Dunkl translations, Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$

TL;DR

This paper studies Dunkl translations and defines Dunkl-type BMO space and Riesz transforms, proving their boundedness on L-infinity and extending support descriptions of Dunkl translations for radial functions.

Contribution

It introduces Dunkl-type BMO space and Riesz transforms, establishing their boundedness and extending support analysis of Dunkl translations to all nonnegative radial functions.

Findings

Boundedness of Riesz transforms from L-infinity to Dunkl-type BMO space.

Extended support description of Dunkl translations to all nonnegative radial functions.

Analysis of Dunkl translations on compactly supported functions.

Abstract

In this paper, we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$, and prove the boundedness of Riesz transforms from $L^\infty$ to Dunkl-type $BMO$ space under the uniform boundedness assumption of Dunkl translations. The proof and the definition in Dunkl setting will be harder than in the classical case for the lack of some similar properties of Dunkl translations to that of classical translations. We will also extend the preciseness of the description of support of Dunkl translations on characteristic functions by Gallardo and Rejeb to that on all nonnegative radial functions in $L^2(m_k)$.

Equations

$$Tf(x)={\int }_{{\mathbb{R}}^N}K(x,y)f(y)dy,x\in {\mathbb{R}}^N\backslash supp(f),$$

$${\int }_{|x-y|>2|x-w|}\left|K(x,y)-K(w,y)\right|dy\le C,$$

$$R_j(f)(x)=\underset{\epsilon \to 0}{\lim }{\int }_{{\mathbb{R}}^N}(K_{\epsilon }(x,y)-K_1(0,y))f(y)dy.$$

$${\Vert {\tau }_{y}f\Vert }_{2,k}\le {\Vert f\Vert }_{2,k},f\in L^2(m_k).$$

$${\Vert {\tau }_{y}f\Vert }_{p,k}\le C{\Vert f\Vert }_{p,k},$$

$$supp{\tau }_{x}f(-\cdot )\subseteq \bigcup _{g\in G}B(gx,r).$$

$$supp{\tau }_{x}f(-\cdot )=\bigcup _{g\in G}B(gx,r).$$

$${\sigma }_{\alpha }(x)=x-2\frac{⟨x,\alpha ⟩}{{\Vert \alpha \Vert }^2}\alpha .$$

$$T_{\xi }f(x)$$

$$={\partial }_{\xi }f(x)+\sum _{\alpha \in R^{+}}k(\alpha )⟨\alpha ,\xi ⟩\frac{f(x)-f({\sigma }_{\alpha }(x))}{⟨\alpha ,x⟩},$$

$$h_k(x)=\prod _{\alpha \in R^{+}}|⟨\alpha ,x⟩{|}^{k(\alpha )}$$

$$m_k(B(x,2r))\le C{m}_k(B(x,r))$$

$$△f\left(x\right)={△}_{eucl}f\left(x\right)+2\sum _{\alpha \in R^{+}}k\left(\alpha \right)\left(\frac{⟨{\nabla }_{eucl}f,\alpha ⟩}{⟨\alpha ,x⟩}-\frac{f\left(x\right)-f\left({\sigma }_{\alpha }\left(x\right)\right)}{{⟨\alpha ,x⟩}^2}\right).$$

$$V_{k}f(x)={\int }_{{\mathbb{R}}^N}f(y)d{\mu }_x(y),$$

$$T_{\xi }\circ V_k=V_k\circ {\partial }_{\xi }.$$

$${\mu }_{rx}\left(B\right)={\mu }_x\left(r^{-1}B\right),{\mu }_{gx}\left(B\right)={\mu }_x\left(g^{-1}B\right).$$

$$E\left(x,y\right)=V_k\left(e^{⟨\cdot ,y⟩}\right)\left(x\right)={\int }_{{\mathbb{R}}^d}{e}^{⟨\eta ,y⟩}d{\mu }_x\left(\eta \right).$$

$$T_{\xi }f=⟨\xi ,y⟩f,f(0)=1.$$

$$F(f)(\xi )=\frac{1}{c_k}{\int }_{{\mathbb{R}}^N}f(x)E(-i\xi ,x)d{m}_k(x),c_k={\int }_{{\mathbb{R}}^N}{e}^{-\frac{{|x|}^2}{2}}d{m}_k(x).$$

$$e^{t△}f\left(x\right)=k_t\ast f={\int }_{{\mathbb{R}}^N}{h}_t(x,y)f(y)d{m}_k(y),$$

$$F({\tau }_x(f))(y)=E\left(ix,y\right)Ff(y),y\in {\mathbb{R}}^N.$$

$${\tau }_{x}f\left(y\right)={\left(V_k\right)}_y{\left(V_k\right)}_x\left[{\left(V_k\right)}^{-1}\left(f\right)\left(x+y\right)\right].$$

$$\begin{array}[]{l}1.\;(identity)\;\tau_{0}=I;\\ 2.\;(Symmetry)\;\tau_{x}f(y)=\tau_{y}f(x),\;x,\;y\in\mathbb{R}^{N},\;f\in S(\mathbb{R}^{N});\\ 3.\;(Scaling)\;\tau_{x}(f_{\lambda})=(\tau_{\lambda^{-1}x}f)_{\lambda},\;\lambda>0,\;x\in\mathbb{R}^{N},\;f\in S(\mathbb{R}^{N});\\ 4.\;(Commutativity)\;T_{\xi}(\tau_{x}f)=\tau_{x}(T_{\xi}f),\;x,\;\xi\in\mathbb{R}^{N};\\ 5.\;(Skew-symmetry)\\ \;\;\;\;\displaystyle{\int_{\mathbb{R}^{N}}\tau_{x}f(y)g(y)dm_{k}(y)}=\displaystyle{\int_{\mathbb{R}^{N}}f(y)\tau_{-x}g(y)dm_{k}(y)},\;x\in\mathbb{R}^{N},\;f,\;g\in S(\mathbb{R}^{N});\end{array}$$

