

For all $f\in L_k^1\left({\mathbb{R}}^N\right)$ with ${\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\in L_k^1\left({\mathbb{R}}^N\right),$

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

(50)

Definition 5.

For each $\mathbf{m}\in SL(2,\mathbb{R})$, we define the linear canonical Dunkl transform on $L_k^p\left({\mathbb{R}}^N\right),1\le p\le 2$, by

$${\mathfrak{F}}_D^{\mathbf{m}}=e^{-i\left(\frac{2\gamma +N}{2}\right)\frac{\pi }{2}\mathrm{sgn}\left(b\right)}\left({\mathbf{L}}_{\frac{d}{b}}\circ {\mathsf{\Delta }}_b\circ {\mathfrak{F}}_D\circ {\mathbf{L}}_{\frac{a}{b}}\right),$$

where ${\mathfrak{F}}_D:L_k^p\left({\mathbb{R}}^N\right)\to L_k^{p^{\prime }}\left({\mathbb{R}}^N\right)$ is the Dunkl transformation on $L_k^p\left({\mathbb{R}}^N\right)$.

Theorem 3

(Young’s inequality). Let $\mathbf{m}\in SL(2,\mathbb{R})$, and let p be real number such that $1\le p\le 2$. Then, ${\mathfrak{F}}_D^{\mathbf{m}}$ extends to a bounded linear operator on $L_k^p\left({\mathbb{R}}^N\right)$, and we have

$$\parallel {\mathfrak{F}}_D^{\mathbf{m}}\left(f\right){\parallel }_{L_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}\le {\left|b\right|}^{\left(\frac{2\gamma +N}{2}\right)\left(2/p^{\prime }-1\right)}{\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^N\right)}.$$

(51)

Proof. 

From Theorem 1, we have for all $f\in L_k^1\left({\mathbb{R}}^N\right)$,

$$\parallel {\mathfrak{F}}_D^{\mathbf{m}}\left(f\right){\parallel }_{L_k^{\infty }\left({\mathbb{R}}^N\right)}\le {\left|b\right|}^{-\left(\frac{2\gamma +N}{2}\right)}{\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^N\right)}.$$

(52)

Moreover, involving Relation (48), we have for all $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$$\parallel {\mathfrak{F}}_D^{\mathbf{m}}\left(f\right){\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(53)

By (52), (53) and the Riesz-Thorin interpolation theorem, ${\mathfrak{F}}_D^{\mathbf{m}}$ may be uniquely extended to a linear operator on $L_k^p\left({\mathbb{R}}^N\right)$, $1\le p\le \infty$ and we have:

$$\parallel {\mathfrak{F}}_D^{\mathbf{m}}\left(f\right){\parallel }_{L_k^{p^{\prime }}\left({\mathbb{R}}^N\right)}\le {\left|b\right|}^{\left(\frac{2\gamma +N}{2}\right)\left(2/p^{\prime }-1\right)}{\parallel f\parallel }_{L_k^p\left({\mathbb{R}}^N\right)}.$$

The proof is complete. □

2.3. Generalized Convolution Product Associated with ${\mathfrak{F}}_D^{\mathbf{m}}$

In this subsection, we recall some results proved in [47].

Definition 6.

Let $\mathbf{m}\in SL(2,\mathbb{R})$ such that $b\ne 0.$ For suitable function f, we define the generalized translation operators associated with the operator ${\mathsf{\Delta }}_k^{\mathbf{m}}$ by:

$$\begin{array}{c}\hfill T_x^{\mathbf{m},k}f\left(y\right)=e^{\frac{i}{2}\frac{d}{b}{(\parallel x\parallel }^2+{\parallel y\parallel }^2)}{\tau }_x\left[e^{-\frac{i}{2}\frac{d}{b}{\left|\right|s\left|\right|}^2}f\left(s\right)\right]\left(y\right),\end{array}$$

(54)

where ${\tau }_x$ is the Dunkl translation operator.

We will rely on this definition for each function whose Dunkl translation is well-defined.

Proposition 7.

Let $\mathbf{m}\in SL(2,\mathbb{R})$ such that $b\ne 0.$ Then, the operators $T_x^{\mathbf{m},k}$, $x\in {\mathbb{R}}^N$ satisfy:

1. 

$T_0^{\mathbf{m},k}=Id$ and $T_x^{\mathbf{m},k}f\left(y\right)=T_y^{\mathbf{m},k}f\left(x\right).$

2. 

For all $x,y,z\in {\mathbb{R}}^N,$ we have the product formula

$$\begin{array}{c}\hfill \begin{array}{c}{T}_x^{\mathbf{m},k}\left[{\mathfrak{B}}_k^{\mathbf{m}}(.,y)\right]\left(z\right)=e^{-\frac{i}{2}\frac{a}{b}{\parallel y\parallel }^2}{\mathfrak{B}}_k^{\mathbf{m}}(x,y){\mathfrak{B}}_k^{\mathbf{m}}(z,y).\hfill \end{array}\end{array}$$

(55)

3. 

The operator $T_x^{\mathbf{m},k}$ is continuous from $C_{b,rad}\left({\mathbb{R}}^N\right)$ into itself, $L_k^2\left({\mathbb{R}}^N\right)$ into itself and on $L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$. More precisely, if $f\in L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right)$, we have

$$\begin{array}{c}\hfill {∥T_x^{\mathbf{m},k}f∥}_{L_k^p\left({\mathbb{R}}^N\right)}\le {∥f∥}_{L_k^p\left({\mathbb{R}}^N\right)}.\end{array}$$

(56)

Similarly, if $f\in L_k^2\left({\mathbb{R}}^N\right)$, we have

$$\begin{array}{c}\hfill {∥T_x^{\mathbf{m},k}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}\le {∥f∥}_{L_k^2\left({\mathbb{R}}^N\right)}.\end{array}$$

(57)

4. 

When $W={\mathbb{Z}}_2^N$, for any $f\in L_k^p\left({\mathbb{R}}^N\right)$, we have

$$\begin{array}{c}\hfill {∥T_x^{\mathbf{m},k}f∥}_{L_k^p\left({\mathbb{R}}^N\right)}\le 2^{\frac{N}{2p}|p-2|}{∥f∥}_{L_k^p\left({\mathbb{R}}^N\right)}.\end{array}$$

(58)

5. 

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

$$\begin{array}{c}\hfill {\mathfrak{F}}_D^{\mathbf{m}}\left[T_x^{{\mathbf{m}}^{-1},k}f\right]\left(\lambda \right)=e^{\frac{i}{2}\frac{d}{b}{\parallel \lambda \parallel }^2}\overline{{\mathfrak{B}}_k^{\mathbf{m}}(x,\lambda )}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right).\end{array}$$

(59)

6. 

For all $f\in L_{k,\mathrm{rad}}^p\left({\mathbb{R}}^N\right),p\in (1,2]$, we have

$$\begin{array}{c}\hfill {\mathfrak{F}}_D^{\mathbf{m}}\left[T_x^{{\mathbf{m}}^{-1},k}f\right]\left(\lambda \right)=e^{\frac{i}{2}\frac{d}{b}{\parallel \lambda \parallel }^2}\overline{{\mathfrak{B}}_k^{\mathbf{m}}(x,\lambda )}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right),a.e.\end{array}$$

(60)

Corollary 1.

Let $\mathbf{m}\in SL(2,\mathbb{R})$ such that $b\ne 0$ and $x\in {\mathbb{R}}^N.$ Then, for each $f\in \mathcal{S}\left({\mathbb{R}}^N\right)$, we have:

$$\begin{array}{ccc}\hfill T_x^{{\mathbf{m}}^{-1},k}f\left(y\right)& =& {\displaystyle \frac{1}{{(-ib)}^{\frac{2\gamma +N}{2}}}{e}^{-\frac{i}{2}\frac{a}{b}{\parallel y\parallel }^2}{\int }_{{\mathbb{R}}^N}K(i\lambda /b,y)\overline{{\mathfrak{B}}_k^{\mathbf{m}}(x,\lambda )}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right)d{\gamma }_k\left(\lambda \right).}\hfill \end{array}$$

Let $\mathbf{m}\in SL(2,\mathbb{R})$ such that $b\ne 0$. The generalized convolution product associated with ${\mathfrak{F}}_D^{\mathbf{m}}$ of two suitable functions f and g on ${\mathbb{R}}^N$ is the function $f\underset{\mathbf{m},k}{\ast }g$ defined by

$$\begin{array}{c}\hfill {\displaystyle f\underset{\mathbf{m},k}{\ast }g\left(x\right)={\int }_{{\mathbb{R}}^N}{T}_x^{\mathbf{m},k}f(-y)e^{-i\frac{d}{b}{\parallel y\parallel }^2}g\left(y\right)d{\gamma }_k\left(y\right)}\end{array}$$

(61)

for all x such that the integral exists. The elementary properties of convolutions are summarized in the following proposition.

Proposition 8.

Assuming that all integrals in question exist, we have

1. 

$f\underset{\mathbf{m},k}{\ast }g=g\underset{\mathbf{m},k}{\ast }f.$

2. 

$T_x^{\mathbf{m},k}\left(f\underset{\mathbf{m},k}{\ast }g\right)=\left(T_x^{\mathbf{m},k}f\right)\underset{\mathbf{m},k}{\ast }g=f\underset{\mathbf{m},k}{\ast }\left(T_x^{\mathbf{m},k}g\right).$

Proposition 9

(Young’s Inequality). Let $\mathbf{m}\in SL(2,\mathbb{R})$ such that $b\ne 0$ and let $1\le p,q,r\le \infty$ such that $p^{-1}+q^{-1}=r^{-1}+1.$ If $W={\mathbb{Z}}_2^N$, $f\in L_k^p\left({\mathbb{R}}^N\right)$ and $g\in L_k^q\left({\mathbb{R}}^N\right),$ then $f\underset{\mathbf{m},k}{\ast }g\in L_k^r\left({\mathbb{R}}^N\right)$ and we have

$$\begin{array}{c}\hfill {∥f\underset{\mathbf{m},k}{\ast }g∥}_{L_k^r\left({\mathbb{R}}^N\right)}\le 2^{\frac{N}{2p}|p-2|}{∥f∥}_{L_k^p\left({\mathbb{R}}^N\right)}{∥g∥}_{L_k^q\left({\mathbb{R}}^N\right)}.\end{array}$$

(62)

Moreover, if we assume that $f\in L_{k,rad}^p\left({\mathbb{R}}^N\right)$ and $g\in L_k^q\left({\mathbb{R}}^N\right),$ then

$$\begin{array}{c}\hfill {∥f\underset{\mathbf{m},k}{\ast }g∥}_{L_k^r\left({\mathbb{R}}^N\right)}\le {∥f∥}_{L_k^p\left({\mathbb{R}}^N\right)}{∥g∥}_{L_k^q\left({\mathbb{R}}^N\right)}.\end{array}$$

(63)

Proposition 10.

Let $\mathbf{m}\in SL(2,\mathbb{R})$ such that $b\ne 0.$

1. 

If $f\in L_{k,rad}^1\left({\mathbb{R}}^N\right)$ and $g\in L_k^1\left({\mathbb{R}}^N\right)$, then for all $x\in {\mathbb{R}}^N$,

$$\begin{array}{c}\hfill \left(1/{\left(ib\right)}^{\frac{2\gamma +N}{2}}\right){\mathfrak{F}}_D^{\mathbf{m}}\left(f\underset{{\mathbf{m}}^{-1},k}{\ast }g\right)\left(x\right)=e^{-\frac{i}{2}\frac{d}{b}{\parallel x\parallel }^2}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(x\right){\mathfrak{F}}_D^{\mathbf{m}}\left(g\right)\left(x\right).\end{array}$$

(64)

2. 

