# Dini Lipschitz functions for the Dunkl transform in the Space $$\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$$

• Mohamed El Hamma
Open Access
Article

## Abstract

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 $$d$$-Dunkl Dini Lipschitz condition in the space $$\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$$, where $$w_{k}$$ is a weight function invariant under the action of an associated reflection group.

## Keywords

Dunkl transform Dunkl kernel Generalized spherical mean operator

42B37

## 1 Introduction and preliminaries

Younis Theorem 5.2 [13] characterized the set of functions in $$\mathrm {L}^{2}(\mathbb {R})$$ 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 $$f\in \mathrm {L}^{2}(\mathbb {R})$$. Then the following are equivalents
1. 1.

$$\Vert f(x+h)-f(x)\Vert _{2}=O\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\beta }}\right) \quad as\, h\longrightarrow 0, ~0<\alpha <1, \beta \ge 0$$,

2. 2.

$$\int _{|x|\ge r}|\mathcal {F}(f)(x)|^{2}dx=O\left( \frac{r^{-2\alpha }}{(\log r)^{2\beta }}\right) \quad as \,r \longrightarrow +\infty$$,

where $$\mathcal {F}$$ stands for the Fourier transform of $$f$$.

In this paper, we obtain an analog of Theorem 1.1 for the Dunkl transform on $$\mathbb {R}^{d}$$. 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 $$\mathrm {D}_{i}$$; $$1\le i\le d$$, on $$\mathbb {R}^{d}$$, which are the differential-difference 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 $$\mathrm {R}$$ be a root system in $$\mathbb {R}^{d}$$, $$W$$ the corresponding reflection group, $$\mathrm {R}_{+}$$ a positive subsystem of $$\mathrm {R}$$ (see [5, 7, 9, 10, 11]) and $$k$$ a non-negative and $$W$$-invariant function defined on $$\mathrm {R}$$.

The Dunkl operators is defined for $$f \in C^{1}(\mathbb {R}^{d})$$ by
\begin{aligned} \mathrm {D}_{j}f(x)=\frac{\partial f}{\partial x_{j}}(x)+\sum _{\alpha \in \mathrm {R}_{+}}k(\alpha )\alpha _{j}\frac{f(x)-f(\sigma _{\alpha }(x))}{\langle \alpha ,x\rangle },\quad x\in \mathbb {R}^{d} ~(1\le j \le d) \end{aligned}
Here $$\langle ,\rangle$$ is the usual euclidean scalar product on $$\mathbb {R}^{d}$$ with the associated norm $$|.|$$ and $$\sigma _{\alpha }$$ the reflection with respect to the hyperplane $$\mathrm {H}_{\alpha }$$ orthogonal to $$\alpha$$, and $$\alpha _{j}=\langle \alpha ,e_{j}\rangle$$, $$(e_{1},e_{2},\ldots ,e_{d})$$ being the canonical basis of $$\mathbb {R}^{d}$$.
The weight function $$w_{k}$$ defined by
\begin{aligned} w_{k}(x)=\prod _{\zeta \in \mathrm {R}_{+}}|\langle \zeta ,x\rangle |^{2k(\alpha )}, \quad x \in \mathbb {R}^{d}, \end{aligned}
where $$w_{k}$$ is $$W$$-invariant and homogeneous of degree $$2\gamma$$ where
\begin{aligned} \gamma = \gamma (\mathrm {R})=\sum _{\zeta \in \mathrm {R}_{+}}k(\zeta ) \ge 0. \end{aligned}
The Dunkl Kernel $$E_{k}$$ on $$\mathbb {R}^{d}\times \mathbb {R}^{d}$$ has been introduced by Dunkl in [8]. For $$y \in \mathbb {R}^{d}$$ the function $$x \mapsto E_{k}(x,y)$$ is the unique solution on $$\mathbb {R}^{d}$$ of
\begin{aligned} \left\{ \begin{array}{ll} \mathrm {D}_{j}u(x,y)=y_{j}u(x,y) &{} \quad \mathrm {for}\,\quad 1\le j\le d \\ u(0,y)=1 &{}\quad \mathrm {for~ all}\, y\in \mathbb {R}^{d} \\ \end{array} \right. \end{aligned}
$$E_{k}$$ is called the Dunkl Kernel.

