<figure><img src="image_1.png" /></figure>

BLAISE PASCAL

BECHIR AMRI & MOHAMED SIFI

Riesz transforms for Dunkl transform

Volume 19, n° 1 (2012), p. 247-262.

<http://ambp.cedram.org/item?id=AMBP_2012__19_1_247_0>

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# Riesz transforms for Dunkl transform

BECHIR AMRI

MOHAMED SIFI

## Abstract

In this paper we obtain the $L^p$-boundedness of Riesz transforms for the Dunkl transform for all $1 < p < \infty$.

## Transformées de Riesz associés à la transformée de Dunkl

## Résumé

Dans cet article, nous étudions la bornitude des transformées de Riesz associées à la transformée de Dunkl sur les espaces $L^p$, $1 < p < \infty$.

## 1. Introduction

On the Euclidean space $\mathbb{R}^N$, $N \ge 1$, the ordinary Riesz transform $R_j$, $j = 1, \dots, N$ is defined as the multiplier operator

$$ \widehat{R_j(f)}(\xi) = -i \frac{\xi_j}{\|\xi\|} \hat{f}(\xi). \qquad (1.1) $$

It can also be defined by the principal value of the singular integral

$$ R_j(f)(x) = d_0 \lim_{\epsilon \to 0} \int_{\|x-y\| > \epsilon} \frac{x_j - y_j}{\|x-y\|} f(y)dy $$

where $d_0 = 2^{\frac{N}{2}} \frac{\Gamma(\frac{N+1}{2})}{\sqrt{\pi}}$. It follows from the general theory of singular integrals that Riesz transforms are bounded on $L^p(\mathbb{R}^N, dx)$ for all $1 < p < \infty$. What is done in this paper is to extend this result to the context of Dunkl theory where a similar operator is already defined.

Dunkl theory generalizes classical Fourier analysis on $\mathbb{R}^N$. It started twenty years ago with Dunkl's seminal work [3] and was further developed by several mathematicians. See for instance the surveys [5, 6, 7, 9] and the

Keywords: Dunkl transforms, Riesz Transforms, Singular integrals.

Math. classification: 17B22, 32A55, 43A32, 42A45.
references cited therein. The study of the $L^p$-boundedness of Riesz transforms for Dunkl transform on $\mathbb{R}^N$ goes back to the work of S. Thangavelu and Y. Xu [10] where they established boundedness result only in a very special case of $N = 1$. It has been noted in [10] that the difficulty arises in the application of the classical $L^p$- theory of Caldéron-Zygmund, since Riesz transforms are singular integral operators. In this paper we describe how this theory can be adapted in Dunkl setting and gives an $L^p$-result for Riesz transforms for all $1 < p < \infty$. More precisely, through the fundamental result of M. Rösler [6] for the Dunkl translation of radial functions, we reformulate a Hörmander type condition for singular integral operators. The Riesz kernel is given by acting Dunkl operator on Dunkl translation of radial function.

This paper is organized as follows. In Section 2 we present some definitions and fundamental results from Dunkl's analysis. The Section 3 is devoted to proving $L^p$-boundedness of Riesz transforms. As applications, we will prove a generalized Riesz and Sobolev inequalities. Throughout this paper $C$ denotes a constant which can vary from line to line.

**Acknowledgments.** The authors are partially supported by the project DGRST 04/UR/15-02 and the cooperation programs PHC Utique / CMCU 07G 1501 and 10G 1503.

## 2. Preliminaries

In this section we collect notations and definitions and recall some basic facts. We refer to [5, 3, 6, 7, 9].

Let $G \subset O(\mathbb{R}^N)$ be a finite reflection group associated to a reduced root system $R$ and $k : R \to [0, +\infty)$ be a $G$-invariant function (called multiplicity function). Let $R_+$ be a positive root subsystem. We shall assume that $R$ is normalized in the sense that $\|\alpha\|^2 = \langle \alpha, \alpha \rangle = 2$ for all $\alpha \in R$, where $\langle \cdot, \cdot \rangle$ is the standard Euclidean scalar product on $\mathbb{R}^N$.

The Dunkl operators $T_\xi, \xi \in \mathbb{R}^N$ are the following $k$-deformations of directional derivatives $\partial_\xi$ by difference operators:

$$T_{\xi}f(x) = \partial_{\xi}f(x) + \sum_{\alpha \in R_{+}} k(\alpha) \langle \alpha, \xi \rangle \frac{f(x) - f(\sigma_{\alpha} \cdot x)}{\langle \alpha, x \rangle}, \quad x \in \mathbb{R}^N$$
where $\sigma_\alpha$ denotes the reflection with respect to the hyperplane orthogonal to $\alpha$. For the standard basis vectors of $\mathbb{R}^N$, we simply write $T_j = T_{e_j}$.

The operators $\partial_\xi$ and $T_\xi$ are intertwined by a Laplace-type operator

$$V_k f(x) = \int_{\mathbb{R}^N} f(y) d\mu_x(y),$$

associated to a family of compactly supported probability measures

$$\{ \mu_x | x \in \mathbb{R}^N \}.$$

Specifically, $\mu_x$ is supported in the convex hull $co(G \cdot x)$.