If $f\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$ and $g\in L_k^p\left({\mathbb{R}}^N\right),p\in (1,2]$, then

$$\begin{array}{c}\hfill \left(1/{\left(ib\right)}^{\frac{2\gamma +N}{2}}\right){\mathfrak{F}}_D^{\mathbf{m}}\left(f\underset{{\mathbf{m}}^{-1},k}{\ast }g\right)\left(x\right)=e^{-\frac{i}{2}\frac{d}{b}{\parallel x\parallel }^2}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(x\right){\mathfrak{F}}_D^{\mathbf{m}}\left(g\right)\left(x\right),a.e.\end{array}$$

(65)

3. 

If $f,g,h\in L_{k,\mathrm{rad}}^1\left({\mathbb{R}}^N\right)$, then we have

$$\left(f\underset{\mathbf{m},k}{\ast }g\right)\underset{\mathbf{m},k}{\ast }h=f\underset{\mathbf{m},k}{\ast }\left(g\underset{\mathbf{m},k}{\ast }h\right).$$

4. 

If $W={\mathbb{Z}}_2^N$, the previous three results are valid without requiring the functions to be radials.

3. The Linear Canonical Dunkl-Gabor Transform

In this section, we introduce the continuous linear canonical deformed Gabor transform associated with the operator ${\mathsf{\Delta }}_k^{\mathbf{m}}$, and we give some harmonic analysis properties for it. We will denote by $L_{{\mu }_k}^p\left({\mathbb{R}}^{2N}\right),p\in [1,\infty ],$ the space of measurable functions $f:{\mathbb{R}}^{2N}\to \mathbb{C}$ such that

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

where

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

Definition 7.

Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$ and $t\in {\mathbb{R}}^N.$ The modulation of the function φ by t is defined as:

$${\mathfrak{M}}_t^{\mathbf{m}}\phi :={\mathfrak{F}}_D^{{\mathbf{m}}^{-1}}\left(\sqrt{|T_t^{\mathbf{m},k}(e^{\frac{id}{2b}{\parallel z\parallel }^2}{\left|\phi \right|}^2\left)\right|}\right).$$

(66)

In the following proposition, we state some properties of the modulation ${\mathfrak{M}}_t^{\mathbf{m}}\phi$.

Proposition 11.

Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$. Then,

1. 

For all $t\in {\mathbb{R}}^N$, we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(y\right){|}^{2}d{\gamma }_k\left(y\right)={\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.}$$

(67)

2. 

For all $y\in {\mathbb{R}}^N$, we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(y\right){|}^{2}d{\gamma }_k\left(t\right)={\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.}$$

(68)

Proof. 

Relation (67) is as an immediate consequence of the relations (27), (48), (54) and (66). On the other hand, from the relations (24), (27), (48), (54) and (66), we have

$${\displaystyle {\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(y\right){|}^{2}d{\gamma }_k\left(t\right)={\int }_{{\mathbb{R}}^N}{\tau }_t{\left(\right|\phi |}^2)\left(y\right)d{\gamma }_k\left(t\right)={\int }_{{\mathbb{R}}^N}{\tau }_y{\left(\right|\phi |}^2)\left(t\right)d{\gamma }_k\left(t\right)={\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.}$$

The proof is complete. □

Definition 8.

Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$. We consider the family ${\phi }_{x,t}^{\mathbf{m}}$, $x,t\in {\mathbb{R}}^N$ defined by:

$$\forall \xi \in {\mathbb{R}}^N,{\phi }_{x,t}^{\mathbf{m}}\left(\xi \right):=T_x^{{\mathbf{m}}^{-1},k}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\xi \right).$$

(69)

For any function $f\in L_k^2\left({\mathbb{R}}^N\right)$, we define its linear canonical Dunkl-Gabor transform (LCDGT) by

$${\displaystyle \forall x,t\in {\mathbb{R}}^N,{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(x,t)={\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{\parallel \xi \parallel }^2}f\left(\xi \right){\phi }_{x,t}^{\mathbf{m}}(-\xi )d{\gamma }_k\left(\xi \right).}$$

(70)

Remark 3.

1. 

We have for all $x,t\in {\mathbb{R}}^N$:

$$\parallel {\phi }_{x,t}^{\mathbf{m}}{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\le {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(71)

2. 

The linear canonical Dunkl-Gabor transform can be written as:

$$\forall (x,t)\in {\mathbb{R}}^{2N},{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(x,t)=f{\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \left(x\right).$$

(72)

3. 

A simple calculation implies that, for any $f\in L_k^2\left({\mathbb{R}}^N\right)$ and any $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$,

$${\mathfrak{G}}_{{\phi }_{1/\lambda }}^{\mathbf{m}}\left(f_{\lambda }\right)(x,t)={\mathfrak{G}}_{\phi }^{{\mathbf{m}}_{\lambda }}\left(f\right)\left({\displaystyle \frac{x}{\lambda }},\lambda t\right),\lambda >0,(x,t)\in {\mathbb{R}}^{2N},$$

(73)

where for all $r>0$,

$$g_r\left(y\right):={\displaystyle \frac{1}{r^{2\gamma +N}}}g(y/r),y\in {\mathbb{R}}^N,$$

and

$${\mathbf{m}}_r:=\left(\begin{array}{cc}a{r}^2& b\\ c& d{r}^{-2}\end{array}\right).$$

Using basic computation, we prove the following result.

Lemma 1.

If $\phi \in L_k^{\infty }\left({\mathbb{R}}^N\right)\cap L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$, then for every $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$${\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(·,t)\right)\left(y\right)={\left(ib\right)}^{\frac{2\gamma +N}{2}}{e}^{-i\frac{d}{2b}{\parallel \xi \parallel }^2}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(y\right)\sqrt{{\tau }_t{\left|\phi \right|}^2\left(y\right)}.$$

(74)

Proposition 12.

Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$. Then, for every $f\in L_k^2\left({\mathbb{R}}^N\right),{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)$ belongs to $L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)$ and we have

$$\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)}\le {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(75)

Proof. 

Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$. Using (70), Cauchy–Schwartz’s inequality and Relation (71), we have for every $f\in L_k^2\left({\mathbb{R}}^N\right)$

$$\begin{array}{ccc}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(x,t)|\hfill & \le \hfill & {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel {\phi }_{x,t}^{\mathbf{m}}\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ \hfill & \le \hfill & {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

Thus, ${\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)$ belongs to $L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)$, and we have

$$\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)}\le {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

This completes the proofs. □

Theorem 4

$(L_k^2$ inversion formula). Let $\phi \in L_k^{\infty }\left({\mathbb{R}}^N\right)\cap L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$ with ${\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1$. Then, for any function $f\in L_k^2\left({\mathbb{R}}^N\right)$, we have

$${\displaystyle f_j\left(x\right)={\int }_{B_N(0,j)}{\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{\parallel y\parallel }^2}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(y,t)T_x^{{\mathbf{m}}^{-1},k}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)(-y)d{\mu }_k(y,t)}$$

(76)

in $L_k^2\left({\mathbb{R}}^N\right)$ and satisfies

$${\displaystyle \underset{j\to \infty }{lim}{\parallel f_j-f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=0.}$$

(77)

For proof this theorem, we need the following Lemmas.

Lemma 2.

Let φ be as above. Then, for any function $f\in L_k^2\left({\mathbb{R}}^N\right)$, we have

$$\begin{array}{c}\hfill f_j={\mathfrak{R}}_j{\ast }_{{\mathbf{m}}^{-1},k}f,\end{array}$$

(78)

where for any positive integer j

$${\displaystyle {\mathfrak{R}}_j\left(x\right):={\int }_{B_N(0,j)}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\right|}^2{\mathfrak{B}}_k^{{\mathbf{m}}^{-1}}(x,\lambda )d{\mu }_k(\lambda ,t),\mathrm{for}x\in {\mathbb{R}}^N.}$$

Proof. 

We have

$$\begin{array}{ccc}\forall x\in {\mathbb{R}}^N,f_j\left(x\right)\hfill & =\hfill & {\displaystyle {\int }_{B_N(0,j)}{\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{\parallel y\parallel }^2}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(y,t)T_x^{{\mathbf{m}}^{-1},k}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\phi (-y)d{\mu }_k(y,t)}\hfill \\ & =\hfill & {\displaystyle {\int }_{B_N(0,j)}\left({\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,t){\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(x\right)d{\gamma }_k\left(t\right)}\hfill \\ & =\hfill & {\displaystyle {\int }_{B_N(0,j)}\left(f{\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi {\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(x\right)d{\gamma }_k\left(t\right)}\hfill \\ & =\hfill & {\displaystyle {\int }_{B_N(0,j)}{\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{\parallel y\parallel }^2}{T}_x^{{\mathbf{m}}^{-1},k}f(-y)\left({\mathfrak{M}}_t^{\mathbf{m}}\phi {\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \left(y\right)\right)d{\mu }_k(t,y).}\hfill \end{array}$$

Then, using the hypothesis on $\phi$ and Fubini’s theorem, we derive that, for all $x\in {\mathbb{R}}^N$,

$$\begin{array}{ccc}{f}_j\left(x\right)\hfill & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{\parallel y\parallel }^2}{T}_x^{{\mathbf{m}}^{-1},k}f(-y)\left[{\int }_{B_N(0,j)}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi {\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(y\right)d{\gamma }_k\left(t\right)\right]d{\gamma }_k\left(y\right).}\hfill \end{array}$$

Moreover, involving the hypothesis on $\phi$ and by a standard analysis, we obtain

$${\displaystyle {\mathfrak{R}}_j\left(j\right)={\int }_{B_N(0,j)}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi {\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(y\right)d{\gamma }_k\left(t\right).}$$

(79)

Thus, we derive

$$\begin{array}{ccc}\forall x\in {\mathbb{R}}^N,f_j\left(x\right)\hfill & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{\parallel y\parallel }^2}{T}_x^{{\mathbf{m}}^{-1},k}f(-y){\mathfrak{R}}_j\left(y\right)d{\gamma }_k\left(y\right)}\hfill \\ & =\hfill & {\mathfrak{R}}_j{\ast }_{{\mathbf{m}}^{-1},k}f\left(x\right).\hfill \end{array}$$

The proof is complete. □

Lemma 3.

Let φ be as above. For any positive integer j, we define the function

$${\displaystyle {\mathfrak{H}}_j\left(\lambda \right):={\int }_{B_N(0,j)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\right|}^{2}d{\gamma }_k\left(t\right),\lambda \in {\mathbb{R}}^N.}$$

Then, we have

$${\mathfrak{R}}_j\in L_k^2\left({\mathbb{R}}^N\right),{\mathfrak{H}}_j\in L_k^1\left({\mathbb{R}}^N\right)\cap L_k^{\infty }\left({\mathbb{R}}^N\right)and{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{R}}_j\right)={\mathfrak{H}}_j.$$

Proof. 

