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.
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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.
\(\Vert f(x+h)-f(x)\Vert _{2}=O\left( \frac{h^{\alpha }}{(\log \frac{1}{h})^{\beta }}\right) \quad as\, h\longrightarrow 0, ~0<\alpha <1, \beta \ge 0\),
-
2.
\(\int _{|x|\ge r}|\mathcal {F}(f)(x)|^{2}dx=O\left( \frac{r^{-2\alpha }}{(\log r)^{2\beta }}\right) \quad as \,r \longrightarrow +\infty \),
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 [5–7]. 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, 9–11]) 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
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
where \(w_{k}\) is \(W\)-invariant and homogeneous of degree \(2\gamma \) where
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
\(E_{k}\) is called the Dunkl Kernel.
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 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
‘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 \(\mathrm {L}_{k}^{2}=\mathrm {L}_{k}^{2}(\mathbb {R}^{d})=\mathrm {L}_{k}^{2}(\mathbb {R}^{d}, w_{k}(x)dx)\), when both \(f\) and \(\widehat{f}\) are in \(\mathrm {L}_{k}^{1}(\mathbb {R}^{d})\), we have the inversion formula
In \(\mathrm {L}_{k}^{2}(\mathbb {R}^{d})\), consider the generalized spherical mean operator defined by
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
for all \(h>0\).
For \(p \ge -\frac{1}{2}\), we introduce the normalized Bessel function of the first kind \(j_{p}\) defined by
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
Lemma 1.4
The following inequality is true
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
Proposition 1.5
Let \(f\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})\). Then
i.e
and
We have
Invoking Parseval’s identity (4) gives
2 Dini Lipschitz condition
Definition 2.1
Let \(f(x)\in \mathrm {L}_{k}^{2}(\mathbb {R}^{d})\), and let
i.e
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
i.e
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
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
Then
Therefore
which tends to zero with \(h \rightarrow 0\). Thus
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 \(\mathbb {R}^{d}\)
and
It is clear that
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
Then \(f\) is equal to the null function in \(\mathbb {R}^{d}\).
Proof
Assume that \(f\in Lip(\alpha , \gamma )\). Then
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
i.e
it follows that
Then
Since \(\alpha >2\) we have
Then
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
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.
\(f\in Lip(\alpha ,\gamma )\)
-
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 have
Proposition 1.5 and Parseval’s identity give
If \(|\xi |\in [\frac{1}{h},\frac{2}{h}]\) then \(h|\xi |\ge 1\) and lemma 1.4 implies that
Then
We obtain
where \(C\) is a positive constant.
So that
where \(K_{\alpha }=C(1-2^{-2\alpha })^{-1}\) since \(2^{-2\alpha }<1.\)
This proves that
\(2)\Longrightarrow 1)\) Suppose now that
We have to show that
where
We write
where
and
Firstly, from (1) in lemma 1.3 we see that
Set
From formula 2, an integration by parts yields
where \(C_{1}\) is a positive constant, and this ends the proof \(\square \)
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Dedicated to Professor François Rouvière for his 69’s birthday.
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Hamma, M.E., Daher, R. Dini Lipschitz functions for the Dunkl transform in the Space \(\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)\) . Rend. Circ. Mat. Palermo 64, 241–249 (2015). https://doi.org/10.1007/s12215-015-0195-9
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DOI: https://doi.org/10.1007/s12215-015-0195-9