For every $\lambda \in \mathbb{C}^N$, the simultaneous eigenfunction problem,

$$T_\xi f = \langle \lambda, \xi \rangle f, \quad \xi \in \mathbb{R}^N$$

has a unique solution $f(x) = E_k(\lambda, x)$ such that $E_k(\lambda, 0) = 1$, which is given by

$$E_k(\lambda, x) = V_k(e^{\langle \lambda, \cdot \rangle})(x) = \int_{\mathbb{R}^N} e^{\langle \lambda, y \rangle} d\mu_x(y), \quad x \in \mathbb{R}^N.$$

Furthermore $\lambda \mapsto E_k(\lambda, x)$ extends to a holomorphic function on $\mathbb{C}^N$.

Let $m_k$ be the measure on $\mathbb{R}^N$, given by

$$dm_k(x) = \prod_{\alpha \in R_+} |\langle \alpha, x \rangle|^{2k(\alpha)} dx.$$

For $f \in L^1(m_k)$ (the Lebesgue space with respect to the measure $m_k$) the Dunkl transform is defined by

$$\mathcal{F}_k(f)(\xi) = \frac{1}{c_k} \int_{\mathbb{R}^N} f(x) E_k(-i\xi, x) dm_k(x), \quad c_k = \int_{\mathbb{R}^N} e^{-\frac{|x|^2}{2}} dm_k(x).$$

This new transform shares many analogous properties of the Fourier transform.

(i) The Dunkl transform is a topological automorphism of $\mathcal{S}(\mathbb{R}^N)$ (Schwartz space).

(ii) (Plancherel Theorem) The Dunkl transform extends to an isometric automorphism of $L^2(m_k)$.

(iii) (Inversion formula) For every $f \in L^1(m_k)$ such that $\mathcal{F}_k f \in L^1(m_k)$, we have

$$f(x) = \mathcal{F}_k^2 f(-x), \quad x \in \mathbb{R}^N.$$
(iv) For all $\xi \in \mathbb{R}^N$ and $f \in S(\mathbb{R}^N)$

$$ \mathcal{F}_k(T_\xi(f))(x) = \langle i\xi, x \rangle \mathcal{F}_k(f)(x), \quad x \in \mathbb{R}^N. \qquad (2.1) $$

Let $x \in \mathbb{R}^N$, the Dunkl translation operator $\tau_x$ is defined on $L^2(m_k)$ by,

$$ \mathcal{F}_k(\tau_x(f))(y) = E_k(ix, y)\mathcal{F}_k f(y), \quad y \in \mathbb{R}^N. \qquad (2.2) $$

If $f$ is a continuous radial function in $L^2(m_k)$ with $f(y) = \tilde{f}(\|y\|)$, then

$$ \tau_x(f)(y) = \int_{\mathbb{R}^N} \tilde{f}\left(\sqrt{\|x\|^2 + \|y\|^2 + 2\langle y, \eta \rangle}\right) d\mu_x(\eta). \qquad (2.3) $$

This formula is first proved by M. Rösler [6] for $f \in S(\mathbb{R}^N)$ and recently is extended to continuous functions by F. and H. Dai Wang [2].

We collect below some useful facts :

(i) For all $x, y \in \mathbb{R}^N$,

$$ \tau_x(f)(y) = \tau_y(f)(x). \qquad (2.4) $$

(ii) For all $x, \xi \in \mathbb{R}^N$ and $f \in S(\mathbb{R}^N)$,

$$ T_\xi \tau_x(f) = \tau_x T_\xi(f). \qquad (2.5) $$

(iii) For all $x \in \mathbb{R}^N$ and $f, g \in L^2(m_k)$,

$$ \int_{\mathbb{R}^N} \tau_x(f)(-y)g(y)dm_k(y) = \int_{\mathbb{R}^N} f(y)\tau_x g(-y)dm_k(y). \qquad (2.6) $$

(iv) For all $x \in \mathbb{R}^N$ and $1 \le p \le 2$, the operator $\tau_x$ can be extended to all radial functions $f$ in $L^p(m_k)$ and the following holds

$$ \|\tau_x(f)\|_{p,k} \le \|f\|_{p,k}. \qquad (2.7) $$

$\|\cdot\|_{p,k}$ is the usual norm of $L^p(m_k)$.

### 3. Riesz transforms for the Dunkl transform.

In Dunkl setting the Riesz transforms (see [10]) are the operators $\mathcal{R}_j$, $j = 1...N$ defined on $L^2(m_k)$ by

$$ \mathcal{R}_j(f)(x) = d_k \lim_{\epsilon \to 0} \int_{|y|>\epsilon} \tau_x(f)(-y) \frac{y_j}{\|y\|^{p_k}} dm_k(y), \quad x \in \mathbb{R}^N $$
RIESZ TRANSFORMS FOR DUNKL TRANSFORM

where

$$d_k = 2^{\frac{p_k-1}{2}} \frac{\Gamma(\frac{p_k}{2})}{\sqrt{\pi}}; \quad p_k = 2\gamma_k + N + 1 \quad \text{and} \quad \gamma_k = \sum_{\alpha \in R^+} k(\alpha).$$