Using the hypothesis on $\phi$, (79) and the Cauchy–Schwarz inequality, we infer that ${\mathfrak{R}}_j\in L_k^2\left({\mathbb{R}}^N\right)$. Now, we will to prove that ${\mathfrak{H}}_j\in L_k^1\left({\mathbb{R}}^N\right)\cap L_k^{\infty }\left({\mathbb{R}}^N\right)$. Firstly, we prove that ${\mathfrak{H}}_j\in L_k^1\left({\mathbb{R}}^N\right)$. Indeed, from (66), we have

$$\begin{array}{ccc}\forall \lambda \in {\mathbb{R}}^N,\left|{\mathfrak{H}}_j\left(\lambda \right)\right|\hfill & =\hfill & {\displaystyle \left|{\int }_{B_N(0,j)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\right|}^{2}d{\gamma }_k\left(t\right)\right|}\hfill \\ & =\hfill & {\displaystyle {\int }_{B_N(0,j)}{\tau }_t{\left|\phi \right|}^2\left(\lambda \right)d{\gamma }_k\left(t\right)}\hfill \\ & \le \hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{\tau }_t{\left|\phi \right|}^2\left(\lambda \right)d{\gamma }_k\left(t\right)}\hfill \\ & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{\tau }_{\lambda }{\left|\phi \right|}^2\left(t\right)d{\gamma }_k\left(t\right)={\left|\right|\phi \left|\right|}_{L_k^2\left({\mathbb{R}}^N\right)}^2<\infty .}\hfill \end{array}$$

Thus, ${\mathfrak{H}}_j$ belongs to $L_k^{\infty }\left({\mathbb{R}}^N\right).$ On the other hand, by Fubini’s theorem and the relation (27), we have

$$\begin{array}{ccc}{\displaystyle \left|\right|{\mathfrak{H}}_j{\left|\right|}_{L_k^1\left({\mathbb{R}}^N\right)}={\int }_{{\mathbb{R}}^N}\left|{\mathfrak{H}}_j\left(\lambda \right)\right|d{\gamma }_k\left(\lambda \right)}\hfill & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}\left|{\int }_{B_N(0,j)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\right|}^{2}d{\gamma }_k\left(t\right)\right|d{\gamma }_k\left(\lambda \right)}\hfill \\ & =\hfill & {\displaystyle {\int }_{B_N(0,j)}\left({\int }_{{\mathbb{R}}^N}{\tau }_t{\left|\phi \right|}^2\left(\lambda \right)d{\gamma }_k\left(\lambda \right)\right)d{\gamma }_k\left(t\right)}\hfill \\ & \le \hfill & {\displaystyle {\left|\right|\phi \left|\right|}_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{B_N(0,j)}d{\gamma }_k\left(t\right)<\infty .}\hfill \end{array}$$

Therefore, ${\mathfrak{H}}_j$ belongs to $L_k^1\left({\mathbb{R}}^N\right).$ Moreover, by Fubini’s theorem, we have

$$\begin{array}{ccc}\forall x\in {\mathbb{R}}^N,{\left({\mathfrak{F}}_D^{\mathbf{m}}\right)}^{-1}\left({\mathfrak{H}}_j\right)\left(x\right)\hfill & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{H}}_j\left(\lambda \right){\mathfrak{B}}_k^{{\mathbf{m}}^{-1}}(x,\lambda )d{\gamma }_k\left(\lambda \right)}\hfill \\ & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{\mathfrak{B}}_k^{{\mathbf{m}}^{-1}}(x,\lambda ){\int }_{B_N(0,j)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\right|}^{2}d{\gamma }_k\left(t\right)d{\gamma }_k\left(\lambda \right)}\hfill \\ & =\hfill & {\displaystyle {\int }_{B_N(0,j)}{\int }_{{\mathbb{R}}^N}{\mathfrak{B}}_k^{{\mathbf{m}}^{-1}}(x,\lambda ){\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\right|}^{2}d{\gamma }_k\left(t\right)d{\gamma }_k\left(\lambda \right)}\hfill \\ & =\hfill & {\mathfrak{R}}_j\left(x\right).\hfill \end{array}$$

The proof is complete. □

Proof of Theorem 4.

It follows from Proposition 4, Lemma 2 and Lemma 3 that $f_j\in L_k^2\left({\mathbb{R}}^N\right)$ and

$$\forall \lambda \in {\mathbb{R}}^N,{\mathfrak{F}}_D^{\mathbf{m}}\left(f_j\right)\left(\lambda \right)={\mathfrak{H}}_j\left(\lambda \right){\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right).$$

Then, by Plancherel’s Formula (48) and the fact that ${\mathfrak{H}}_j\to 1$ pointwise as $j\to \infty$, we have:

$$\begin{array}{ccc}\parallel f-f_j{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right)-{\mathfrak{H}}_j\left(\lambda \right){\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right)|}^{2}d{\gamma }_k\left(\lambda \right)}\hfill \\ & =\hfill & {\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right)(1-{\mathfrak{H}}_j\left(\lambda \right))\right|}^{2}d{\gamma }_k\left(\lambda \right)\to 0}\hfill \end{array}$$

as $j\to \infty$, which achieves the proof. □

Proposition 13.

If $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$, then for every $f,g\in L_k^2\left({\mathbb{R}}^N\right)$,

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(x,t)\overline{{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(g\right)(x,t)}d{\mu }_k(x,t)={\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}f\left(x\right)\overline{g\left(x\right)}d{\gamma }_k\left(x\right).}$$

(80)

Proof. 

Using the relations (72), (47) and (65), we have

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(x,t)\overline{{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(g\right)(x,t)}d{\mu }_k(x,t)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}f{\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \left(x\right)\overline{g{\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \left(x\right)}d{\gamma }_k\left(x\right)d{\gamma }_k\left(t\right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^{\mathbf{m}}\left(f{\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^{\mathbf{m}}\left(g{\ast }_{{\mathbf{m}}^{-1},k}{\mathfrak{M}}_t^{\mathbf{m}}\phi \right)}\left(\lambda \right)d{\gamma }_k\left(\lambda \right)d{\gamma }_k\left(t\right)}\hfill \\ & =& {\displaystyle {\left|b\right|}^{2\gamma +N}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^{\mathbf{m}}\left(g\right)\left(\lambda \right)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{M}}_t^{\mathbf{m}}\phi \right)\left(\lambda \right)\right|}^{2}d{\gamma }_k\left(\lambda \right)d{\gamma }_k\left(t\right)}\hfill \\ & =& {\displaystyle {\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \right)\overline{{\mathfrak{F}}_D^{\mathbf{m}}\left(g\right)\left(\lambda \right)}d{\gamma }_k\left(\lambda \right)}\hfill \\ & =& {\displaystyle {\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}f\left(x\right)\overline{g\left(x\right)}d{\gamma }_k\left(x\right).}\hfill \end{array}$$

The proof is complete. □

Corollary 2

(Plancherel’s Formula). Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$. If $f,g\in L_k^2\left({\mathbb{R}}^N\right)$, then ${\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)\in L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)$ and we have

$$\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}={\left|b\right|}^{\frac{2\gamma +N}{2}}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$$

(81)

Proposition 14.

Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)$ and $2\le p<\infty$. Then, for all $f\in L_k^2\left({\mathbb{R}}^N\right)$

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(x,t)|}^{p}d{\mu }_k(x,t)\le {\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^p{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^p.}$$

(82)

Proof. 

We have

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(x,t)|}^{p}d{\mu }_k(x,t)\le \parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2{\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)}^{p-2}.}$$

Using Proposition 12 and Corollary 2, we obtain the desired result. □

4. Heisenberg-Type Uncertainty Principles for the LCDGT

The window function $\phi$ will be non trivial and radial in $L_k^2\left({\mathbb{R}}^N\right).$

4.1. $L^2$-Heisenberg-Type Uncertainty Inequalities

We begin this subsection by the following Heisenberg uncertainty principle for the LCDT.

Proposition 15.

For $s,t>0$, there exists a positive constant $C_{k,b}(s,t)$, such that for every $f\in L_k^2\left({\mathbb{R}}^N\right),$ the following inequality holds

$$\begin{array}{c}\hfill {\parallel \parallel \xi \parallel }^s{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right){\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{t}{s+t}}{\parallel \left|\right|x\left|\right|}^{t}f{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{s}{s+t}}\ge C_{k,b}(s,t){\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.\end{array}$$

(83)

For $s,t\ge 1$, $C_{k,b}(s,t)={\left(\left(\frac{2\gamma +N}{2}\right)\left|b\right|\right)}^{{\displaystyle \frac{st}{s+t}}}$.

Proof. 

Involving the Heisenberg uncertainty principle for the Dunkl transform ${\mathfrak{F}}_D$ (see [48]) and the identity (34), we derive the result. □

Theorem 5.

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

$$\begin{array}{ccc}\hfill & & {\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2t}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{s}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2s}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\right)}^{\frac{t}{s+t}}\hfill \\ & \ge & {\left(C_{k,b}(s,t)\right)}^2{\left|b\right|}^{\frac{(2\gamma +N)s}{s+t}}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{2s}{s+t}}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

(84)

Proof. 

By fixing $\omega$, we have by Inequality (83)

$$\begin{array}{ccc}& & {\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2s}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\right)}^{\frac{t}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel \theta \parallel }^{2t}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\gamma }_k\left(\theta \right)}\right)}^{\frac{s}{s+t}}\hfill \\ & \ge & {\displaystyle {\left(C_{k,b}(s,t)\right)}^2{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\gamma }_k\left(\theta \right).}\hfill \end{array}$$

By integrating over $\omega$, we obtain

$$\begin{array}{ccc}& & {\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \xi \parallel }^{2s}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right)\left(\xi \right)\right|}^{2}d{\mu }_k(\xi ,\omega )}\right)}^{\frac{t}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2t}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{s}{s+t}}\hfill \\ & \ge & {\displaystyle {\left(C_{k,b}(s,t)\right)}^2{\int }_{{\mathbb{R}}^{2N}}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega ).}\hfill \end{array}$$

Further, using the identity (74), we derive

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \xi \parallel }^{2s}|{\mathfrak{F}}_D^{\mathbf{m}}\left({\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right){\left(\xi \right)|}^{2}d{\mu }_k(\xi ,\omega )={\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2s}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

Thus, we deduce that

$$\begin{array}{ccc}& & {\left|b\right|}^{\left(\frac{(2\gamma +N)t}{s+t}\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{2t}{s+t}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2s}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\mu }_k\left(\xi \right)}\right)}^{\frac{t}{t+s}}\times \hfill \\ & & {\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2t}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{s}{s+t}}\hfill \\ & \ge & {\displaystyle {\left(C_{k,b}(s,t)\right)}^2{\int }_{{\mathbb{R}}^{2N}}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )={\left(C_{k,b}(s,t)\right)}^2{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.}\hfill \end{array}$$

As desired. □

Proposition 16

(Nash-type Inequality for ${\mathfrak{G}}_{\phi }^{\mathbf{m}}$). Let $s>0$. Then, there is a constant $\mathcal{C}(k,s,b)>0$ such that, for every $f\in L_k^2\left({\mathbb{R}}^N\right),$

$${\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}⩽\mathcal{C}(k,s,b)\parallel {\parallel (\theta ,\omega )\parallel }^s{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}.$$

(85)

Proof. 

