Dini Lipschitz functions for the Dunkl transform in the Space \(\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)\)
Open Access
ArticleFirst Online:
Received:
Accepted:
- 596 Downloads
- 2 Citations
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 operatorMathematics Subject Classification
42B371 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 equivalentswhere \(\mathcal {F}\) stands for the Fourier transform of \(f\).
- 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.
\(\int _{|x|\ge r}|\mathcal {F}(f)(x)|^{2}dx=O\left( \frac{r^{-2\alpha }}{(\log r)^{2\beta }}\right) \quad as \,r \longrightarrow +\infty \),
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 \(f \in C^{1}(\mathbb {R}^{d})\) byHere \(\langle ,\rangle \) 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 \(\mathbb {R}^{d}\).
$$\begin{aligned} \mathrm {D}_{j}f(x)=\frac{\partial f}{\partial x_{j}}(x)+\sum _{\alpha \in \mathrm {R}_{+}}k(\alpha )\alpha _{j}\frac{f(x)-f(\sigma _{\alpha }(x))}{\langle \alpha ,x\rangle },\quad x\in \mathbb {R}^{d} ~(1\le j \le d) \end{aligned}$$
The weight function \(w_{k}\) defined bywhere \(w_{k}\) is \(W\)-invariant and homogeneous of degree \(2\gamma \) whereThe 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 \(E_{k}\) is called the Dunkl Kernel.
$$\begin{aligned} w_{k}(x)=\prod _{\zeta \in \mathrm {R}_{+}}|\langle \zeta ,x\rangle |^{2k(\alpha )}, \quad x \in \mathbb {R}^{d}, \end{aligned}$$
$$\begin{aligned} \gamma = \gamma (\mathrm {R})=\sum _{\zeta \in \mathrm {R}_{+}}k(\zeta ) \ge 0. \end{aligned}$$
$$\begin{aligned} \left\{ \begin{array}{ll} \mathrm {D}_{j}u(x,y)=y_{j}u(x,y) &{} \quad \mathrm {for}\,\quad 1\le j\le d \\ u(0,y)=1 &{}\quad \mathrm {for~ all}\, y\in \mathbb {R}^{d} \\ \end{array} \right. \end{aligned}$$
Proposition 1.2
[5] Let \(z,w \in \mathbb {C}\) and \(\lambda \in \mathbb {C}\). Then
- 1.
\(E_{k}(z,0)=1\).
- 2.
\(E_{k}(z,w)=E_{k}(w,z)\).
- 3.
\(E_{k}(\lambda z,w)= E_{k}(z,\lambda w)\).
- 4.For all \(\nu =(\nu _{1},\ldots ,\nu _{d})\in {\mathbb N}^{d},~ x\in {\mathbb R}^{d},~ z\in {\mathbb C}^{d}\), we havewhere$$\begin{aligned} |\partial _{z}^{\nu }E_{k}(x,z)|\le |x|^{|\nu |}exp(|x||Re(z)|), \end{aligned}$$In particular$$\begin{aligned} \partial _{z}^{\nu }=\frac{\partial ^{|\nu |}}{\partial z_{1}^{\nu _{1}}\ldots \partial z_{d}^{\nu _{d}}},\quad |\nu |=\nu _{1}+\cdots +\nu _{d}. \end{aligned}$$for all \(x,z\in {\mathbb R}^{d}.\)$$\begin{aligned} |\partial _{z}^{\nu }E_{k}(ix,z)|\le |x|^{\nu }, \end{aligned}$$
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‘where the constant \(c_{k}\) is given byThe 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 formulaIn \(\mathrm {L}_{k}^{2}(\mathbb {R}^{d})\), consider the generalized spherical mean operator defined bywhere \(\tau _{x}\) 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 \(d_{k}=\eta _{k}(\mathrm {S}^{d-1})\).
$$\begin{aligned} \widehat{f}(\xi )=c_{k}^{-1}\int _{\mathbb {R}^{d}}f(x)E_{k}(-i \xi ,x)w_{k}(x)dx, \end{aligned}$$
$$\begin{aligned} c_{k}=\int _{\mathbb {R}^{d}}e^{\frac{-|z|^{2}}{2}}w_{k}(z)dz. \end{aligned}$$
$$\begin{aligned} f(x)=\int _{\mathbb {R}^{d}}\widehat{f}(\xi )E_{k}(ix,\xi )w_{k}(\xi )d \xi ,\quad x \in \mathbb {R}^{d}. \end{aligned}$$
$$\begin{aligned} \mathrm {M}_{h}f(x)=\frac{1}{d_{k}}\int \limits _{\mathbb {S}^{d-1}}\tau _{x}(f)(hy)d\eta _{k}(y),\quad (x\in {\mathbb {R}}^{d}, h>0) \end{aligned}$$
We see that \(\mathrm {M}_{h}f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})\) whenever \(f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})\) andfor all \(h>0\).
$$\begin{aligned} \Vert \mathrm {M}_{h}f\Vert _{\mathrm {L}_{k}^{2}} \le \Vert f\Vert _{\mathrm {L}_{k}^{2}}, \end{aligned}$$
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.