### Proposition 1.2

[5] Let $$z,w \in \mathbb {C}$$ and $$\lambda \in \mathbb {C}$$. Then
1. 1.

$$E_{k}(z,0)=1$$.

2. 2.

$$E_{k}(z,w)=E_{k}(w,z)$$.

3. 3.

$$E_{k}(\lambda z,w)= E_{k}(z,\lambda w)$$.

4. 4.
For all $$\nu =(\nu _{1},\ldots ,\nu _{d})\in {\mathbb N}^{d},~ x\in {\mathbb R}^{d},~ z\in {\mathbb C}^{d}$$, we have
\begin{aligned} |\partial _{z}^{\nu }E_{k}(x,z)|\le |x|^{|\nu |}exp(|x||Re(z)|), \end{aligned}
where
\begin{aligned} \partial _{z}^{\nu }=\frac{\partial ^{|\nu |}}{\partial z_{1}^{\nu _{1}}\ldots \partial z_{d}^{\nu _{d}}},\quad |\nu |=\nu _{1}+\cdots +\nu _{d}. \end{aligned}
In particular
\begin{aligned} |\partial _{z}^{\nu }E_{k}(ix,z)|\le |x|^{\nu }, \end{aligned}
for all $$x,z\in {\mathbb R}^{d}.$$

The Dunkl transform is defined for $$f\in \mathrm {L}_{k}^{1}(\mathbb {R}^{d})=\mathrm {L}^{1}(\mathbb {R}^{d},w_{k}(x)dx)$$ by
\begin{aligned} \widehat{f}(\xi )=c_{k}^{-1}\int _{\mathbb {R}^{d}}f(x)E_{k}(-i \xi ,x)w_{k}(x)dx, \end{aligned}
‘where the constant $$c_{k}$$ is given by
\begin{aligned} c_{k}=\int _{\mathbb {R}^{d}}e^{\frac{-|z|^{2}}{2}}w_{k}(z)dz. \end{aligned}
The Dunkl transform shares several properties with its counterpart in the classical case, we mention here in particular that Parseval Theorem holds in $$\mathrm {L}_{k}^{2}=\mathrm {L}_{k}^{2}(\mathbb {R}^{d})=\mathrm {L}_{k}^{2}(\mathbb {R}^{d}, w_{k}(x)dx)$$, when both $$f$$ and $$\widehat{f}$$ are in $$\mathrm {L}_{k}^{1}(\mathbb {R}^{d})$$, we have the inversion formula
\begin{aligned} f(x)=\int _{\mathbb {R}^{d}}\widehat{f}(\xi )E_{k}(ix,\xi )w_{k}(\xi )d \xi ,\quad x \in \mathbb {R}^{d}. \end{aligned}
In $$\mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$, consider the generalized spherical mean operator defined by
\begin{aligned} \mathrm {M}_{h}f(x)=\frac{1}{d_{k}}\int \limits _{\mathbb {S}^{d-1}}\tau _{x}(f)(hy)d\eta _{k}(y),\quad (x\in {\mathbb {R}}^{d}, h>0) \end{aligned}
where $$\tau _{x}$$ Dunkl translation operator (see [11, 12]), $$\eta$$ is the normalized surface measure on the unit sphere $$\mathbb {S}^{d-1}$$ in $${\mathbb R}^{d}$$ and set $$d\eta _{k}(y)=w_{k}(x)d\eta (y)$$, $$\eta _{k}$$ is a $$W$$-invariant measure on $$\mathrm {S}^{d-1}$$ and $$d_{k}=\eta _{k}(\mathrm {S}^{d-1})$$.
We see that $$\mathrm {M}_{h}f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$ whenever $$f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$ and
\begin{aligned} \Vert \mathrm {M}_{h}f\Vert _{\mathrm {L}_{k}^{2}} \le \Vert f\Vert _{\mathrm {L}_{k}^{2}}, \end{aligned}
for all $$h>0$$.
For $$p \ge -\frac{1}{2}$$, we introduce the normalized Bessel function of the first kind $$j_{p}$$ defined by
\begin{aligned} j_{p}(z)=\Gamma (p +1) \sum \limits _{n=0}^{\infty }\frac{(-1)^{n}(\frac{z}{2})^{2n}}{n!\Gamma (n+p +1)}, \quad z\in \mathbb {C}. \end{aligned}
(1)

### Lemma 1.3

[1] The following inequalities are fulfilled
1. 1.

$$|j_{p}(x)|\le 1$$,

2. 2.

$$1-j_{p}(x)=O(1),~x\ge 1$$.