Let $f\in L_k^2\left({\mathbb{R}}^N\right)$ be a nonzero function and let $R>0$. Then, by (81), we have

$$\begin{array}{ccc}\hfill {\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2& =& \parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2={\parallel {\chi }_{B_{2N}(0,R)}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2\hfill \\ & +& \parallel (1-{\chi }_{B_{2N}(0,R)}){\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2,\hfill \end{array}$$

where

$$B_{2N}(0,R):=\left\{(\theta ,\omega )\in {\mathbb{R}}^{2N}:\parallel (\theta ,\omega )\parallel ⩽R\right\}.$$

By (75), we have

$$\begin{array}{ccc}\hfill \parallel {\chi }_{B_{2N}(0,R)}{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2& ⩽& {\displaystyle {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^{2N}}{\chi }_{B_{2N}(0,R)}d{\mu }_k(\theta ,\omega )}\hfill \\ & ⩽& C{R}^{4\gamma +2N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

Moreover,

$$\begin{array}{ccc}\parallel (1-{\chi }_{B_{2N}(0,R)}){\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2\hfill & ⩽\hfill & R^{-2s}\parallel (1-{\chi }_{B_{2N}(0,R)}){\parallel (\theta ,\omega )\parallel }^s{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2\hfill \\ & ⩽\hfill & R^{-2s}\parallel {\parallel (\theta ,\omega )\parallel }^s{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2.\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}\hfill {\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2& ⩽& C{R}^{4\gamma +2N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & +& R^{-2s}\parallel {\parallel (\theta ,\omega )\parallel }^s{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2.\hfill \end{array}$$

By minimizing over $R>0$, the previous inequality implies

$${\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2⩽{C(k,s)\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{2s}{2\gamma +N+s}}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{2s}{2\gamma +N+s}}\parallel {\parallel (\theta ,\omega )\parallel }^s{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{2(2\gamma +N)}{2\gamma +N+s}}.$$

Then, we obtain the desired result. □

4.2. Entropic Uncertainty Inequality

Consider $\rho$ to be a probability density function on ${\mathbb{R}}^{2N},$ that is,

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

According to Shannon [49],

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

defines the k-entropy of the probability density function $\rho$ on ${\mathbb{R}}^{2N}$.

Therefore, whenever the preceding integral on the right side is properly defined, we can extend the definition of the k-entropy of a nonnegative measurable function $\rho$ on ${\mathbb{R}}^{2N}$. Then, we have the following result.

Proposition 17.

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

$$E_k\left(\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)|}^2{)⩾-2|b|}^{2\gamma +N}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{2}ln\left({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\right).$$

(86)

Proof. 

Assume that ${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1.$ By (75), we have

$$|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|⩽\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1.$$

(87)

Thus, $E_k\left(\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){|}^2)⩾0.$ Next, let

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

Therefore $\varphi ,\psi \in L_k^2\left({\mathbb{R}}^N\right)$ and ${\parallel \varphi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \psi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1.$ Then, $E_k\left(\right|{\mathfrak{G}}_{\psi }^{\mathbf{m}}\left(\varphi \right){|}^2)⩾0.$ Moreover,

$${\mathfrak{G}}_{\psi }^{\mathbf{m}}\left(\varphi \right)={\displaystyle \frac{1}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}}}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right),$$

which implies

$$E_k\left(\right|{\mathfrak{G}}_{\psi }^{\mathbf{m}}{\left(\varphi \right)|}^2)={\displaystyle \frac{1}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}}{E}_k\left(\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)|}^2{)+2|b|}^{2\gamma +N}ln\left({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\right).$$

Using the fact that $E_k\left(\right|{\mathfrak{G}}_{\psi }^{\mathbf{m}}\left(\varphi \right){|}^2)⩾0,$ we deduce that

$$E_k\left(\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)|}^2{)⩾-2|b|}^{2\gamma +N}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{2}ln\left({\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\right).$$

The proof is complete. □

Using the k-entropy of the linear canonical Dunkl-Gabor transform, we can obtain another version of the Heisenberg uncertainty principle for ${\mathfrak{G}}_{\phi }^{\mathbf{m}}$.

Theorem 6.

Let $p,q>0$. Then, for every $f\in L_k^2\left({\mathbb{R}}^N\right)$, we have

$$\begin{array}{c}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^p{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{q}{p+q}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \omega \parallel }^q{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{p}{p+q}}\\ ⩾{\mathcal{C}}_{p,q}{(k,b)\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2,\end{array}$$

where

$${\displaystyle {\mathcal{C}}_{p,q}(k,b)={\left|b\right|}^{2\gamma +N}\frac{2\gamma +N}{p^{\frac{q}{p+q}}{q}^{\frac{p}{p+q}}}exp\left(\frac{pq}{\left(2\gamma +N\right)(p+q)}ln\left(\frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}\right)-1\right).}$$

Proof. 

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

$${\eta }_{t,p,q}^k(\theta ,\omega ):={\displaystyle \frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}}{\displaystyle \frac{exp\left(-\frac{{\parallel \theta \parallel }^p+{\parallel \omega \parallel }^q}{t}\right)}{t^{\frac{\left(2\gamma +N\right)(p+q)}{pq}}}}.$$

Using basic computation, we obtain

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

In particular, the measure $d{\sigma }_{t,p,q}^k(\theta ,\omega ):={\eta }_{t,p,q}^k(\theta ,\omega )d{\mu }_k(\theta ,\omega )$ is a probability measure on ${\mathbb{R}}^{2N}$. As $\phi \left(t\right)=tln\left(t\right)$ is a convex function over $(0,\infty )$, we can use Jensen’s condition for convex functions to obtain

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}\frac{|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^2}{{\eta }_{t,p,q}^k(\theta ,\omega )}ln\left(\frac{|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^2}{{\eta }_{t,p,q}^k(\theta ,\omega )}\right)d{\sigma }_{t,p,q}^k(\theta ,\omega )⩾0.}$$

Therefore,

$$\begin{array}{ccc}& & E_k\left(\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)|}^2)+ln\left({\displaystyle \frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}}\right){\left|b\right|}^{2\gamma +N}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & \le & ln\left(t^{\frac{\left(2\gamma +N\right)(p+q)}{pq}}\right){\left|b\right|}^{2\gamma +N}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & +& {\displaystyle \frac{1}{t}{\int }_{{\mathbb{R}}^{2N}}{(\parallel \theta \parallel }^p+{\parallel \omega \parallel }^q\left)\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega ).}\hfill \end{array}$$

Assume that ${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel h\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1.$ Then, by Proposition 17, we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}{(\parallel \theta \parallel }^p+{\parallel \omega \parallel }^q\left)\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & ⩾& {t\left|b\right|}^{2\gamma +N}\left(ln\left({\displaystyle \frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}}\right)-ln\left(t^{\frac{\left(2\gamma +N\right)(p+q)}{pq}}\right)\right).\hfill \end{array}$$

However, the expression

$$t\left(ln\left({\displaystyle \frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}}\right)-ln\left(t^{\frac{\left(2\gamma +N\right)(p+q)}{pq}}\right)\right)$$

attains its upper bound at

$$t_0=exp\left({\displaystyle \frac{pq}{\left(2\gamma +N\right)(p+q)}}ln\left({\displaystyle \frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}}\right)-1\right),$$

and consequently

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}{(\parallel \theta \parallel }^p+{\parallel \omega \parallel }^q\left)\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )⩾r_{p,q}(k,b),}$$

where

$$r_{p,q}(k,b)={\left|b\right|}^{2\gamma +N}{\displaystyle \frac{\left(2\gamma +N\right)(p+q)}{pq}}exp\left({\displaystyle \frac{pq}{\left(2\gamma +N\right)(p+q)}}ln\left({\displaystyle \frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}}\right)-1\right).$$

Hence,

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^p|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )+{\int }_{{\mathbb{R}}^{2N}}{\parallel \omega \parallel }^q{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )⩾r_{p,q}(k,b).}$$

Now by substituting f by $f_{\lambda }$, $\phi$ by ${\phi }_{\frac{1}{\lambda }}$, $\mathbf{m}$ by ${\mathbf{m}}_{\frac{1}{\lambda }}$ and since $\parallel f_{\lambda }{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\parallel {\phi }_{{\displaystyle \frac{1}{\lambda }}}{\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1,$ the above inequality gives

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^p|{\mathfrak{G}}_{{\phi }_{\frac{1}{\lambda }}}^{{\mathbf{m}}_{\frac{1}{\lambda }}}\left(f_{\lambda }\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )+{\int }_{{\mathbb{R}}^{2N}}{\parallel \omega \parallel }^q{|{\mathfrak{G}}_{{\phi }_{\frac{1}{\lambda }}}^{{\mathbf{m}}_{\frac{1}{\lambda }}}\left(f_{\lambda }\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )⩾r_{p,q}(k,b).}$$

By (73), we have

$${\displaystyle {\lambda }^{-p}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^p|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )+{\lambda }^q{\int }_{{\mathbb{R}}^{2N}}{\parallel \omega \parallel }^q{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )⩾r_{p,q}(k,b).}$$

If we choose

$$\lambda ={\left({\displaystyle \frac{{\displaystyle p{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^p{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}}{{\displaystyle q{\int }_{{\mathbb{R}}^{2N}}{\parallel \omega \parallel }^q{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}}}\right)}^{\frac{1}{p+q}},$$

then we obtain

$${\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^p{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{q}{p+q}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \omega \parallel }^q{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{p}{p+q}}⩾{\mathcal{C}}_{p,q}(k,b),$$

where

$$\begin{array}{ccc}{\mathcal{C}}_{p,q}(k,b)\hfill & =\hfill & r_{p,q}(k,b)\frac{p^{\frac{p}{p+q}}{q}^{\frac{q}{p+q}}}{p+q}\hfill \\ & =\hfill & {\left|b\right|}^{2\gamma +N}\frac{2\gamma +N}{p^{\frac{q}{p+q}}{q}^{\frac{p}{p+q}}}exp\left(\frac{pq}{\left(2\gamma +N\right)(p+q)}ln\left(\frac{pq}{4{\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{p}\right)\mathsf{\Gamma }\left(\frac{2\gamma +N}{q}\right)}\right)-1\right).\hfill \end{array}$$

Hence, the desired result follows from the previous inequality by substituting f by $f/{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}$ and $\phi$ by $\phi /{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}.$ □

Remark 4.

If $p=q=2$, then

$$\begin{array}{cc}\hfill & {∥\parallel \theta \parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}{∥\parallel \omega \parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}\hfill \\ & ⩾{\displaystyle \frac{2\gamma +N}{2e{\left({\left(d_k\right)}^2\mathsf{\Gamma }\left(\frac{2\gamma +N}{2}\right)\right)}^{\frac{2}{2\gamma +N}}}}{\left|b\right|}^{2\gamma +N}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

(88)

4.3. $L^p$-Heisenberg’s Uncertainty Principle

In this subsection, we will prove a general form of the Heisenberg-type uncertainty inequality for the linear canonical Dunkl-Gabor transform in the $L^p$-setting. For $\lambda >0$, we set

$${\mathsf{\Gamma }}_{\lambda }(\theta ,\omega ):=e^{-{\lambda \parallel (\theta ,\omega )\parallel }^2},(\theta ,\omega )\in {\mathbb{R}}^{2N}.$$

By simple calculations, it is easy to check that for every $1⩽q<\infty ,$ there exists a positive constant C, such that

$$\parallel {\mathsf{\Gamma }}_{\lambda }{\parallel }_{L_{{\mu }_k}^q\left({\mathbb{R}}^{2N}\right)}=C{\lambda }^{-{\displaystyle \frac{2\gamma +N}{q}}}.$$

(89)

Lemma 4.

Let $0<s<\frac{2\gamma +N}{2p^{\prime }}$ with $1<p⩽2$. Then, there is a constant $C(k,b)>0$ such that, for all $\lambda >0$ and all $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$${∥{\mathsf{\Gamma }}_{\lambda }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}⩽C(k,b){\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\lambda }^{-2s}\left({∥{\parallel \theta \parallel }^{s}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^{2p}\left({\mathbb{R}}^N\right)}\right).$$

(90)

Proof. 

