Dini Lipschitz functions for the Dunkl transform in the Space L2(Rd,wk(x)dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$$\end{document}

Using a generalized spherical mean operator, we obtain an analog of Theorem 5.2 in Younis (J Math Sci 9(2),301–312 1986) for the Dunkl transform for functions satisfying the d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-Dunkl Dini Lipschitz condition in the space L2(Rd,wk(x)dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$$\end{document}, where wk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{k}$$\end{document} is a weight function invariant under the action of an associated reflection group.


Introduction and preliminaries
Younis Theorem 5.2 [13] characterized the set of functions in L 2 (R) satisfying the Cauchy Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely, we have the following Theorem 1.1 [13] Let f ∈ L 2 (R). Then the following are equivalents where F stands for the Fourier transform of f .  Radouan Daher rjdaher024@gmail.com In this paper, we obtain an analog of Theorem 1.1 for the Dunkl transform on R d . For this purpose, we use a generalized spherical mean operator. We point out that similar results have been established in the Bessel transform [4].
We consider the Dunkl operators D i ; 1 ≤ i ≤ d, on R d , which are the differentialdifference operators introduced by Dunkl in [6]. These operators are very important in pure mathematics and in physics. The theory of Dunkl operators provides generalizations of various multivariable analytic structures, among others we cite the exponential function, the Fourier transform and the translation operator. For more details about these operators see [5][6][7]. The Dunkl Kernel E k has been introduced by Dunkl in [8]. This Kernel is used to define the Dunkl transform.
Let R be a root system in R d , W the corresponding reflection group, R + a positive subsystem of R (see [5,7,[9][10][11]) and k a non-negative and W -invariant function defined on R.
The Dunkl operators is defined for Here , is the usual euclidean scalar product on R d with the associated norm |.| and σ α the reflection with respect to the hyperplane H α orthogonal to α, and α j = α, e j , (e 1 , e 2 , . . . , e d ) being the canonical basis of R d .
The weight function w k defined by where w k is W -invariant and homogeneous of degree 2γ where The Dunkl Kernel E k on R d × R d has been introduced by Dunkl in [8]. For y ∈ R d the function 'where the constant c k is given by The Dunkl transform shares several properties with its counterpart in the classical case, we mention here in particular that Parseval Theorem holds in L 2 In L 2 k (R d ), consider the generalized spherical mean operator defined by where τ x Dunkl translation operator (see [11,12]), η is the normalized surface measure on the unit sphere For p ≥ − 1 2 , we introduce the normalized Bessel function of the first kind j p defined by , z ∈ C. (1)

Lemma 1.3 [1]
The following inequalities are fulfilled The following inequality is true Then We have Invoking Parseval's identity (4) gives

Dini Lipschitz condition
for all x in R d and for all sufficiently small h, C being a positive constant. Then we say that f satisfies a d-Dunkl Dini Lipschitz of order α, or f belongs to Li p(α, γ ).
where M = max(K 1 C f , K 2 C g ). Then f g ∈ Li p(α, γ ) Then f is equal to the null function in R d .

New results on Dini Lipschitz class
Proof Assume that f ∈ Li p(α, γ ). Then We have to recall that the Dunkl transform of f (x) satisfies the Parseval's identity f L 2 Since α > 2 we have and also from the formula (3) and Fatou's theorem, we obtain |ξ | 2 f (ξ ) L 2 Analog of the theorem 3.1, we obtain this theorem Then f is equal to null function in R d .
Now, we give another the main result of this paper analog of theorem 1.1.

We obtain
s≤|ξ |≤2s where C is a positive constant.
We have to show that We write where and Firstly, from (1)  h 2α (log 1 h ) 2γ where C 1 is a positive constant, and this ends the proof Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.