3. 3.

$$1-j_{p}(x)=O(x^{2});~0\le x \le 1$$.

From lemma 1.3, we have
\begin{aligned} |1-j_{p}(x)|\le C_{p} x, \quad \forall x\in \mathbb {R}^{+} \end{aligned}
(2)

### Lemma 1.4

The following inequality is true
\begin{aligned} |1-j_{p}(x)|\ge c, \end{aligned}
with $$|x|\ge 1$$, where $$c>0$$ is a certain constant which depends only on $$p$$.

### Proof

(Analog of lemma 2.9 in [3]) $$\square$$

Moreover, from (1) we see that
\begin{aligned} \lim \limits _{z \rightarrow 0}\frac{\left( j_{\gamma +\frac{d}{2}-1}(z)-1\right) }{z^{2}}\ne 0 \end{aligned}
(3)

### Proposition 1.5

Let $$f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$. Then
\begin{aligned} \widehat{(\mathrm {M}_{h}f)}(\xi )=j_{\gamma +\frac{d}{2}-1}(h|\xi |)\widehat{f}(\xi ). \end{aligned}
i.e
\begin{aligned} \mathrm {M}_{h}f(x)=\int \limits _{\mathbb {R}^{d}}j_{\gamma +\frac{d}{2}-1}(h|\xi |)\widehat{f}(\xi ) E_{k}(ix,\xi )w_{k}(\xi )d\xi \end{aligned}
and
\begin{aligned} f(x)=\int \limits _{\mathbb {R}^{d}}\widehat{f}(\xi )E_{k}(ix,\xi )w_{k}(\xi )d\xi . \end{aligned}
We have
\begin{aligned} \mathrm {M}_{h}f(x)-f(x)=\int \limits _{\mathbb {R}^{d}}(j_{\gamma +\frac{d}{2}-1}(h|\xi |)-1) \widehat{f}(\xi )E_{k}(ix,\xi )w_{k}(\xi )d\xi . \end{aligned}
(4)
Invoking Parseval’s identity (4) gives
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}^{2}=\int \limits _{\mathbb {R}^{d}}|j_{\gamma +\frac{d}{2}-1}(h|\xi |)-1|^{2} |\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi . \end{aligned}

## 2 Dini Lipschitz condition

### Definition 2.1

Let $$f(x)\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$, and let
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}, \quad \gamma \ge 0 \end{aligned}
i.e
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=O\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\right) , \end{aligned}
for all $$x$$ in $$\mathbb {R}^{d}$$ and for all sufficiently small $$h$$, $$C$$ being a positive constant. Then we say that $$f$$ satisfies a $$d$$-Dunkl Dini Lipschitz of order $$\alpha$$, or $$f$$ belongs to $$Lip(\alpha ,\gamma )$$.

### Definition 2.2

If however
\begin{aligned} \frac{\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}}{\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}} \rightarrow 0 \quad as\, h \rightarrow 0 \end{aligned}
i.e
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=o\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\right) \quad as~ h \rightarrow 0,~\gamma \ge 0 \end{aligned}
then $$f$$ is said to be belong to the little $$d$$-Dunkl Dini Lipschitz class $$lip(\alpha , \gamma )$$.

### Remark

It follows immediately from these definitions that
\begin{aligned} lip(\alpha ,\gamma )\subset Lip(\alpha ,\gamma ). \end{aligned}

### Theorem 2.3

Let $$\alpha >1$$. If $$f\in Lip(\alpha ,\gamma )$$, then $$f\in lip(1,\gamma )$$.

### Proof

For $$x\in \mathbb {R}^{d}$$ and $$h$$ small, $$f\in Lip(\alpha ,\gamma )$$ we have
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}. \end{aligned}
Then
\begin{aligned} \left( \log \frac{1}{h}\right) ^{\gamma }\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le Ch^{\alpha } \end{aligned}
Therefore
\begin{aligned} \frac{(\log \frac{1}{h})^{\gamma }}{h}\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le Ch^{\alpha -1}, \end{aligned}
which tends to zero with $$h \rightarrow 0$$. Thus
\begin{aligned} \frac{(\log \frac{1}{h})^{\gamma }}{h}\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}} \rightarrow 0 \quad as\, h\rightarrow 0 \end{aligned}
Then $$f\in lip(1,\gamma )$$. $$\square$$

### Theorem 2.4

If $$\alpha <\beta$$, then $$Lip(\alpha ,0)\supset Lip(\beta , 0)$$ and $$lip(\alpha , 0)\supset lip(\beta , 0)$$.