Let us assume that

$${∥{\parallel \theta \parallel }^{s}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^{2p}\left({\mathbb{R}}^N\right)}<\infty .$$

For $r>0,$ let $f_r={\chi }_{B_N(0,r)}f$ and $f^r=f-f_r$. Since

$$|f^r\left(y\right)|⩽r^{-s}{|\parallel \theta \parallel }^{s}f\left(y\right)|,$$

we deduce from Proposition 14 that

$$\begin{array}{ccc}{∥{\mathsf{\Gamma }}_{\lambda }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left({\chi }_{B_N^c(0,r)}f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}\hfill & ⩽\hfill & {∥{\mathsf{\Gamma }}_{\lambda }∥}_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)}{∥{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left({\chi }_{B_N^c(0,r)}f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}\hfill \\ & ⩽\hfill & {\left|b\right|}^{{\displaystyle \frac{2\gamma +N}{p^{\prime }}}}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{∥{\chi }_{B_N^c(0,r)}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & ⩽\hfill & r^{-s}{\left|b\right|}^{{\displaystyle \frac{2\gamma +N}{p^{\prime }}}}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

Moreover, by (75)

$$\begin{array}{ccc}{∥{\mathsf{\Gamma }}_{\lambda }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left({\chi }_{B_N(0,r)}f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}\hfill & ⩽\hfill & {∥{\mathsf{\Gamma }}_{\lambda }∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}{∥{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left({\chi }_{B_N(0,r)}f\right)∥}_{L_k^{\infty }\left({\mathbb{R}}^N\right)}\hfill \\ & ⩽\hfill & {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{∥{\mathsf{\Gamma }}_{\lambda }∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}{∥{\chi }_{B_N(0,r)}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & ⩽\hfill & {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{∥{\mathsf{\Gamma }}_{\lambda }∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}\hfill \\ & \times \hfill & {∥{\parallel \theta \parallel }^{-s}{\chi }_{B_N(0,r)}∥}_{L_k^{2p^{\prime }}\left({\mathbb{R}}^N\right)}{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^{2p}\left({\mathbb{R}}^N\right)}.\hfill \end{array}$$

A straightforward computation shows that there is a constant $C>0$ such that

$${∥{\parallel \theta \parallel }^{-s}{\chi }_{B_N(0,r)}∥}_{L_k^{2p^{\prime }}\left({\mathbb{R}}^N\right)}=C{r}^{-s+\frac{2\gamma +N}{2p^{\prime }}}.$$

Therefore,

$$\begin{array}{ccc}{∥{\mathsf{\Gamma }}_{\lambda }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}\hfill & ⩽\hfill & {∥{\mathsf{\Gamma }}_{\lambda }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f_r\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}+{∥{\mathsf{\Gamma }}_{\lambda }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f^r\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}\hfill \\ & ⩽\hfill & C{r}^{-s}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\hfill \\ & \times \hfill & \left({\left|b\right|}^{\frac{2\gamma +N}{p^{\prime }}}{\parallel \parallel \theta \left|\right|}^s{f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}+r^{\frac{2\gamma +N}{2p^{\prime }}}{\parallel {\mathsf{\Gamma }}_{\lambda }\parallel }_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}{∥{\parallel \theta \left|\right|}^{s}f∥}_{L_k^{2p}\left({\mathbb{R}}^N\right)}\right).\hfill \end{array}$$

If we take $r={\left(\lambda \right|b\left|\right)}^2$, then by (89), we obtain the result. □

Theorem 7.

Let $t>0$ and $0<s<\frac{2\gamma +N}{2p^{\prime }}$, with $1<p⩽2$. Then, there is a constant $\tilde{C}=\tilde{C}(k,b)>0$ such that for every $f\in L_k^2\left({\mathbb{R}}^N\right),$

$$\begin{array}{ccc}\hfill {∥{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}& ⩽& \tilde{C}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{t}{s+t}}{\left({∥{\parallel \theta \parallel }^{s}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^{2p}\left({\mathbb{R}}^N\right)}\right)}^{\frac{t}{s+t}}\times \hfill \\ & & {∥{\parallel (\theta ,\omega )\parallel }^{4t}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}^{\frac{s}{s+t}}.\hfill \end{array}$$

(91)

Proof. 

Let $0<s<\frac{2\gamma +N}{2p^{\prime }}$ such that $1<p⩽2$ and suppose that ${\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)\ne 0$. If $t⩽\frac{1}{2}$, then by the previous lemma, we have for all $\lambda >0$,

$$\begin{array}{ccc}\hfill {∥{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}& ⩽& {∥{\mathsf{\Gamma }}_{\lambda }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}+{∥(1-{\mathsf{\Gamma }}_{\lambda }){\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}\hfill \\ & ⩽& {C(k,b)\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\lambda }^{-2s}\left[{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^{2p}\left({\mathbb{R}}^N\right)}\right]\hfill \\ & +& {∥(1-{\mathsf{\Gamma }}_{\lambda }){\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}.\hfill \end{array}$$

On the other hand,

$${∥(1-{\mathsf{\Gamma }}_{\lambda }){\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}={\lambda }^{2t}{∥{(\lambda \parallel (\theta ,\omega )\parallel }^2{)}^{-2t}(1-{\mathsf{\Gamma }}_{\lambda }){\parallel (\theta ,\omega )\parallel }^{4t}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}.$$

As $(1-e^{-z})z^{-2t}$ is bounded for $z\ge 0$ if $t⩽\frac{1}{2}$, we obtain

$$\begin{array}{ccc}\hfill {∥{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}& ⩽& {C\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\lambda }^{-2s}\left[{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^{2p}\left({\mathbb{R}}^N\right)}\right]\hfill \\ & +& C{\lambda }^{2t}{∥{\parallel (\theta ,\omega )\parallel }^{4t}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)},\hfill \end{array}$$

from which, optimizing in $\lambda$, we obtain (91) for $0<s<{\displaystyle \frac{2\gamma +N}{2p^{\prime }}}$ and $t⩽{\displaystyle \frac{1}{2}}$.

Next, we assume that $t>\frac{1}{2}.$ For $u\ge 0$ and $t^{\prime }⩽\frac{1}{2}<t$, we have $u^{4t^{\prime }}⩽1+u^{4t}$, which is for $u={\displaystyle \frac{\parallel (\theta ,\omega )\parallel }{\epsilon }}$ becomes

$${\left({\displaystyle \frac{\parallel (\theta ,\omega )\parallel }{\epsilon }}\right)}^{4t^{\prime }}<1+{\left({\displaystyle \frac{\parallel (\theta ,\omega )\parallel }{\epsilon }}\right)}^{4t},\mathrm{for}\mathrm{all}\epsilon >0.$$

It follows that

$${∥{\parallel (\theta ,\omega )\parallel }^{4t^{\prime }}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}⩽{\epsilon }^{4t^{\prime }}{∥{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}+{\epsilon }^{4(t^{\prime }-t)}{∥{\parallel (\theta ,\omega )\parallel }^{4t}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}.$$

Minimizing over $\epsilon$, we obtain

$${∥{\parallel (\theta ,\omega )\parallel }^{4t^{\prime }}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}⩽C{∥{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}^{\frac{t-t^{\prime }}{t}}{∥{\parallel (\theta ,\omega )\parallel }^{4t}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^{p^{\prime }}\left({\mathbb{R}}^{2N}\right)}^{\frac{t^{\prime }}{t}}.$$

Together with (91) for $t^{\prime }$, we obtain the result for $t>\frac{1}{2}$. □

Corollary 3.

Let $t>0$ and $0<s<\frac{2\gamma +N}{4}$. Then, there is a constant $\mathfrak{C}(k,b)>0$ such that, for every $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}& ⩽& {\mathfrak{C}(k,b)\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{-s}{s+t}}{\left({∥{\parallel \theta \parallel }^{s}f∥}_{L_k^2\left({\mathbb{R}}^N\right)}+{∥{\parallel \theta \parallel }^{s}f∥}_{L_k^4\left({\mathbb{R}}^N\right)}\right)}^{\frac{t}{s+t}}\hfill \\ & \times & {∥{\parallel (\theta ,\omega )\parallel }^{4t}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)∥}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{s}{s+t}}.\hfill \end{array}$$

Proof. 

The result follows from (81) and Theorem 7 for $p=2$. □

5. Pitt and Logarithmic Inequalities for ${\mathfrak{G}}_{\phi }^{\mathbf{m}}$

Using (34) and the Pitt-type inequality in the Dunkl setting proved by Gorbachev et al. in [50], we obtain that for each $f\in \mathcal{S}\left({\mathbb{R}}^N\right)\subseteq L_k^2\left({\mathbb{R}}^N\right)$

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel y\parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(y\right)|}^{2}d{\gamma }_k\left(y\right)\le C_{k,b}\left(\lambda \right){\int }_{{\mathbb{R}}^N}{\parallel x\parallel }^{2\lambda }{\left|f\left(x\right)\right|}^{2}d{\gamma }_k\left(x\right),0\le \lambda <\frac{2\gamma +N}{2},}$$

(92)

where

$$C_{k,b}\left(\lambda \right):={\left({\displaystyle \frac{1}{2\left|b\right|}}\right)}^{2\lambda }{\left({\displaystyle \frac{\mathsf{\Gamma }\left(\frac{2\gamma +N-2\lambda }{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N+2\lambda }{4}\right)}}\right)}^2$$

(93)

and $\mathsf{\Gamma }$ is the Euler’s Gamma function.

Our first aim here is to prove an analogue of Relation (92) for the LCDGT.

Theorem 8.

For any arbitrary $f\in \mathcal{S}\left({\mathbb{R}}^N\right)\subseteq L_k^2\left({\mathbb{R}}^N\right),$ the Pitt inequality for the linear canonical Dunkl-Gabor transform is given for $0\le \lambda <\frac{2\gamma +N}{2}$ by:

$${\displaystyle {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le \frac{C_{k,b}\left(\lambda \right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ),}$$

(94)

where $C_{k,b}\left(\lambda \right)$ is given by (93).

Proof. 

As a consequence of the inequality (92), we can write

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le C_{k,b}\left(\lambda \right){\int }_{{\mathbb{R}}^N}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\gamma }_k\left(\theta \right),\omega \in {\mathbb{R}}^N}$$

(95)

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

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\mu }_k(\xi ,\omega )\le C_{k,b}\left(\lambda \right){\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ).}$$

(96)

Invoking Lemma 1, we can express the inequality (96) in the following manner:

$$\begin{array}{ccc}& & {\displaystyle {\left|b\right|}^{2\gamma +N}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^2{\tau }_{\omega }{\left|\phi \right|}^2\left(\xi \right)d{\mu }_k(\xi ,\omega )}\hfill \\ & \le & {\displaystyle C_{k,b}\left(\lambda \right){\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ).}\hfill \end{array}$$

Then,

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^2\left\{{\int }_{{\mathbb{R}}^N}{\tau }_{\omega }{\left|\phi \right|}^2\left(\xi \right)d{\gamma }_k\left(\omega \right)\right\}d{\gamma }_k\left(\xi \right)}\hfill \\ & \le & {\displaystyle \frac{C_{k,b}\left(\lambda \right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \end{array}$$

Using the hypothesis on $\phi$, the relation (27) becomes

$${\displaystyle {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le \frac{C_{k,b}\left(\lambda \right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}$$

(97)

which gives the desired result. □

Remark 5.

According to the Plancherel Formula (81), equality holds for $\lambda =0$ in (94).

Involving the Beckner-type inequality for the Dunkl transform (see [50]) and by (34), we derive that

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{{\mathbb{R}}^N}{log\left|t\right|\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right)+{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \left(\frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}+log\left(2\left|b\right|\right)\right){\int }_{{\mathbb{R}}^N}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right),}\hfill \end{array}$$

for all $f\in \mathcal{S}\left({\mathbb{R}}^N\right)$. This inequality is related to the Heisenberg’s uncertainty principle and for that reason, it is often referred as the logarithmic uncertainty principle.