\(|j_{p}(x)|\le 1\),
- 2.
\(1-j_{p}(x)=O(1),~x\ge 1\).
- 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 truewith \(|x|\ge 1\), where \(c>0\) is a certain constant which depends only on \(p\).
$$\begin{aligned} |1-j_{p}(x)|\ge c, \end{aligned}$$
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.eandWe haveInvoking Parseval’s identity (4) gives
$$\begin{aligned} \mathrm {M}_{h}f(x)=\int \limits _{\mathbb {R}^{d}}j_{\gamma +\frac{d}{2}-1}(h|\xi |)\widehat{f}(\xi ) E_{k}(ix,\xi )w_{k}(\xi )d\xi \end{aligned}$$
$$\begin{aligned} f(x)=\int \limits _{\mathbb {R}^{d}}\widehat{f}(\xi )E_{k}(ix,\xi )w_{k}(\xi )d\xi . \end{aligned}$$
$$\begin{aligned} \mathrm {M}_{h}f(x)-f(x)=\int \limits _{\mathbb {R}^{d}}(j_{\gamma +\frac{d}{2}-1}(h|\xi |)-1) \widehat{f}(\xi )E_{k}(ix,\xi )w_{k}(\xi )d\xi . \end{aligned}$$
(4)
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}^{2}=\int \limits _{\mathbb {R}^{d}}|j_{\gamma +\frac{d}{2}-1}(h|\xi |)-1|^{2} |\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi . \end{aligned}$$
2 Dini Lipschitz condition
Definition 2.1
Let \(f(x)\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})\), and leti.efor 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 \(Lip(\alpha ,\gamma )\).
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}, \quad \gamma \ge 0 \end{aligned}$$
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=O\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\right) , \end{aligned}$$
Definition 2.2
If howeveri.ethen \(f\) is said to be belong to the little \(d\)-Dunkl Dini Lipschitz class \(lip(\alpha , \gamma )\).
$$\begin{aligned} \frac{\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}}{\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}} \rightarrow 0 \quad as\, h \rightarrow 0 \end{aligned}$$
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=o\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\right) \quad as~ h \rightarrow 0,~\gamma \ge 0 \end{aligned}$$
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 haveThenThereforewhich tends to zero with \(h \rightarrow 0\). ThusThen \(f\in lip(1,\gamma )\). \(\square \)
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}. \end{aligned}$$
$$\begin{aligned} \left( \log \frac{1}{h}\right) ^{\gamma }\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le Ch^{\alpha } \end{aligned}$$
$$\begin{aligned} \frac{(\log \frac{1}{h})^{\gamma }}{h}\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le Ch^{\alpha -1}, \end{aligned}$$
$$\begin{aligned} \frac{(\log \frac{1}{h})^{\gamma }}{h}\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}} \rightarrow 0 \quad as\, h\rightarrow 0 \end{aligned}$$
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 \(\mathbb {R}^{d}\) andIt is clear thatwhere \(M=\max (K_{1}C_{f}, K_{2}C_{g})\). Then \(fg\in Lip(\alpha ,\gamma )\) \(\square \)
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }} \end{aligned}$$
$$\begin{aligned} \Vert \mathrm {M}_{h}g(x)-g(x)\Vert _{\mathrm {L}_{k}^{2}}\le C_{g}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }} \end{aligned}$$
$$\begin{aligned}&\Vert \mathrm {M}_{h}(fg)(x)-f(x)g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad = \Vert \mathrm {M}_{h}(fg)(x)-f(x)\mathrm {M}_{h}g(x)+f(x)\mathrm {M}_{h}g(x)-f(x)g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad =\Vert \mathrm {M}_{h}f(x)\mathrm {M}_{h}g(x)-f(x)\mathrm {M}_{h}g(x)+f(x)\mathrm {M}_{h}g(x)-f(x)g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad =\Vert \mathrm {M}_{h}g(x)(\mathrm {M}_{h}f(x)-f(x))+f(x)(\mathrm {M}_{h}g(x)-g(x))\Vert _{\mathrm {L}_{k}^{2}}\\&\quad \le \Vert \mathrm {M}_{h}g(x)\Vert _{\mathrm {L}_{k}^{2}}\Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}+\Vert f(x)\Vert _{\mathrm {L}_{k}^{2}}\Vert \mathrm {M}_{h}g(x)-g(x)\Vert _{\mathrm {L}_{k}^{2}}\\&\quad \le K_{1}C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}+K_{2}C_{g}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\\&\quad \le M\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}, \end{aligned}$$
3 New results on Dini Lipschitz class
Theorem 3.1
Let \(\alpha >2\). If \(f\) belong to the \(d\)-Dunkl Dini Lipschitz class, i.eThen \(f\) is equal to the null function in \(\mathbb {R}^{d}\).