It has been proved by S. Thangavelu and Y. Xu [10], that $\mathcal{R}_j$ is a multiplier operator given by

$$\mathcal{F}_k(\mathcal{R}_j(f))(\xi) = -i \frac{\xi_j}{\|\xi\|} \mathcal{F}_k(f)(\xi), \quad f \in S(\mathbb{R}^N), \xi \in \mathbb{R}^N, \quad (3.1)$$

The authors state that if $N = 1$ and $2\gamma_k \in \mathbb{N}$ the operator $\mathcal{R}_j$ is bounded on $L^p(m_k)$, $1 < p < \infty$. In [1] this result is improved by removing $2\gamma_k \in \mathbb{N}$, where Riesz transform is called Hilbert transform. If $\gamma_k = 0$ ($k = 0$), this operator coincides with the usual Riesz transform $\mathcal{R}_j$ given by (1.1). Our interest is to prove the boundedness of this operator for $N \ge 2$ and $k \ge 0$. To do this, we invoke the theory of singular integrals. Our basic is the following,

**Theorem 3.1.** Let $\mathcal{K}$ be a measurable function on $\mathbb{R}^N \times \mathbb{R}^N \setminus \{(x, g \cdot x); x \in \mathbb{R}^N, g \in G\}$ and $S$ be a bounded operator from $L^2(m_k)$ into itself, associated with a kernel $\mathcal{K}$ in the sense that

$$S(f)(x) = \int_{\mathbb{R}^N} \mathcal{K}(x, y)f(y)dm_k(y), \quad (3.2)$$

for all compactly supported function $f$ in $L^2(m_k)$ and for a.e $x \in \mathbb{R}^N$ satisfying $g \cdot x \notin \text{supp}(f)$, for all $g \in G$. If $\mathcal{K}$ satisfies

$$\int_{\min_{g \in G} \|g \cdot x - y\| > 2\|y - y_0\|} |\mathcal{K}(x, y) - \mathcal{K}(x, y_0)| dm_k(x) \le C, \quad y, y_0 \in \mathbb{R}^N, \quad (3.3)$$

then $S$ extends to a bounded operator from $L^p(m_k)$ into itself for all $1 < p \le 2$.

*Proof.* We first note that $(\mathbb{R}^N, m_k)$ is a space of homogenous type, that is, there is a fixed constant $C > 0$ such that

$$m_k(B(x, 2r)) \le Cm_k(B(x, r)), \quad \forall x \in \mathbb{R}^N, r > 0 \quad (3.4)$$

where $B(x, r)$ is the closed ball of radius $r$ centered at $x$ (see [8], Ch 1). Then we can adapt to our context the classical technic which consist to show that $S$ is weak type (1,1) and conclude by Marcinkiewicz interpolation theorem.
In fact, the Calderón-Zygmund decomposition says that for all $f \in L^1(m_k) \cap L^2(m_k)$ and $\lambda > 0$, there exist a decomposition of $f$, $f = h + b$ with $b = \sum_j b_j$ and a sequence of balls $(B(y_j, r_j))_j = (B_j)_j$ such that for some constant $C$, depending only on the multiplicity function $k$

(i) $\|h\|_{\infty} \le C\lambda$;

(ii) $\mathrm{supp}(b_j) \subset B_j$;

(iii) $\int_{B_j} b_j(x) dm_k(x) = 0$;

(iv) $\|b_j\|_{1,k} \le C \lambda m_k(B_j)$;

(v) $\sum_j m_k(B_j) \le C \frac{\|f\|_{1,k}}{\lambda}$.

The proof consists in showing the following inequality hold for $w = h$ and $w = b$:

$$ \rho_{\lambda}(S(w)) = m_k\left(\{x \in \mathbb{R}^N; |S(w)(x)| > \frac{\lambda}{2}\}\right) \le C \frac{\|f\|_{1,k}}{\lambda}. \quad (3.5) $$

By using the $L^2$-boundedness of $S$ we get

$$ \rho_{\lambda}(S(h)) \le \frac{4}{\lambda^2} \int_{\mathbb{R}^N} |S(h)(x)|^2 dm_k(x) \le \frac{C}{\lambda^2} \int_{\mathbb{R}^N} |h(x)|^2 dm_k(x). \quad (3.6) $$

From (i) and (v),

$$ \int_{\cup B_j} |h(x)|^2 dm_k(x) \le C \lambda^2 \mu_k(\cup B_j) \le C \lambda \|f\|_{1,k}. \quad (3.7) $$

Since on $(\cup B_j)^c$, $f(x) = h(x)$, then

$$ \int_{(\cup B_j)^c} |h(x)|^2 dm_k(x) \le C\lambda \|f\|_{1,k}. \quad (3.8) $$

From (3.6), (3.7) and (3.8), the inequality (3.5) is satisfied for $h$.