Our second main goal of this section is to establish an analogue of Inequality (98) for the LCDGT.

Theorem 9.

For all $f\in \mathcal{S}\left({\mathbb{R}}^N\right)$, we have

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )+{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\left|b\right|}^{2\gamma +N}{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ \\ \ge \left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\right|b\left|\right)\right){\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\left|b\right|}^{2\gamma +N}.\hfill \end{array}$$

(98)

Proof. 

Replacing f with ${\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )$ in (98) gives

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^N}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\gamma }_k\left(\theta \right)+{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\ge }\hfill \\ {\displaystyle \left(\frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}+log\left(2\left|b\right|\right)\right){\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\gamma }_k\left(\theta \right),\omega \in {\mathbb{R}}^N.}\hfill \end{array}$$

(99)

Integrating (99) with respect to the measure $d{\gamma }_k\left(\omega \right)$, we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )+{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\mu }_k(\xi ,\omega )}\hfill \\ & \ge & {\displaystyle \left(\frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}+log\left(2\left|b\right|\right)\right){\int }_{{\mathbb{R}}^{2N}}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ).}\hfill \end{array}$$

Using Plancherel’s Formula (81), we obtain

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )+{\int }_{{\mathbb{R}}^{2N}}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]{\left(\xi \right)|}^{2}d{\mu }_k(\theta ,\omega )\ge }\hfill \\ \left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right){\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

(100)

By (27) and Lemma 1, we have

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]{\left(\xi \right)|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}^N}\left({\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\right)d{\gamma }_k\left(\omega \right)}\hfill \\ & =& {\displaystyle \left({\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\right){\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\left|b\right|}^{2\gamma +N}.}\hfill \end{array}$$

Plugging the estimate (101) in (100) gives the desired inequality for the linear canonical Dunkl-Gabor transforms as

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )+{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ \ge \left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right){\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

The proof is complete. □

A different proof of Theorem 9 is now presented by using Inequality (94).

Proof of Theorem 9.

For $0\le \lambda <\frac{2\gamma +N}{2}$, let

$${\displaystyle S\left(\lambda \right)={\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-\frac{C_{k,b}\left(\lambda \right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ).}$$

When it is differentiated with respect to $\lambda$, we obtain

$$\begin{array}{ccc}\hfill S^{\prime }\left(\lambda \right)& =& {\displaystyle -{2\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{-2\lambda }log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & -& {\displaystyle \frac{2C_{k,b}\left(\lambda \right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & -& {\displaystyle \frac{C_{k,b}^{\prime }\left(\lambda \right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2\lambda }|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ),}\hfill \end{array}$$

where

$$\begin{array}{ccc}{C}_{k,b}^{\prime }\left(\lambda \right)\hfill & =\hfill & -C_{k,b}\left(\lambda \right)\left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N-2\lambda }{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N-2\lambda }{4}\right)}+\frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N+2\lambda }{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N+2\lambda }{4}\right)}+2log\left(2\left|b\right|\right)}\right).\hfill \end{array}$$

(101)

For $\lambda =0,$ Equation (101) yields

$$C_{k,b}^{\prime }\left(0\right)=-2\left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right).$$

(102)

Inequality (94) implies that

$$\forall \lambda \in \left[0,{\displaystyle \frac{2\gamma +N}{2}}\right),S\left(\lambda \right)\le 0$$

and

$$\begin{array}{ccc}S\left(0\right)\hfill & =\hfill & {\displaystyle {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-\frac{C_{k,b}\left(0\right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & =\hfill & {\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2-{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2=0.\hfill \end{array}$$

Thus,

$$S^{\prime }\left(0^{+}\right):=\underset{\lambda \to 0^{+}}{lim}{\displaystyle \frac{S\left(\lambda \right)}{\lambda }}\le 0.$$

Likewise, we have

$$\begin{array}{c}{\displaystyle -{2\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-\frac{2C_{k,b}\left(0\right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ {\displaystyle -\frac{C_{k,b}^{\prime }\left(0\right)}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )\le 0.}\hfill \end{array}$$

Applying Plancherel’s Formula (81) and the obtained estimate (102) of $C_{k,b}^{\prime }\left(0\right)$, we obtain

$$\begin{array}{c}\hfill {\displaystyle -{2\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)-\frac{2}{{\left|b\right|}^{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\\ \hfill +2\left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right){\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\le 0\end{array}$$

or equivalently,

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel |{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )+{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel |{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ \\ \ge \left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right){\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

This completes the second proof of Theorem 9. □

Corollary 4.

Let $\phi \in L_{k,\mathrm{rad}}^2\left({\mathbb{R}}^N\right)\cap L_k^{\infty }\left({\mathbb{R}}^N\right)$. For any function f belongs to $\mathcal{S}\left({\mathbb{R}}^N\right)$, the following inequality holds:

$$\begin{array}{c}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^2|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\right)}^{1/2}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^2{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\right)}^{1/2}\hfill \\ \ge exp\left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right){\left|b\right|}^{\frac{2\gamma +N}{2}}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

Proof. 

Using Jensen’s inequality in (98), we obtain

$$\begin{array}{ccc}& & {\displaystyle log{\left\{{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^2\frac{|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^2}{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}d{\mu }_k(\theta ,\omega ){\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^2\frac{|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^2}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}d{\gamma }_k\left(\xi \right)\right\}}^{1/2}}\hfill \\ & =& {\displaystyle log{\left\{{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^2\frac{|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^2}{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}d{\mu }_k(\theta ,\omega )\right\}}^{1/2}}\hfill \\ & +& {\displaystyle log{\left\{{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^2\frac{|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^2}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}d{\gamma }_k\left(\xi \right)\right\}}^{1/2}}\hfill \\ & \ge & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}log\parallel \theta \parallel \frac{|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^2}{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}d{\mu }_k(\theta ,\omega )+{\int }_{{\mathbb{R}}^N}log\parallel \xi \parallel \frac{|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^2}{{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & \left({\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right),\hfill \end{array}$$

which gives the desired result. □

Remark 6.

1. 

By this approximation relation (see [51])

$${\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(z\right)}{\mathsf{\Gamma }\left(z\right)}=logz-\frac{1}{2z}-2{\int }_0^{\infty }\frac{t}{(t^2+z^2)(e^{2\pi t}-1)}dt}$$

(103)

we have

$$exp\left[{\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)\right]\approx \left({\displaystyle \frac{2\gamma +N}{2}}\left|b\right|\right),\mathrm{for}{\displaystyle \frac{2\gamma +N}{2}}\left|b\right|\gg 1,$$

(104)

which coincides with the constant as in Theorem 5.

2. 

Proceeding as in (98), one can obtain the following Heisenberg-type uncertainty relation

$$\begin{array}{ccc}\hfill & & {\displaystyle {\left({\int }_{{\mathbb{R}}^N}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right)\right)}^{\frac{1}{2}}{\left({\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)\right)}^{\frac{1}{2}}}\hfill \\ & \ge & {\displaystyle exp\left[\frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}+log\left(2\left|b\right|\right)\right]{\int }_{{\mathbb{R}}^N}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right).}\hfill \end{array}$$

(105)

3. 

By (103), we notice that the constant in (50),

$$exp\left[{\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(\frac{2\gamma +N}{4}\right)}{\mathsf{\Gamma }\left(\frac{2\gamma +N}{4}\right)}}+log\left(2\left|b\right|\right)]\right]\approx \left({\displaystyle \frac{2\gamma +N}{2}}\left|b\right|\right),\mathrm{for}{\displaystyle \frac{2\gamma +N}{2}}\left|b\right|\gg 1$$

coincides with the constant as in Proposition 15.

6. Uncertainty Inequalities for the LCDGT on Subsets of Finite Measures

The aim of this section is to prove some uncertainty inequalities for the LCDGT on subsets of finite measures, such as a Benedicks-type and a local-type uncertainty principles.

6.1. Benedicks Uncertainty Principle

Involving the Benedicks uncertainty principle for the Dunkl transform [52] and the identity (34), if $E_1$ and $E_2$ are two subsets of ${\mathbb{R}}^N$ with finite measure, then there exists a positive constant $C_k(E_1,E_2)$ such that for any $f\in L_k^2\left({\mathbb{R}}^N\right)$

$${\displaystyle {\int }_{{\mathbb{R}}^N}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right)\le C_k(E_1,E_2)\left({\displaystyle {\int }_{{\mathbb{R}}^N\setminus E_1}{\left|f\left(t\right)\right|}^{2}d{\gamma }_k\left(t\right)+{\int }_{{\mathbb{R}}^N\setminus \left|b\right|E_2}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\right).}$$

(106)

Our primary interest in this section is to establish a Benedick–Amrein–Berthier’s uncertainty principle for the LCDGT.

Theorem 10.

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

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{{\mathbb{R}}^N\setminus E_1}{\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )+{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N\setminus \left|b\right|E_2}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{C_k(E_1,E_2)}.}\hfill \end{array}$$

(107)

Proof. 

Replacing f with ${\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )$ in (106), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle {\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\gamma }_k\left(\theta \right)}& \le & {\displaystyle C_k(E_1,E_2)\left\{{\int }_{{\mathbb{R}}^N\setminus E_1}\right|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\gamma }_k\left(\theta \right)}\hfill \\ & +& {\displaystyle {\int }_{{\mathbb{R}}^N\setminus \left|b\right|E_2}\left|{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\right\}.}\hfill \end{array}$$

(108)

Taking the integral of (108) with respect to $d{\gamma }_k\left(\omega \right)$, we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )\le C_k(E_1,E_2)}\hfill \\ & \times & \left\{{\displaystyle {\int }_{{\mathbb{R}}^N\setminus E_1}{\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )+{\int }_{{\mathbb{R}}^N\setminus \left|b\right|E_2}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right)\right|}^{2}d{\mu }_k(\xi ,\omega )}\right\}.\hfill \end{array}$$

From (81) and Lemma 1, we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^N\setminus E_1}{\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & +& {\displaystyle {\left|b\right|}^{2\gamma +N}{\int }_{{\mathbb{R}}^N\setminus \left|b\right|E_2}{\int }_{{\mathbb{R}}^N}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\sqrt{{\tau }_{\omega }{\left|\phi \right|}^2\left(\xi \right)}\right|}^{2}d{\mu }_k(\xi ,\omega )}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{C_k(E_1,E_2)}},\hfill \end{array}$$

which further implies

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^N\setminus E_1}{\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & +& {\displaystyle {\left|b\right|}^{2\gamma +N}{\int }_{{\mathbb{R}}^N\setminus \left|b\right|E_2}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^2\left\{{\int }_{{\mathbb{R}}^N}{\tau }_{\omega }{\left|\phi \right|}^2\left(\xi \right)d{\gamma }_k\left(\omega \right)\right\}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{C_k(E_1,E_2)}}.\hfill \end{array}$$

Hence, by (27), we derive that

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^N\setminus E_1}{\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )+{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N\setminus \left|b\right|E_2}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{C_k(E_1,E_2)}},\hfill \end{array}$$

which is the desired result. □

By Theorem 10, we can derive another type of Heisenberg’s uncertainty relation for the LCDGT.

Corollary 5.