$$\begin{aligned} f\in Lip(\alpha , \gamma ),\quad ~\alpha >2,~\gamma \ge 0. \end{aligned}$$
Proof
Assume that \(f\in Lip(\alpha , \gamma )\). ThenWe have to recall that the Dunkl transform of \(f(x)\) satisfies the Parseval’s identity \(\Vert f\Vert _{\mathrm {L}_{k}^{2}}=\Vert \widehat{f}\Vert _{\mathrm {L}_{k}^{2}}\).
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}\le C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}. \end{aligned}$$
Soi.eit follows thatThenSince \(\alpha >2\) we haveThenand 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.\(\square \)
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=\Vert \widehat{\mathrm {M}_{h}f-f}\Vert _{\mathrm {L}_{k}^{2}} \end{aligned}$$
$$\begin{aligned} \left\| (1-j_{\gamma +\frac{d}{2}-1}(h|\xi |))\widehat{f}(\xi )\right\| _{\mathrm {L}_{k}^{2}}\le C_{f}\frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}. \end{aligned}$$
$$\begin{aligned} \int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \le C^{2}_{f}\frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}. \end{aligned}$$
$$\begin{aligned} \frac{\int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi }{h^{4}}\le C^{2}_{f}\frac{h^{2\alpha -4}}{(\log \frac{1}{h})^{2\gamma }}. \end{aligned}$$
$$\begin{aligned} \lim \limits _{h \rightarrow 0}\frac{h^{2\alpha -4}}{(\log \frac{1}{h})^{2\gamma }}=0 \end{aligned}$$
$$\begin{aligned} \lim \limits _{h \rightarrow 0}\int _{\mathbb {R}^{d}} \left( \frac{|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|}{|\xi |^{2}h^{2}}\right) ^{2}|\xi |^{4}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi =0. \end{aligned}$$
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.eThen \(f\) is equal to null function in \(\mathbb {R}^{d}\).
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=o(h^{2}) \quad as\, h \rightarrow 0. \end{aligned}$$
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.
\(f\in Lip(\alpha ,\gamma )\)
- 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 haveProposition 1.5 and Parseval’s identity giveIf \(|\xi |\in [\frac{1}{h},\frac{2}{h}]\) then \(h|\xi |\ge 1\) and lemma 1.4 implies thatThenWe obtainwhere \(C\) is a positive constant.
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}=O\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\gamma }}\right) \quad as\, h\longrightarrow 0, \end{aligned}$$
$$\begin{aligned} \Vert \mathrm {M}_{h}f(x)-f(x)\Vert _{\mathrm {L}_{k}^{2}}^{2}=\int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi . \end{aligned}$$
$$\begin{aligned} 1\le \frac{1}{c^{2}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}. \end{aligned}$$
$$\begin{aligned} \int _{\frac{1}{h}\le |\xi |\le \frac{2}{h}}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi\le & {} \frac{1}{c^{2}}\int _{\frac{1}{h}\le |\xi |\le \frac{2}{h}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \\\le & {} \frac{1}{c^{2}}\int _{\mathbb {R}^{d}}|1-j_{\gamma +\frac{d}{2}-1}(h|\xi |)|^{2}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \\= & {} O\left( \frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}\right) . \end{aligned}$$
$$\begin{aligned} \int _{s\le |\xi |\le 2s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \le C \frac{s^{-2\alpha }}{(\log s)^{2\gamma }}. \end{aligned}$$
So thatwhere \(K_{\alpha }=C(1-2^{-2\alpha })^{-1}\) since \(2^{-2\alpha }<1.\)
$$\begin{aligned} \int _{|\xi |\ge s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi= & {} \left( \int _{s\le |\xi |\le 2s}+\int _{2s \le |\xi |\le 4s}+\int _{4s \le |\xi |\le 8s}+\cdots \right) |\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi \\\le & {} C\left( \frac{s^{-2\alpha }}{(\log s)^{2\gamma }}+\frac{(2s)^{-2\alpha }}{(\log 2s)^{2\gamma }}+\frac{(4s)^{-2\alpha }}{(\log 4s)^{2\gamma }}+\cdots \right) \\\le & {} C\frac{s^{-2\alpha }}{(\log s)^{2\gamma }}(1+2^{-2\alpha }+(2^{-2\alpha })^{2}+(2^{-2\alpha })^{3}\cdots )\\\le & {} C K_{\alpha }\frac{s^{-2\alpha }}{(\log s)^{2\gamma }}, \end{aligned}$$