Next we turn to the inequality (3.5) for the function $b$. Consider

$$ B_j^* = B(y_j, 2r_j); \quad \text{and} \quad Q_j^* = \bigcup_{g \in G} g \cdot B_j^*. $$
Then

$$ \rho_{\lambda}(S(b)) \leq m_k\left(\bigcup_j Q_j^*\right) + m_k\left\{x \in \left(\bigcup_j Q_j^*\right)^c; |S(b)(x)| > \frac{\lambda}{2}\right\}. $$

Now by (3.4) and (v)

$$ m_k\left(\bigcup_j Q_j^*\right) \leq |G| \sum_j m_k(B_j^*) \leq C \sum_j m_k(B_j) \leq C \frac{\|f\|_{1,k}}{\lambda}. $$

Furthermore if $x \notin Q_j^*$, we have

$$ \min_{g \in G} \|g \cdot x - y_j\| > 2 \|y - y_j\|, \quad y \in B_j. $$

Thus, from (3.2), (iii), (ii), (3.3), (iv) and (v)

$$
\begin{align*}
& \int_{(\cup Q_j^*)^c} |S(b)(x)| dm_k(x) \\
& \leq \sum_j \int_{(Q_j^*)^c} |S(b_j)(x)| dm_k(x) \\
& = \sum_j \int_{(Q_j^*)^c} \left| \int_{\mathbb{R}^N} \mathcal{K}(x, y) b_j(y) dm_k(y) \right| dm_k(x) \\
& = \sum_j \int_{(Q_j^*)^c} \left| \int_{\mathbb{R}^N} b_j(y) (\mathcal{K}(x, y) - \mathcal{K}(x, y_j)) dm_k(y) \right| dm_k(x) \\
& \leq \sum_j \int_{\mathbb{R}^N} |b_j(y)| \int_{(Q_j^*)^c} |\mathcal{K}(x, y) - \mathcal{K}(x, y_j)| dm_k(x) dm_k(y) \\
& \leq \sum_j \int_{\mathbb{R}^N} |b_j(y)| \int_{\min_{g \in G} \|g \cdot x - y_j\| > 2 \|y - y_j\|} |\mathcal{K}(x, y) - \mathcal{K}(x, y_j)| dm_k(x) dm_k(y) \\
& \leq C \sum_j \|b_j\|_{1,k} \\
& \leq C \|f\|_{1,k}.
\end{align*}
$$

Therefore,

$$ m_k\{x \in \left(\bigcup_j Q_j^*\right)^c; |S(b)(x)| > \frac{\lambda}{2}\} \\ \leq \frac{2}{\lambda} \int_{(\cup Q_j^*)^c} |S(b)(x)| dm_k(x) \leq C \frac{\|f\|_{1,k}}{\lambda}. $$
This achieves the proof of (3.5) for b.

<figure><img src="image_2.png" /></figure>

Now, we will give an integral representation for the Riesz transform $\mathcal{R}_j$. For this end, we put for $x, y \in \mathbb{R}^N$ and $\eta \in \text{co}(G \cdot x)$

$$A(x, y, \eta) = \sqrt{\|x\|^2 + \|y\|^2 - 2 \langle y, \eta \rangle} = \sqrt{\|y - \eta\|^2 + \|x\|^2 - \|\eta\|^2}.$$

It is easy to check that

$$\min_{g \in G} \|g \cdot x - y\| \le A(x, y, \eta) \le \max_{g \in G} \|g \cdot x - y\|. \quad (3.9)$$

The following inequalities are clear

$$\begin{aligned} \left| \frac{\partial A^\ell}{\partial y_r}(x, y, \eta) \right| &\le C A^{\ell-1}(x, y, \eta), \\ \left| \frac{\partial^2 A^\ell}{\partial y_r \partial y_s}(x, y, \eta) \right| &\le C A^{\ell-2}(x, y, \eta) \end{aligned} \quad (3.10)$$

and for $\alpha \in R_+$,

$$\begin{aligned} \left| \frac{\partial A^\ell}{\partial y_r}(x, \sigma_\alpha \cdot y, \eta) \right| &\le C A^{\ell-1}(x, \sigma_\alpha \cdot y, \eta), \\ \left| \frac{\partial^2 A^\ell}{\partial y_r \partial y_s}(x, \sigma_\alpha \cdot y, \eta) \right| &\le C A^{\ell-2}(x, \sigma_\alpha \cdot y, \eta), \end{aligned} \quad (3.11)$$

where $r, s = 1, \dots, N$ and $\ell \in \mathbb{R}$.

Let us set

$$\begin{aligned} \mathcal{K}_j^{(1)}(x, y) &= \int_{\mathbb{R}^N} \frac{\eta_j - y_j}{A^{p_k}(x, y, \eta)} d\mu_x(\eta) \\ \mathcal{K}_j^{(\alpha)}(x, y) &= \frac{1}{\langle y, \alpha \rangle} \int_{\mathbb{R}^N} \left[ \frac{1}{A^{p_k-2}(x, y, \eta)} - \frac{1}{A^{p_k-2}(x, \sigma_\alpha \cdot y, \eta)} \right] d\mu_x(\eta), \\ \mathcal{K}_j(x, y) &= d_k \left\{ \mathcal{K}_j^{(1)}(x, y) + \sum_{\alpha \in R_+} \frac{k(\alpha)\alpha_j}{p_k-2} \mathcal{K}_j^{(\alpha)}(x, y) \right\}, \end{aligned}$$

where $\alpha \in R_+$.