### Proof

We have $$0\le h \le 1$$ and $$\alpha <\beta$$, then $$h^{\beta }\le h^{\alpha }$$.

Then the proof of this theorem. $$\square$$

### Theorem 2.5

Let $$f, g\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$ such that $$\mathrm {M}_{h}(fg)(x)=\mathrm {M}_{h}f(x)\mathrm {M}_{h}g(x)$$. If $$f,g\in Lip(\alpha , \gamma )$$, then $$fg\in Lip(\alpha ,\gamma )$$.

### Proof

Since $$f,g\in Lip(\alpha , \gamma )$$, we have for all $$x$$ in $$\mathbb {R}^{d}$$
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }} \end{aligned}
and
\begin{aligned} \Vert \mathrm {M}_{h}g(x)-g(x)\Vert _{\mathrm {L}_{k}^{2}}\le C_{g}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }} \end{aligned}
It is clear that
\begin{aligned}&\Vert \mathrm {M}_{h}(fg)(x)-f(x)g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad = \Vert \mathrm {M}_{h}(fg)(x)-f(x)\mathrm {M}_{h}g(x)+f(x)\mathrm {M}_{h}g(x)-f(x)g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad =\Vert \mathrm {M}_{h}f(x)\mathrm {M}_{h}g(x)-f(x)\mathrm {M}_{h}g(x)+f(x)\mathrm {M}_{h}g(x)-f(x)g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad =\Vert \mathrm {M}_{h}g(x)(\mathrm {M}_{h}f(x)-f(x))+f(x)(\mathrm {M}_{h}g(x)-g(x))\Vert _{\mathrm {L}_{k}^{2}}\\&\quad \le \Vert \mathrm {M}_{h}g(x)\Vert _{\mathrm {L}_{k}^{2}}\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}+\Vert f(x)\Vert _{\mathrm {L}_{k}^{2}}\Vert \mathrm {M}_{h}g(x)-g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad \le K_{1}C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}+K_{2}C_{g}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\\&\quad \le M\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}, \end{aligned}
where $$M=\max (K_{1}C_{f}, K_{2}C_{g})$$. Then $$fg\in Lip(\alpha ,\gamma )$$ $$\square$$

## 3 New results on Dini Lipschitz class

### Theorem 3.1

Let $$\alpha >2$$. If $$f$$ belong to the $$d$$-Dunkl Dini Lipschitz class, i.e
\begin{aligned} f\in Lip(\alpha , \gamma ),\quad ~\alpha >2,~\gamma \ge 0. \end{aligned}
Then $$f$$ is equal to the null function in $$\mathbb {R}^{d}$$.

### Proof

Assume that $$f\in Lip(\alpha , \gamma )$$. Then
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}. \end{aligned}
We have to recall that the Dunkl transform of $$f(x)$$ satisfies the Parseval’s identity $$\Vert f\Vert _{\mathrm {L}_{k}^{2}}=\Vert \widehat{f}\Vert _{\mathrm {L}_{k}^{2}}$$.
So
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=\Vert \widehat{\mathrm {M}_{h}f-f}\Vert _{\mathrm {L}_{k}^{2}} \end{aligned}
i.e
\begin{aligned} \left\| (1-j_{\gamma +\frac{d}{2}-1}(h|\xi |))\widehat{f}(\xi )\right\| _{\mathrm {L}_{k}^{2}}\le C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}. \end{aligned}
it follows that
\begin{aligned} \int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \le C^{2}_{f}\frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}. \end{aligned}
Then
\begin{aligned} \frac{\int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi }{h^{4}}\le C^{2}_{f}\frac{h^{2\alpha -4}}{(\log \frac{1}{h})^{2\gamma }}. \end{aligned}
Since $$\alpha >2$$ we have
\begin{aligned} \lim \limits _{h \rightarrow 0}\frac{h^{2\alpha -4}}{(\log \frac{1}{h})^{2\gamma }}=0 \end{aligned}
Then
\begin{aligned} \lim \limits _{h \rightarrow 0}\int _{\mathbb {R}^{d}} \left( \frac{|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|}{|\xi |^{2}h^{2}}\right) ^{2}|\xi |^{4}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi =0. \end{aligned}
and also from the formula (3) and Fatou’s theorem, we obtain $$\Vert |\xi |^{2}\widehat{f}(\xi )\Vert _{\mathrm {L}_{k}^{2}}=0$$. Thus $$|\xi |^{2}\widehat{f}(\xi )=0$$ for all $$\xi \in \mathbb {R}^{d}$$, then $$f(x)$$ is the null function.$$\square$$