For $q,p>0,$ there is a constant ${\mathcal{C}}_{k,b}(p,q)>0$, such that for every $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$$\begin{array}{ccc}\hfill & & {\displaystyle {\left({\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2p}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )\right)}^{\frac{q}{2}}{\left({\int }_{{\mathbb{R}}^N}{\parallel y\parallel }^{2q}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(y\right)\right|}^{2}d{\gamma }_k\left(y\right)\right)}^{\frac{p}{2}}}\hfill \\ & \ge & {\mathcal{C}}_{k,b}(p,q){\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^q{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{p+q}.\hfill \end{array}$$

(109)

Proof. 

Let $f\in L_k^2\left({\mathbb{R}}^N\right)$, $E_1=B_N(0,1)$ and $E_2=B_N\left(0,{\displaystyle \frac{1}{\left|b\right|}}\right)$. Then, by (107), we have

$$\begin{array}{ccc}& & {\displaystyle {\int }_{B_N^c(0,1)}{\int }_{{\mathbb{R}}^N}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )+{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{B_N^c(0,1)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{\mathfrak{C}(k,b)}},\hfill \end{array}$$

where $\mathfrak{C}(k,b):=C_k(E_1,E_2)$. Therefore,

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2p}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )+{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2q}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{\mathfrak{C}(k,b)}}.\hfill \end{array}$$

Now replacing f by $f_{\lambda }$, $\phi$ by ${\phi }_{{\displaystyle \frac{1}{\lambda }}}$ and $\mathbf{m}$ by ${\mathbf{m}}_{{\displaystyle \frac{1}{\lambda }}}$, we obtain by (73)

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2p}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta /\lambda ,\lambda \omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & +& {\displaystyle {\left|b\right|}^{2\gamma +N}{\lambda }^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2q}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\lambda \xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{\mathfrak{C}(k,b)}}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & {\displaystyle {\lambda }^{2p}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2p}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & +& {\displaystyle {\left|b\right|}^{2\gamma +N}{\lambda }^{-2q}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2q}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \ge & {\displaystyle \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{\mathfrak{C}(k,b)}}.\hfill \end{array}$$

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

6.2. Local-Type Uncertainty Principles

We begin this subsection by stating the following local uncertainty principle.

Proposition 18.

Let $0<s<\frac{2\gamma +N}{2}$ and let $E\subset {\mathbb{R}}^N$ be a subset of finite measure $0<{\gamma }_k\left(E\right)<\infty$. Then, there is a constant $\mathfrak{C}(k,s)>0$ such that for every $f\in L_k^2\left({\mathbb{R}}^N\right)$

$${\displaystyle {\int }_E|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le \mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}{\parallel \parallel x\parallel }^s{f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.}$$

(110)

Proof. 

Involving the local uncertainty principle for the Dunkl transform [52] and the identity (34), we derive the result. □

Consequently, we have the following local-type uncertainty principle for the LCDGT.

Theorem 11.

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

$${\displaystyle {\int }_E|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\le \frac{\mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}}{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2s}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ).}$$

(111)

Proof. 

Replacing f with ${\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )$ in (110), we obtain

$${\displaystyle {\int }_E|{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\le \mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}{\parallel \parallel \theta \parallel }^s{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(.,\omega )\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.}$$

We shall integrate this inequality with respect to the measure $d{\gamma }_k\left(\omega \right)$ to obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_E{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^{\mathbf{m}}\left[{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\right]\left(\xi \right){|}^{2}d{\mu }_k(\xi ,\omega )}\hfill \\ & \le & {\displaystyle \mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2s}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \end{array}$$

which together with Lemma 1 gives

$$\begin{array}{c}{\displaystyle {\left|b\right|}^{2\gamma +N}{\int }_E{\int }_{{\mathbb{R}}^N}|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^2{\tau }_{\omega }{\left|\phi \right|}^2\left(\xi \right)d{\gamma }_k\left(\xi \right)d{\gamma }_k\left(\omega \right)\le }\\ {\displaystyle \mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2s}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega ).}\end{array}$$

(112)

Using the hypothesis on $\phi$, inequality (112) reduces to

$$\begin{array}{ccc}& & {\displaystyle {\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_E|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \le & {\displaystyle \mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2s}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)(\theta ,\omega )|}^{2}d{\mu }_k(\theta ,\omega ).}\hfill \end{array}$$

Or equivalently, for any $s\in (0,\frac{2\gamma +N}{2})$

$${\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2s}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )\ge \frac{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{\mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}}{\int }_E{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

(113)

This completes the proof of (111). □

For a subset E of ${\mathbb{R}}^N$, we set the following Paley–Wiener-type space $P{W}_k^{\mathbf{m}}\left(E\right)$:

$$P{W}_k^{\mathbf{m}}\left(E\right):=\left\{f\in L_k^2\left({\mathbb{R}}^N\right):\mathrm{supp}{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\subset E\right\}.$$

By (18) and Theorem 11, we obtain the following result.

Corollary 6.

Let $0<s<{\displaystyle \frac{2\gamma +N}{2}}$ and let $E\subset {\mathbb{R}}^N$ be a subset of finite measure $0<{\gamma }_k\left(E\right)<\infty$. Then, for any $f\in P{W}_k^{\mathbf{m}}\left(E\right)$,

$${\displaystyle {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\le \frac{\mathfrak{C}(k,s){\left({\gamma }_k\left(\frac{E}{\left|b\right|}\right)\right)}^{\frac{2s}{2\gamma +N}}}{{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2s}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ).}$$

(114)

The following result can be obtained by interchanging the roles of f and ${\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)$ in Proposition 18.

Corollary 7.

Let $0<t<{\displaystyle \frac{2\gamma +N}{2}}$ and $F\subset {\mathbb{R}}^N$ be a subset of finite measure $0<{\gamma }_k\left(F\right)<\infty$. Then, for all $f\in L_k^2\left({\mathbb{R}}^N\right)$,

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

(115)

By utilizing Corollary 7 and analogous concepts as presented in the proof of Theorem 11, we establish the following result.

Corollary 8.

Let $0<t<{\displaystyle \frac{2\gamma +N}{2}}$ and let $F\subset {\mathbb{R}}^N$ be a subset of finite measure $0<{\gamma }_k\left(F\right)<\infty$. Then, for all $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{{\mathbb{R}}^N}{\int }_F|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )}\hfill \\ & \le & {\displaystyle \mathfrak{C}(k,t){\left({\gamma }_k\left(\frac{F}{\left|b\right|}\right)\right)}^{\frac{2t}{2\gamma +N}}{\left|b\right|}^{2\gamma +N}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2t}|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right).}\hfill \end{array}$$

(116)

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

$$GP{W}_k^{\mathbf{m}}\left(F\right):=\left\{f\in L_k^2\left({\mathbb{R}}^N\right):\forall \omega \in {\mathbb{R}}^N,\mathrm{supp}{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(.,\omega )\subset F\right\}.$$

Applying Plancherel’s Formula (81), the definition of generalized Paley–Wiener space $GP{W}_k^{\mathbf{m}}\left(F\right)$, and the previous corollary, we obtain the following:

Corollary 9.

Let E and F be two subsets of ${\mathbb{R}}^N$ such that $0<{\gamma }_k\left(E\right),{\gamma }_k\left(F\right)<\infty$.

Let $0<s,t<\frac{2\gamma +N}{2}$.

1. 

For any $f\in GP{W}_k^{\mathbf{m}}\left(F\right)$, we have

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

2. 

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

$$\begin{array}{ccc}\hfill & & {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^{s+t}\le {\left(\mathfrak{C}(k,t)\right)}^{{\displaystyle \frac{s}{2}}}{\left(\mathfrak{C}(k,s)\right)}^{{\displaystyle \frac{t}{2}}}{\left({\gamma }_k\left({\displaystyle \frac{E}{\left|b\right|}}\right){\gamma }_k\left({\displaystyle \frac{F}{\left|b\right|}}\right)\right)}^{{\displaystyle \frac{ts}{2\gamma +N}}}\times \hfill \\ & & {\displaystyle {\left({\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2t}|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right){|}^{2}d{\gamma }_k\left(\xi \right)\right)}^{\frac{s}{2}}{\left({\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2s}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega )\right)}^{\frac{t}{2}}.}\hfill \end{array}$$

We end this section by the following Heisenberg-type uncertainty relation for the LCDGT.

Theorem 12.

For $q>0$ and $0<p<{\displaystyle \frac{2\gamma +N}{2}}$, we have for every $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\le {\mathcal{C}(k,b,p,q)\left|\right|\parallel \theta \parallel }^p{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)|{|}_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{2q}{p+q}}{\left|\right|\parallel \xi \left|\right|}^q{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)|{|}_{L_k^2\left({\mathbb{R}}^N\right)}^{\frac{2p}{p+q}},$$

(117)

where

$$\mathcal{C}(k,b,p,q)=\begin{array}{c}{\left({\displaystyle \frac{{\left(\frac{d_k}{2\gamma +N}\right)}^{\frac{2p}{2\gamma +N}}\mathfrak{C}(k,p)}{{\left|b\right|}^{2\gamma +N+2p}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}}\right)}^{\frac{q}{p+q}}\left[{\left({\displaystyle \frac{p}{q}}\right)}^{{\displaystyle \frac{q}{p+q}}}+{\left({\displaystyle \frac{q}{p}}\right)}^{{\displaystyle \frac{p}{p+q}}}\right].\hfill \end{array}$$

Proof. 

For $r>0$, we have

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2& =& \parallel {\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill \\ & =& {\displaystyle {\int }_{B_N(0,r)}|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)+{\int }_{B_N^c(0,r)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}\hfill \end{array}$$

(118)

From Theorem 11 and by simple calculation, we have

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{B_N(0,r)}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right)}\hfill \\ & \le & {\displaystyle \frac{{\left(\frac{d_k}{2\gamma +N}\right)}^{\frac{2p}{2\gamma +N}}\mathfrak{C}(k,p)}{{\left|b\right|}^{2\gamma +N+2p}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{r}^{2p}{\int }_{{\mathbb{R}}^{2N}}{\parallel \theta \parallel }^{2p}|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega ){|}^{2}d{\mu }_k(\theta ,\omega ).}\hfill \end{array}$$

(119)

Moreover, it is easy to see that

$${\displaystyle {\int }_{B_N^c(0,r)}|{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\left(\xi \right)|}^{2}d{\gamma }_k\left(\xi \right)\le r^{-2q}{\int }_{{\mathbb{R}}^N}{\parallel \xi \parallel }^{2q}{\left|{\mathfrak{F}}_D^{\mathbf{m}}\left(f\right)\left(\xi \right)\right|}^{2}d{\gamma }_k\left(\xi \right).}$$

(120)

Combining the relations (118), (119) and (120), we obtain

$$\begin{array}{ccc}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2\hfill & \le \hfill & {\displaystyle \frac{{\left(\frac{d_k}{2\gamma +N}\right)}^{\frac{2p}{2\gamma +N}}\mathfrak{C}(k,p)}{{\left|b\right|}^{2\gamma +N+2p}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}}{r}^{2p}\parallel {\parallel \theta \parallel }^p{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^2+r^{-2q}{\parallel \parallel \xi \parallel }^q{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2.\hfill \end{array}$$

If

$$r={\left[{\displaystyle \frac{{q\left|b\right|}^{2\gamma +N+2p}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}^2}{p{\left(\frac{d_k}{2\gamma +N}\right)}^{\frac{2p}{2\gamma +N}}\mathfrak{C}(k,p)}}\right]}^{\frac{1}{2p+2q}}{\left({\displaystyle \frac{{\parallel \parallel \theta \parallel }^p{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}}{{\parallel \parallel \xi \parallel }^q{\mathfrak{F}}_D^{\mathbf{m}}{\left(f\right)\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}}}\right)}^{\frac{1}{p+q}},$$

then we obtain the result. □

We end this paragraph by the following Faris–Price-type uncertainty inequality.