**Proposition 3.2.** If $f \in L^2(m_k)$ with compact support, then for all $x \in \mathbb{R}^N$ satisfying $g \cdot x \notin \text{supp}(f)$ for all $g \in G$, we have

$$\mathcal{R}_j(f)(x) = \int_{\mathbb{R}^N} \mathcal{K}_j(x, y)f(y)dm_k(y).$$
RIESZ TRANSFORMS FOR DUNKL TRANSFORM

*Proof.* Let $f \in L^2(m_k)$ be a compact supported function and $x \in \mathbb{R}^N$, such that $g \cdot x \notin \text{supp}(f)$ for all $g \in G$. For $0 < \varepsilon < \min_{g \in G} \min_{y \in \text{supp}(f)} |g \cdot x - y|$ and $n \in \mathbb{N}$, we consider $\tilde{\varphi}_{n,\varepsilon}$ a $C^\infty$-function on $\mathbb{R}$, such that:

* $\tilde{\varphi}_{n,\varepsilon}$ is odd.

* $\tilde{\varphi}_{n,\varepsilon}$ is supported in $\{t \in \mathbb{R}; \varepsilon \le |t| \le n+1\}$.

* $\tilde{\varphi}_{n,\varepsilon} = 1$ in $\{t \in \mathbb{R}; \varepsilon + \frac{1}{n} \le t \le n\}$.

* $|\tilde{\varphi}_{n,\varepsilon}| \le 1$.

Let

$$ \tilde{\phi}_{n, \varepsilon}(t) = \int_{-\infty}^{t} \frac{\tilde{\varphi}_{n, \varepsilon}(u)}{|u|^{p_k-1}} du \quad \text{and} \quad \phi_{n, \varepsilon}(y) = \tilde{\phi}_{n, \varepsilon}(\|y\|), $$

for $t \in \mathbb{R}$ and $y \in \mathbb{R}^N$. Clearly, $\phi_{n,\varepsilon}$ is a $C^\infty$ radial function supported in the ball $B(0, n+1)$ and

$$ \lim_{n \to +\infty} \tilde{\varphi}_{n, \varepsilon}(\|y\|) = 1, \quad \forall y \in \mathbb{R}^N, \|y\| > \varepsilon. $$

The dominated convergence theorem, (2.5) and (2.6) yield

$$ \begin{align*} \int_{\|y\| > \varepsilon} \tau_x(f)(-y) \frac{y_j}{\|y\|^{p_k}} dm_k(y) & \\ &= \lim_{n \to \infty} \int_{\mathbb{R}^N} \tau_x(f)(-y) \frac{y_j}{\|y\|^{p_k}} \tilde{\varphi}_{n, \varepsilon}(\|y\|) dm_k(y) \\ &= \lim_{n \to \infty} \int_{\mathbb{R}^N} \tau_x(f)(-y) T_j(\phi_{n, \varepsilon})(y) dm_k(y) \\ &= \lim_{n \to \infty} \int_{\mathbb{R}^N} f(y) T_j \tau_x(\phi_{n, \varepsilon})(-y) dm_k(y). \end{align*} $$

Now we have

$$ \begin{align*} T_j \tau_x(\phi_{n,\varepsilon})(-y) & = \int_{\mathbb{R}^N} \frac{(\eta_j - y_j) \tilde{\varphi}_{n,\varepsilon}(A(x, y, \eta))}{A^{p_k}(x, y, \eta)} d\mu_x(\eta) \\ & + \sum_{\alpha \in R_+} k(\alpha) \alpha_j \int_{\mathbb{R}^N} \frac{\tilde{\phi}_{n,\varepsilon}(A(x, \sigma_\alpha \cdot y, \eta)) - \tilde{\phi}_{n,\varepsilon}(A(x, y, \eta))}{\langle y, \alpha \rangle} d\mu_x(\eta), \end{align*} $$
where from (3.9)

$$ \varepsilon < A(x, y, \eta); \quad \varepsilon < A(x, \sigma_{\alpha}.y, \eta), \quad y \in \text{supp}(f), \eta \in \text{co}(G.x). $$

Then with the aid of dominated convergence theorem

$$ \lim_{n \to \infty} T_j \tau_x(\phi_{n, \varepsilon})(-y) = \frac{1}{d_k} \mathcal{K}_j(x, y), $$

and

$$ d_k \int_{\|y\| \ge \varepsilon} \tau_x(f)(-y) \frac{y_j}{\|y\|^{p_k}} dm_k(y) = \int_{\mathbb{R}^N} \mathcal{K}_j(x, y) f(y) dm_k(y). $$

Letting $\varepsilon \to 0$, it follows that

$$ \mathcal{R}_j(f)(x) = \int_{\mathbb{R}^N} \mathcal{K}_j(x, y) f(y) dm_k(y), $$

which proves the result.

<figure><img src="image_3.png" /></figure>

Now, we are able to state our main result.

**Theorem 3.3.** The Riesz transform $\mathcal{R}_j$, $j = 1...N$, is a bounded operator from $L^p(m_k)$ into itself, for all $1 < p < \infty$.

*Proof.* Clearly, from (3.1) and Plancherel's theorem $\mathcal{R}_j$ is bounded from $L^2(m_k)$ into itself, with adjoint operator $\mathcal{R}_j^* = -\mathcal{R}_j$. Thus, via duality it's enough to consider the range $1 < p \le 2$ and apply Theorem 3.1. In view of Proposition 3.2 it only remains to show that $\mathcal{K}_j$ satisfies condition (3.3).