Analog of the theorem 3.1, we obtain this theorem

### Theorem 3.2

Let $$f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$. If $$f$$ belong to $$lip(2,0)$$, i.e
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=o(h^{2}) \quad as\, h \rightarrow 0. \end{aligned}
Then $$f$$ is equal to null function in $$\mathbb {R}^{d}$$.

Now, we give another the main result of this paper analog of theorem 1.1.

### Theorem 3.3

Let $$\alpha \in (0,1)$$$$\gamma \ge 0$$ and $$f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})$$. Then the following are equivalents
1. 1.

$$f\in Lip(\alpha ,\gamma )$$

2. 2.

$$\int _{|\xi |\ge s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi =O\left( \frac{s^{-2\alpha }}{(\log s)^{2\gamma }}\right) ~as~ s \rightarrow +\infty$$

### Proof

$$1)\Longrightarrow 2)$$ Assume that $$f\in Lip(\alpha ,\gamma )$$. Then we have
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=O\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\right) \quad as\, h\longrightarrow 0, \end{aligned}
Proposition 1.5 and Parseval’s identity give
\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}^{2}=\int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi . \end{aligned}
If $$|\xi |\in [\frac{1}{h},\frac{2}{h}]$$ then $$h|\xi |\ge 1$$ and lemma 1.4 implies that
\begin{aligned} 1\le \frac{1}{c^{2}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}. \end{aligned}
Then
\begin{aligned} \int _{\frac{1}{h}\le |\xi |\le \frac{2}{h}}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi\le & {} \frac{1}{c^{2}}\int _{\frac{1}{h}\le |\xi |\le \frac{2}{h}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \\\le & {} \frac{1}{c^{2}}\int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \\= & {} O\left( \frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}\right) . \end{aligned}
We obtain
\begin{aligned} \int _{s\le |\xi |\le 2s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \le C \frac{s^{-2\alpha }}{(\log s)^{2\gamma }}. \end{aligned}
where $$C$$ is a positive constant.
So that
\begin{aligned} \int _{|\xi |\ge s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi= & {} \left( \int _{s\le |\xi |\le 2s}+\int _{2s \le |\xi |\le 4s}+\int _{4s \le |\xi |\le 8s}+\cdots \right) |\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \\\le & {} C\left( \frac{s^{-2\alpha }}{(\log s)^{2\gamma }}+\frac{(2s)^{-2\alpha }}{(\log 2s)^{2\gamma }}+\frac{(4s)^{-2\alpha }}{(\log 4s)^{2\gamma }}+\cdots \right) \\\le & {} C\frac{s^{-2\alpha }}{(\log s)^{2\gamma }}(1+2^{-2\alpha }+(2^{-2\alpha })^{2}+(2^{-2\alpha })^{3}\cdots )\\\le & {} C K_{\alpha }\frac{s^{-2\alpha }}{(\log s)^{2\gamma }}, \end{aligned}
where $$K_{\alpha }=C(1-2^{-2\alpha })^{-1}$$ since $$2^{-2\alpha }<1.$$
This proves that
\begin{aligned} \int _{|\xi |\ge s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi =O\left( \frac{s^{-2\alpha }}{(\log s)^{2\gamma }} \right) \quad as\, s\longrightarrow +\infty \end{aligned}
$$2)\Longrightarrow 1)$$ Suppose now that
\begin{aligned} \int _{|\xi |\ge s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi =O\left( \frac{s^{-2\alpha }}{(\log s)^{2\gamma }}\right) \quad as\, s\longrightarrow +\infty . \end{aligned}
We have to show that
\begin{aligned} \int _{0}^{\infty }r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr=O\left( \frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}\right) , \quad as\, h \rightarrow 0 \end{aligned}
where
\begin{aligned} \phi (r)=\int _{\mathbb {S}^{d-1}}|\widehat{f}(ry)|^{2}w_{k}(y)dy. \end{aligned}
We write
\begin{aligned} \int _{0}^{\infty }r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr=\mathrm {I}_{1}+\mathrm {I}_{2}, \end{aligned}
where
\begin{aligned} \mathrm {I}_{1}=\int _{0}^{1/h}r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr, \end{aligned}
and
\begin{aligned} \mathrm {I}_{2}=\int _{1/h}^{\infty }r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr. \end{aligned}
Firstly, from (1) in lemma 1.3 we see that
\begin{aligned} \mathrm {I}_{2}\le 4\int _{1/h}^{\infty }r^{2\gamma +d-1}\phi (r)dr=O\left( \frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}\right) \quad as\,h \longrightarrow 0. \end{aligned}
Set
\begin{aligned} \psi (r)=\int _{r}^{\infty }x^{2\gamma +d-1}\phi (x)dx. \end{aligned}
From formula 2, an integration by parts yields
\begin{aligned} \mathrm {I}_{1}= & {} \int _{0}^{1/h}r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr\\\le & {} -C_{p}h^{2}\int _{0}^{1/h}r^{2}\psi ^{\prime }(r)dr\\\le & {} -C_{p}\psi (1/h)+2C_{p}h^{2}\int _{0}^{1/h}r\psi (r)dr\\\le & {} 2C_{p}h^{2}\int _{0}^{1/h}r \frac{r^{-2\alpha }}{(\log r)^{2\gamma }}dr\\\le & {} C_{1}\frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }} \end{aligned}
where $$C_{1}$$ is a positive constant, and this ends the proof $$\square$$