This proves that \(2)\Longrightarrow 1)\) Suppose now thatWe have to show thatwhereWe writewhereandFirstly, from (1) in lemma 1.3 we see thatSetFrom formula 2, an integration by parts yieldswhere \(C_{1}\) is a positive constant, and this ends the proof \(\square \)
$$\begin{aligned} \int _{|\xi |\ge s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi =O\left( \frac{s^{-2\alpha }}{(\log s)^{2\gamma }} \right) \quad as\, s\longrightarrow +\infty \end{aligned}$$
$$\begin{aligned} \int _{|\xi |\ge s}|\widehat{f}(\xi )|^{2}w_{k}(\xi )d\xi =O\left( \frac{s^{-2\alpha }}{(\log s)^{2\gamma }}\right) \quad as\, s\longrightarrow +\infty . \end{aligned}$$
$$\begin{aligned} \int _{0}^{\infty }r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr=O\left( \frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}\right) , \quad as\, h \rightarrow 0 \end{aligned}$$
$$\begin{aligned} \phi (r)=\int _{\mathbb {S}^{d-1}}|\widehat{f}(ry)|^{2}w_{k}(y)dy. \end{aligned}$$
$$\begin{aligned} \int _{0}^{\infty }r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr=\mathrm {I}_{1}+\mathrm {I}_{2}, \end{aligned}$$
$$\begin{aligned} \mathrm {I}_{1}=\int _{0}^{1/h}r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr, \end{aligned}$$
$$\begin{aligned} \mathrm {I}_{2}=\int _{1/h}^{\infty }r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr. \end{aligned}$$
$$\begin{aligned} \mathrm {I}_{2}\le 4\int _{1/h}^{\infty }r^{2\gamma +d-1}\phi (r)dr=O\left( \frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }}\right) \quad as\,h \longrightarrow 0. \end{aligned}$$
$$\begin{aligned} \psi (r)=\int _{r}^{\infty }x^{2\gamma +d-1}\phi (x)dx. \end{aligned}$$
$$\begin{aligned} \mathrm {I}_{1}= & {} \int _{0}^{1/h}r^{2\gamma +d-1}|1-j_{\gamma +\frac{d}{2}-1}(hr)|^{2}\phi (r)dr\\\le & {} -C_{p}h^{2}\int _{0}^{1/h}r^{2}\psi ^{\prime }(r)dr\\\le & {} -C_{p}\psi (1/h)+2C_{p}h^{2}\int _{0}^{1/h}r\psi (r)dr\\\le & {} 2C_{p}h^{2}\int _{0}^{1/h}r \frac{r^{-2\alpha }}{(\log r)^{2\gamma }}dr\\\le & {} C_{1}\frac{h^{2\alpha }}{(\log \frac{1}{h})^{2\gamma }} \end{aligned}$$
References
- 1.Abilov, V.A., Abilova, F.V.: Approximation of functions by Fourier-Bessel sums. Izv. Vyssh. Uchebn. Zaved. Mat. 8, 3–9 (2001)MathSciNetGoogle Scholar
- 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]MathSciNetGoogle Scholar
- 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)MathSciNetGoogle Scholar
- 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.de Jeu, M.F.E.: The Dunkl transform. Inv. Math. 113, 147–162 (1993)CrossRefGoogle Scholar
- 6.Dunkl, C.F.: Differential- difference operators associated to reflection groups. Trans. Am. Math Soc. 311, 167–183 (1989)MathSciNetCrossRefGoogle Scholar
- 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.Dunkl, C.F.: Integral Kernels with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991)MathSciNetCrossRefGoogle Scholar
- 9.Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, vol. 81. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
- 10.Rösler, M., Voit, M.: Markov processes with Dunkl operators. Adv. Appl. Math. 21, 575–643 (1998)CrossRefGoogle Scholar
- 11.Thangavelu, S., Xu, Y.: Convolution operator and maximal function for Dunkl transform. J. Anal. Math. 97, 25–56 (2005)MathSciNetCrossRefGoogle Scholar
- 12.Trimèche, K.: Paley-Wiener theorems for the Dunkl transform and Dunkl transform operators. Integral Transf. Spec. Funct. 13, 17–38 (2002)CrossRefGoogle Scholar
- 13.Younis, M.S.: Fourier transforms of Dini-Lipschitz Functions. Int. J. Math. Sci. 9(2), 301–312 (1986)MathSciNetCrossRefGoogle Scholar
Copyright information
© The Author(s) 2015
Open AccessThis 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.