Let $y, y_0 \in \mathbb{R}^N$, $y \ne y_0$ and $x \in \mathbb{R}^N$, such that

$$ \min_{g \in G} \|g \cdot x - y\| > 2 \|y - y_0\|. \qquad (3.12) $$

By mean value theorem,
$$
\begin{align*}
|\mathcal{K}_j^{(1)}(x, y) - \mathcal{K}_j^{(1)}(x, y_0)|
  &= \left| \sum_{i=0}^{N} (y_i - (y_0)_i) \int_0^1 \frac{\partial \mathcal{K}_j^{(1)}}{\partial y_i}(x, y_t) \, dt \right| \\
  &= \left| \sum_{i=0}^{N} (y_i - (y_0)_i) \int_0^1 \int_{\mathbb{R}^N} \left( \frac{\delta_{i,j}}{A^{p_k}(x, y_t, \eta)} \right. \right. \\
  &\qquad \left. \left. + \frac{p_k((y_t)_i - \eta_i)(\eta_j - (y_t)_j)}{A^{p_k+2}(x, y_t, \eta)} \right) d\mu_x(\eta) \right| \\
  &\le C \|y - y_0\| \int_0^1 \int_{\mathbb{R}^N} \frac{1}{A^{p_k}(x, y_t, \eta)} d\mu_x(\eta) dt.
\end{align*}
$$

where $y_t = y_0 + t(y - y_0)$ and $\delta_{i,j}$ is the Kronecker symbol.
In view of (3.9) and (3.12), we obtain

$$
\|y - y_0\| < A(x, y_t, \eta), \quad \eta \in \co(G \times x).
$$

Therefore,

$$
\begin{align*}
\left| \mathcal{K}_j^{(1)}(x, y) - \mathcal{K}_j^{(1)}(x, y_0) \right|
  &\le C \|y - y_0\| \int_0^1 \int_{\mathbb{R}^N} \frac{1}{(\|y - y_0\|^2 + A^2(x, y_t, \eta))^{\frac{p_k}{2}}} d\mu_x(\eta) dt \\
  &\le C \|y - y_0\| \int_0^1 \tau_x(\psi)(y_t) dt
\end{align*}
$$

where $\psi$ is the function defined by

$$
\psi(z) = \frac{1}{(\|y - y_0\|^2 + \|z\|^2)^{\frac{p_k}{2}}}, \quad z \in \mathbb{R}^N.
$$

Using Fubini's theorem, (2.4) and (2.7), we get

$$
\begin{align*}
\int_{\min_{g \in G} \|g \cdot x - y\| > 2 \|y - y_0\|} |\mathcal{K}_j^{(1)}(x, y) - \mathcal{K}_j^{(1)}(x, y_0)| dm_k(x) \\
&\le C \|y - y_0\| \int_0^1 \int_{\mathbb{R}^N} \tau_{-y_t}(\psi)(x) dm_k(x) dt \\
&\le C |y - y_0| \int_{\mathbb{R}^N} \psi(z) dm_k(z) = C \int_{\mathbb{R}^N} \frac{du}{(1+u^2)^{\frac{p_k}{2}}} = C'.
\end{align*}
$$
This established the condition (3.3) for $\mathcal{K}_j^{(1)}$.

To deal with $\mathcal{K}_j^{(\alpha)}$, $\alpha \in R_+$, we put for $x, y \in \mathbb{R}^N$, $\eta \in \text{co}(G \cdot x)$ and $t \in [0, 1]$

$$
\begin{align*}
U(x, y, \eta) &= A^{2p_k-4}(x, y, \eta), \\
V_{\alpha}(x, y, \eta) &= A^{p_k-2}A_{\alpha}^{p_k-2}(A^{p_k-2} + A_{\alpha}^{p_k-2}), \\
h_{\alpha,t}(y) &= y + t(\sigma_{\alpha} \cdot y - y) = y - t <y, \alpha> \alpha,
\end{align*}
$$

By mean value theorem we have

$$
\begin{align*}
\mathcal{K}_j^{(\alpha)}(x, y) &= \int_{\mathbb{R}^N} \frac{1}{\langle y, \alpha \rangle} \frac{U(x, \sigma_\alpha \cdot y, \eta) - U(x, y, \eta)}{V_\alpha(x, y, \eta)} d\mu_x(\eta) \\
&= - \int_{\mathbb{R}^N} \int_0^1 \frac{\partial_\alpha U(x, h_{\alpha,t}(y), \eta)}{V_\alpha(x, y, \eta)} dt \, d\mu_x(\eta)
\end{align*}
$$

and

$$
\begin{align}
& \mathcal{K}_j^{(\alpha)}(x, y) - \mathcal{K}_j^{(\alpha)}(x, y_0) \nonumber \\
& \quad = \int_{\mathbb{R}^N} \int_0^1 \int_0^1 \partial_{y-y_0} \left( \frac{\partial_\alpha U(x, h_{\alpha,t}(\cdot), \eta)}{V_\alpha(x, \cdot, \eta)} \right) (y_\theta) \, d\theta \, dt \, d\mu_x(\eta). \tag{3.13}
\end{align}
$$

Here the derivations are taken with respect to the variable $y$.