## References

1. 1.
Abilov, V.A., Abilova, F.V.: Approximation of functions by Fourier-Bessel sums. Izv. Vyssh. Uchebn. Zaved. Mat. 8, 3–9 (2001)
2. 2.
Abilov, V.A., Abilova, F.V., Kerimov, M.K.: Some remarks concerning the Fourier transform in the space $${\rm {l}}_{2}(\mathbb{R})$$ Zh. Vychisl. Mat. Mat. Fiz. 48, 939–945 (2008). [Comput. Math. Math. Phys. 48, 885–891]
3. 3.
Belkina, E.S., Platonov, S.S.: Equivalence of $$K$$-functionals and modulus of smoothness constructed by generalized Dunkl translations. Izv. Vyssh. Uchebn. Zaved. Mat 8, 3–15 (2008)
4. 4.
Daher, R., El Hamma, M.: Bessel transform of $$(k, \gamma )$$-Bessel Lipschitz functions, Hindawi Publishing Corporation. J. Math., vol. 2013, Article ID 418546Google Scholar
5. 5.
de Jeu, M.F.E.: The Dunkl transform. Inv. Math. 113, 147–162 (1993)
6. 6.
Dunkl, C.F.: Differential- difference operators associated to reflection groups. Trans. Am. Math Soc. 311, 167–183 (1989)
7. 7.
Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Proceedings of Special Session on Hypergeometric Functions in Domains of Positivity. Jack Polynomials and Applications (Tampa, 1991), Contemp. Math. 138(1992), 123–138 (1991)Google Scholar
8. 8.
Dunkl, C.F.: Integral Kernels with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991)
9. 9.
Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, vol. 81. Cambridge University Press, Cambridge (2001)
10. 10.
Rösler, M., Voit, M.: Markov processes with Dunkl operators. Adv. Appl. Math. 21, 575–643 (1998)
11. 11.
Thangavelu, S., Xu, Y.: Convolution operator and maximal function for Dunkl transform. J. Anal. Math. 97, 25–56 (2005)
12. 12.
Trimèche, K.: Paley-Wiener theorems for the Dunkl transform and Dunkl transform operators. Integral Transf. Spec. Funct. 13, 17–38 (2002)
13. 13.
Younis, M.S.: Fourier transforms of Dini-Lipschitz Functions. Int. J. Math. Sci. 9(2), 301–312 (1986)