Theorem 13

(Faris–Price-type uncertainty principle for ${\mathfrak{G}}_{\phi }^{\mathbf{m}}$). Let $p⩾1$ and $0<\eta <2\gamma +N$. Then, there exists a constant $C_{k,b}(\eta ,p)>0$ such that for any measurable subset $T\subset {\mathbb{R}}^{2N}$ of finite measure $0<{\mu }_k\left(T\right)<\infty$, and all $f\in L_k^2\left({\mathbb{R}}^N\right)$,

$$\begin{array}{ccc}& & {\left({\displaystyle {\int }_T{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{p}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{1}{p}}\hfill \\ & ⩽& C_{k,b}(\eta ,p){\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}\parallel {\parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{2(2\gamma +N)}{\left(2\gamma +N+\eta \right)(p+1)}}\hfill \\ & \times & {\left({\left|b\right|}^{\frac{2\gamma +N}{2}}{\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}{\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}\right)}^{\frac{\left(2\gamma +N+\eta \right)(p+1)-\left(2\left(2\gamma +N\right)\right)}{\left(2\gamma +N+\eta \right)(p+1)}}.\hfill \end{array}$$

Proof. 

Suppose that ${\parallel f\parallel }_{L_k^2\left({\mathbb{R}}^N\right)}={\parallel \phi \parallel }_{L_k^2\left({\mathbb{R}}^N\right)}=1$. Therefore, for all $r>1$,

$$\begin{array}{c}\hfill \parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^p\left(T\right)}⩽\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\chi }_{B_{2N}(0,r)}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}+{\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\chi }_{B_{2N}^c(0,r)}\parallel }_{L_{{\mu }_k}^p\left(T\right)},\end{array}$$

where $B_{2N}(0,r)$ denotes the ball of ${\mathbb{R}}^{2N}$ of radius r given by

$$B_{2N}(0,r):=\left\{(\theta ,\omega )\in {\mathbb{R}}^{2N}:\parallel (\theta ,\omega )\parallel \le r\right\}.$$

By (75), we have for all $\eta \in (0,2\gamma +N)$,

$$\begin{array}{cc}& \parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\chi }_{B_{2N}(0,r)}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}\hfill \\ & ={\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^p{\chi }_{B_{2N}(0,r)}(\theta ,\omega ){\chi }_T(\theta ,\omega )d{\mu }_k(\theta ,\omega )}\right)}^{\frac{1}{p}}\hfill \\ & ⩽\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)}^{\frac{p}{p+1}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{\frac{p}{p+1}}{\chi }_{B_{2N}(0,r)}(\theta ,\omega ){\chi }_T(\theta ,\omega )d{\mu }_k(\theta ,\omega )}\right)}^{\frac{1}{p}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\chi }_{B_{2N}(0,r)}\parallel }_{L_{{\mu }_k}^1\left({\mathbb{R}}^{2N}\right)}^{\frac{1}{p+1}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{{\displaystyle \frac{1}{p(p+1)}}}{\parallel \parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{1}{p+1}}\parallel {\parallel (\theta ,\omega )\parallel }^{-\eta }{\chi }_{B_{2N}(0,r)}{\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{1}{p+1}}.\hfill \end{array}$$

On the other hand, by simple calculation, we see that

$${\parallel \parallel (\theta ,\omega )\parallel }^{-\eta }{\chi }_{B_{2N}(0,r)}{\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}\le \sqrt{{\displaystyle \frac{{\left(\mathsf{\Gamma }\left(\frac{2\gamma +N}{2}\right)\right)}^2}{\left(2\gamma +N-\eta \right)\mathsf{\Gamma }\left(2\gamma +N\right)}}}{s}^{2\gamma +N-\eta }.$$

Thus, we obtain

$$\begin{array}{ccc}\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\chi }_{B_{2N}(0,s)}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}\hfill & ⩽\hfill & {\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}{\left({\displaystyle \frac{{\left(\mathsf{\Gamma }\left(\frac{2\gamma +N}{2}\right)\right)}^2}{\left(2\gamma +N-\eta \right)\mathsf{\Gamma }\left(2\gamma +N\right)}}\right)}^{\frac{1}{2(p+1)}}\hfill \\ & & s^{\frac{2\gamma +N-\eta }{(p+1)}}\parallel {\parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{1}{p+1}}.\hfill \end{array}$$

On the other hand, and again by Hölder’s inequality and Relation (75), we deduce that

$$\begin{array}{cc}& \parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\chi }_{B_{2N}^c(0,s)}{\parallel }_{L_{{\mu }_k}^p\left(T\right)}\hfill \\ & ⩽\parallel {\mathfrak{G}}_{\phi }^{\mathbf{m}}{\left(f\right)\parallel }_{L_{{\mu }_k}^{\infty }\left({\mathbb{R}}^{2N}\right)}^{\frac{p-1}{p+1}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{\frac{2p}{p+1}}{\chi }_{B_{2N}^c(0,s)}(\theta ,\omega ){\chi }_T(\theta ,\omega )d{\mu }_k(\theta ,\omega )}\right)}^{\frac{1}{p}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{{\displaystyle \frac{1}{p(p+1)}}}{\left({\displaystyle {\int }_{{\mathbb{R}}^{2N}}{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^2{\chi }_{B_{2N}^c(0,s)}(\theta ,\omega )d{\mu }_k(\theta ,\omega )}\right)}^{\frac{1}{p+1}}\hfill \\ & ⩽{\left({\mu }_k\left(T\right)\right)}^{{\displaystyle \frac{1}{p(p+1)}}}\parallel {\parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{2}{p+1}}{s}^{-\frac{2\eta }{p+1}}.\hfill \end{array}$$

Thus, for any $\eta \in (0,2\gamma +N)$,

$$\begin{array}{cc}& {\left({\displaystyle {\int }_T{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{p}d{\mu }_k(\theta ,\omega )}\right)}^{\frac{1}{p}}⩽{\left({\mu }_k\left(T\right)\right)}^{\frac{1}{p(p+1)}}\parallel {\parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{1}{p+1}}\hfill \\ & \left({\left({\displaystyle \frac{{\left(\mathsf{\Gamma }\left(\frac{2\gamma +N}{2}\right)\right)}^2}{\left(2\gamma +N-\eta \right)\mathsf{\Gamma }\left(2\gamma +N\right)}}\right)}^{\frac{1}{2(p+1)}}{s}^{\frac{2\gamma +N-\eta }{(p+1)}}+\parallel {\parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{\frac{1}{p+1}}{s}^{-\frac{2\eta }{p+1}}\right).\hfill \end{array}$$

If we choose

$$\begin{array}{ccc}s\hfill & =\hfill & {\left({\displaystyle \frac{\left(2\gamma +N-\eta \right)\mathsf{\Gamma }\left(2\gamma +N\right)}{{\left(\mathsf{\Gamma }\left(\frac{2\gamma +N}{2}\right)\right)}^2}}\right)}^{{\displaystyle \frac{1}{2\gamma +N+\eta }}}{\left({\displaystyle \frac{2\eta }{2\gamma +N-\eta }}\right)}^{{\displaystyle \frac{(p+1)}{2\gamma +N+\eta }}}\parallel {\parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{{\displaystyle \frac{1}{2\gamma +N+\eta }}},\hfill \end{array}$$

we obtain

$$\begin{array}{cc}& {\left({\displaystyle {\int }_T{\left|{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right)(\theta ,\omega )\right|}^{p}d{\mu }_k(\theta ,\omega )}\right)}^{{\displaystyle \frac{1}{p}}}⩽{\left({\mu }_k\left(T\right)\right)}^{{\displaystyle \frac{1}{p(p+1)}}}{\left({\displaystyle \frac{\left(2\gamma +N-\eta \right)\mathsf{\Gamma }\left(2\gamma +N\right)}{{\left(\mathsf{\Gamma }\left(\frac{2\gamma +N}{2}\right)\right)}^2}}\right)}^{{\displaystyle \frac{-2\eta }{\left(2\gamma +N+\eta \right)(p+1)}}}\hfill \\ & {\parallel \parallel (\theta ,\omega )\parallel }^{\eta }{\mathfrak{G}}_{\phi }^{\mathbf{m}}\left(f\right){\parallel }_{L_{{\mu }_k}^2\left({\mathbb{R}}^{2N}\right)}^{{\displaystyle \frac{2(2\gamma +N)}{\left(2\gamma +N+\eta \right)(p+1)}}}{\left({\displaystyle \frac{2\gamma +N-\eta }{2\eta }}\right)}^{{\displaystyle \frac{2\eta }{2\gamma +N+\eta }}}\left({\displaystyle \frac{2\gamma +\eta +N}{2\gamma -\eta +N}}\right).\hfill \end{array}$$

The proof is complete. □

Remark 7.

1. 

Let ϕ be in $L_{k,rad}^2\left({\mathbb{R}}^N\right)$. We proceed as in [6], to define the modulation of ϕ by t otherwise, as follow:

$${\mathcal{M}}_t^{\mathbf{m}}\left(\varphi \right):={\mathfrak{F}}_D^{{\mathbf{m}}^{-1}}\left(\sqrt{|T_t^{\mathbf{m},k}(e^{{\displaystyle \frac{id}{2b}}{\parallel z\parallel }^2}|{\mathfrak{F}}_D\left(\varphi \right){|}^2)|}\right).$$

(121)

Subsequently, we define the generalized Gabor transform ${\mathcal{G}}_{\varphi }^{\mathbf{m}}$ as follow: For $(y,t)\in {\mathbb{R}}^{2N}$,

$${\displaystyle {\mathcal{G}}_{\varphi }^{\mathbf{m}}\left(f\right)(y,t):={\int }_{{\mathbb{R}}^N}{e}^{i\frac{a}{b}{\parallel \xi \parallel }^2}f\left(\xi \right)T_y^{{\mathbf{m}}^{-1},k}\left({\mathcal{M}}_t^{\mathbf{m}}\varphi \right)(-\xi )d{\gamma }_k\left(\xi \right)=f{\ast }_{{\mathbf{m}}^{-1},k}{\mathcal{M}}_t^{\mathbf{m}}\varphi \left(y\right).}$$

(122)

It is clear that

$${\mathcal{G}}_{\varphi }^{\mathbf{m}}={\mathfrak{G}}_{{\mathfrak{F}}_D\left(\varphi \right)}^{\mathbf{m}}.$$

(123)

Thus, by involving Plancherel’s Formula (18), we derive that the two integral transforms are equivalent and then all results proved for ${\mathfrak{G}}_{\varphi }^{\mathbf{m}}$ are valuables for ${\mathcal{G}}_{\varphi }^{\mathbf{m}}$. Therefore, we reclaim that all results proved in this paper for the LCDGT are valuables for the integral transform ${\mathcal{G}}_{\varphi }^{\mathbf{m}}$ by replacing ϕ by ${\mathfrak{F}}_D\left(\varphi \right)$ to derive analogues results.

2. 

If $W={\mathbb{Z}}_2^N$, then all the results in this paper for the LCDGT are true without the assumption that the function φ is radial. It is enough to choose a function φ such that ${\tau }_y{\left|\phi \right|}^2\ge 0$.

7. Conclusions and Perspectives

In the present paper, we accomplished two major objectives. We first studied the concept of the linear canonical Dunkl-Gabor transform and investigated its fundamental properties. Then, we examined other versions of the weighted uncertainty principles for this new transformation.

In future work, we will investigate further applications of the theory of time-frequency analysis for the linear canonical Dunkl transform and for other generalized linear canonical integral transforms.