To simplify, let us denote by

$$
A = A(x, y_{\theta}, \eta); \quad A_{\alpha} = A(x, \sigma_{\alpha} \cdot y_{\theta}, \eta)
$$

Then using (3.10) and the fact

$$
\|\eta - h_{\alpha,t}(y_{\theta})\| \leq \max(\|\eta - y_{\theta}\|, \|\eta - \sigma_{\alpha}(y_{\theta})\|),
$$

we obtain

$$
\left| \frac{\partial U}{\partial y_r}(x, h_{\alpha,t}(y_\theta), \eta) \right| \le C (A^{2p_k-5} + A_\alpha^{2p_k-5}) \\
\left| \frac{\partial^2 U}{\partial y_r \partial y_s}(x, h_{\alpha,t}(y_\theta), \eta) \right| \le C (A^{2p_k-6} + A_\alpha^{2p_k-6}), \quad r,s=1,\dots,N.
$$

This gives us the following estimates

$$
\left| \partial_{\alpha} U(x, h_{\alpha,t}(y_{\theta}), \eta) \right| \le C \left( A^{2p_k-5} + A_{\alpha}^{2p_k-5} \right), \quad (3.14)
$$
RIESZ TRANSFORMS FOR DUNKL TRANSFORM

$$
\left| \partial_{y-y_0} \left( \partial_{\alpha} U(x, h_{\alpha,t}(\cdot), \eta) \right) (y_{\theta}) \right| \le C \|y-y_0\| (A^{2p_k-6} + A_{\alpha}^{2p_k-6}). \quad (3.15)
$$

By (3.10) and (3.11), we also have

$$
\left| \frac{\partial V_{\alpha}}{\partial y_r}((x, y_{\theta}, \eta)) \right| \le C A^{p_k-3} A_{\alpha}^{p_k-3} (A^{p_k-2} + A_{\alpha}^{p_k-2}) (A + A_{\alpha}).
$$

The elementary inequality $\frac{u+v}{u^\ell + v^\ell} \le \frac{3}{u^{\ell-1} + v^{\ell-1}}$, $u, v > 0$, $\ell \ge 1$, leads to

$$
\begin{align}
\frac{\left| \partial_{y-y_0} V_{\alpha}(x, y_{\theta}, \eta) \right|}{V_{\alpha}^2(x, y_{\theta}, \eta)} & \nonumber \\
&\le C \|y-y_0\| \frac{A_{\alpha} + A}{A^{p_k-1} A_{\alpha}^{p_k-1} (A^{p_k-2} + A_{\alpha}^{p_k-2})} \nonumber \\
&\le C \|y-y_0\| \frac{1}{A^{p_k-1} A_{\alpha}^{p_k-1} (A^{p_k-3} + A_{\alpha}^{p_k-3})}. \tag{3.16}
\end{align}
$$

Now (3.14), (3.15) and (3.16) yield

$$
\begin{align*}
\left| \partial_{y-y_0} \left( \frac{\partial_{\alpha} U(x, h_{\alpha,t}(\cdot), \eta)}{V_{\alpha}(x, \cdot, \eta)} \right) (y_{\theta}) \right|
&\le C \|y-y_0\| \frac{A^{2p_k-6} + A_{\alpha}^{2p_k-6}}{A^{p_k-2} A_{\alpha}^{p_k-2} (A^{p_k-2} + A_{\alpha}^{p_k-2})} \\
&\quad + C \|y-y_0\| \frac{A^{2p_k-5} + A_{\alpha}^{2p_k-5}}{A^{p_k-1} A_{\alpha}^{p_k-1} (A^{p_k-3} + A_{\alpha}^{p_k-3})} \\
&\le C \|y-y_0\| \left( \frac{1}{A^2 A_{\alpha}^{p_k-2}} + \frac{1}{A^{p_k-2} A_{\alpha}^2} \right) \\
&\quad + C \|y-y_0\| \left( \frac{1}{A A_{\alpha}^{p_k-1}} + \frac{1}{A^{p_k-1} A_{\alpha}} \right) \\
&\le C \|y-y_0\| \left( \frac{1}{A^{p_k}} + \frac{1}{A_{\alpha}^{p_k}} \right)
\end{align*}
$$

where in the last equality we have used the fact that $\frac{1}{uv^{\ell-1}} \le \frac{1}{u^{\ell}} + \frac{1}{v^{\ell}}$, $u, v > 0$ and $\ell \ge 1$.
Thus, in view of (3.13),

$$ \begin{aligned} & |\mathcal{K}_j^{(\alpha)}(x, y) - \mathcal{K}_j^{(\alpha)}(x, y_0)| \\ & \le C \|y - y_0\| \int_0^1 \int_{\mathbb{R}^N} \left[ \frac{1}{A^{p_k}(x, y_\theta, \eta)} + \frac{1}{A^{p_k}(x, \sigma_\alpha y_\theta, \eta)} \right] d\mu_x(\eta) d\theta. \end{aligned} $$

Then by same argument as for $\mathcal{K}_j^{(1)}$ we obtain

$$ \int_{\min_{g \in G} \|g \cdot x - y\| > 2 \|y - y_0\|} |\mathcal{K}_j^{(2)}(x, y) - \mathcal{K}_j^{(2)}(x, y_0)| dm_k(x) \le C, $$

which established the condition (3.3) for the kernel $\mathcal{K}_j^{(\alpha)}$ and furnishes the proof. □

As applications, we will prove a generalized Riesz and Sobolev inequalities

**Corollary 3.4** (Generalized Riesz inequalities). For all $1 < p < \infty$ there exists a constant $C_p$ such that

$$ \|T_r T_s(f)\|_{k,p} \le C_p \|\Delta_k f\|_{k,p}, \quad \text{for all } f \in S(\mathbb{R}^N), \qquad (3.17) $$

where $\Delta_k$ is the Dunkl laplacian: $\Delta_k f = \sum_{r=1}^{N} T_r^2(f)$

*Proof.* From (2.1) and (3.1) one can see that

$$ T_r T_s(f) = \mathcal{R}_r \mathcal{R}_s(-\Delta_k)(f), \quad r, s = 1, \dots, N, \quad f \in S(\mathbb{R}^N). $$

Then (3.17) is concluded by Theorem 3.3. □

**Corollary 3.5** (Generalized Sobolev inequality). For all $1 < p \le q < 2\gamma(k) + N$ with $\frac{1}{q} = \frac{1}{p} - \frac{1}{2\gamma(k) + N}$, we have

$$ \|f\|_{q,k} \le C_{p,q} \|\nabla_k f\|_{p,k} \qquad (3.18) $$

for all $f \in S(\mathbb{R}^N)$. Here $\nabla_k f = (T_1 f, \dots, T_N f)$ and $|\nabla_k f| = (\sum_{r=1}^N |T_r f|^2)^{\frac{1}{2}}$.
RIESZ TRANSFORMS FOR DUNKL TRANSFORM

Proof. For all $f \in S(\mathbb{R}^N)$, we write

$$ 
\begin{aligned}
\mathcal{F}_k(f)(\xi) &= \frac{1}{\|\xi\|} \sum_{r=1}^{N} \frac{-i\xi_r}{\|\xi\|} (i\xi_r \mathcal{F}_k(f)(\xi)) \\
&= \frac{1}{\|\xi\|} \sum_{r=1}^{d} \frac{-i\xi_r}{\|\xi\|} (\mathcal{F}_k(T_r f)(\xi)).
\end{aligned}
$$

This yields to the following identity

$$ f = I_k^1 \left( \sum_{j=1}^{N} \mathcal{R}_j(T_j f) \right), $$

where

$$ I_k^\beta(f)(x) = (d_k^\beta)^{-1} \int_{\mathbb{R}^N} \frac{\tau_y f(x)}{\|y\|^{2\gamma(k)+N-\beta}} dm_k(y), $$

here

$$ d_k^\beta = 2^{-\gamma(k)-N/2+\beta} \frac{\Gamma(\frac{\beta}{2})}{\Gamma(\gamma(k) + \frac{N-\beta}{2})}. $$

Theorem 1.1 of [4] asserts that $I_k^\beta$ a bounded operator from $L^p(m_k)$ to $L^q(m_k)$. Then (3.18) follows from Theorem 3.3. $\square$

References

[1] B. AMRI, A. GASMI & M. SIFI – “Linear and bilinear multiplier operators for the dunkl transform”, *Mediterranean Journal of Mathematics* **7** (2010), p. 503–521.

[2] F. DAI & H. WANG – “A transference theorem for the dunkl transform and its applications”, *Journal of Functional Analysis* **258** (2010), p. 4052–4074.

[3] C. F. DUNKL – “Differential-difference operators associated to reflection groups”, *Trans. Amer. Math. 311* (1989), p. 167–183.

[4] S. HASSANI, S. MUSTAPHA & M. SIFI – “Riesz potentials and fractional maximal function for the dunkl transform”, *J. Lie Theory* **19** (2009, no. 4), p. 725–734.

[5] M. DE JEU – “The dunkl transform”, *Invent. Math.* **113** (1993), p. 147–162.
B. AMRI & M. SIFI

[6] M. ROSLER – “Dunkl operators: theory and applications, in orthogonal polynomials and special functions (leuven, 2002), $\mathbb{R}^n$”, *Lect. Notes Math. 1817* (2003), p. 93–135.

[7] ————, “A positive radial product formula for the dunkl kernel”, *Trans. Amer. Math. Soc. 355* (2003), p. 2413–2438.

[8] E. M. STEIN – *Harmonic analysis: Reals-variable methods, orthogonality and oscillatory integrals*, PrincetonS, New Jersey, 1993.

[9] S. THANGAVELU & Y. XU – “Convolution operator and maximal function for dunkl transform”, *J. Anal. Math. 97* (2005), p. 25–55.

[10] ————, “Riesz transforms and riesz potentials for the dunkl transform”, *J. Comp. and Appl. Math. 199* (2007), p. 181–195.

BECHIR AMRI

Department of Mathematics

University of Tunis

Preparatory Institute of Engineer Studies
of Tunis

1089 Montfleury, Tunis, Tunisia

bechir.amri@ipeit.rnu.tn

MOHAMED SIFI

Department of Mathematics

University of Tunis El Manar

Faculty of Sciences of Tunis

2092 Tunis El Manar, Tunis, Tunisia

mohamed.sifi@fst.run.tn