$${\int }_{{\mathbb{R}}^N}{\tau }_{x}f(y)d{m}_k(y)={\int }_{{\mathbb{R}}^N}f(y)d{m}_k(y),x\in {\mathbb{R}}^N,f\in S({\mathbb{R}}^N).$$

$${\tau }_{x}f(-y)={\int }_{{\mathbb{R}}^N}(\text{mathaccent}869f\circ A)(x,y,\eta )d{\mu }_x(\eta ),x,y\in {\mathbb{R}}^N,$$

$$A(x,y,\eta )=\sqrt{{\Vert x\Vert }^2+{\Vert y\Vert }^2-2⟨y,\eta ⟩}=\sqrt{{\Vert x\Vert }^2-{\Vert \eta \Vert }^2+{\Vert y-\eta \Vert }^2},$$

$${\tau }_{-x}f(y)={\tau }_{y}f(-x)={\tau }_{x}f(-y),x,y\in {\mathbb{R}}^N,f\in S({\mathbb{R}}^N{)}_{rad}.$$

$$(f\ast g)(x)={\int }_{{\mathbb{R}}^N}f(y){\tau }_{x}g(-y)d{m}_k(y),$$

$$(f\ast g)(x)={\int }_{{\mathbb{R}}^N}(Ff)(\xi )(Fg)(\xi )E(ix,\xi )d{m}_k(\xi ).$$

$$supp{\tau }_{x}f(-\cdot )\supseteq \bigcup _{g\in G}B(gx,r).$$

$$0={\tau }_{x}f(-z)={\int }_{{\mathbb{R}}^N}\text{mathaccent}869f(\sqrt{{\Vert x\Vert }^2+{\Vert z\Vert }^2-2⟨z,\eta ⟩})d{\mu }_x(\eta ),$$

Full text

1. Introduction

Let $T$ be bounded operator on $L^{2}(\mathbb{R}^{N})$ and $K$ be a function on $\mathbb{R}^{N}\times\mathbb{R}^{N}\backslash$ $\left\{\left(x,x\right):\;x\in\mathbb{R}^{N}\right\}$, such that for any $f\in L^{2}(\mathbb{R}^{N})$ with a compact support,

[TABLE]

where $K$ satisfies

[TABLE]

then $T$ is a bounded operator from $L^{\infty}$ to $BMO$ space, or the space of bounded mean oscillation functions. Let $K(x,y)=c_{N}(x_{j}-y_{j})/|x-y|^{N+1},\;j=1,\>\dots,\;N$. For any $\varepsilon>0$ consider the truncation $K_{\varepsilon}$ defined by $K_{\varepsilon}(x,y)=K(x,y)$ if $|x-y|>\varepsilon$, and $K_{\varepsilon}(x,y)=0$ if $|x-y|\leq\varepsilon$. If $f$ is a bounded function, the ordinary Riesz transform is defined by

[TABLE]

It is well known the Riesz transform is bounded on $L^{2}(\mathbb{R}^{N})$ and that the kernel $K(x,y)$ satisfies (1.1), and so is a bounded operator from $L^{\infty}$ to $BMO$ space. In this paper we will extend analogous results to the context of Dunkl theory.

In [2], the $L^{p}$-boundedness, $1<p<\infty$ and weak $L^{1}$ boundedness of Riesz transforms for Dunkl transform was proved by adapting the classical $L^{p}$-theory of Calderón–Zygmund, and so the Riesz transforms can be defined as bounded operators on $L^{p},\;1<p<\infty$ and weakly bounded operators on $L^{1}$ (see also [8] for the $L^{p}$ boundedness of Dunkl–Riesz transforms with radial power weights). But there is no reasonable and coincident definition of Riesz transforms by integral on $L^{1}(m_{k})$ in Dunkl setting. Recently, the Riesz transforms were defined in a weak sense on $L^{1}(m_{k})$ (see [3]) using a test function space containing a Poisson kernel, and it was shown in [3] and [6] that in Dunkl setting, the real Hardy space $H_{\triangle}^{1}$ associated to the Dunkl Laplacian $\triangle$ can be characterized by Riesz transforms and also coincide with $H_{atom}^{1}$.

The formula that Dunkl translation operators $\tau_{x}$ are contractions on $L^{2}(m_{k})$ is well-known:

[TABLE]

Assume the uniform $L^{1}$-boundedness of the Dunkl translations (see [7]). Then by Riesz-Törin interpolation and skew-symmetry of Dunkl translations, the uniform $L^{p}$-boundedness ($1\leq p\leq\infty$) can be get immediately, that is, for any root system $R$ and multiplicity function $k\geq 0$ and for any $f\in L^{p}(m_{k})$,

[TABLE]

where $C$ is a constant independent of $y$. It has been known that this assumption holds for radial functions and one-dimensional case, and hence for $G=Z_{2}^{N}$ case. Here we call this assumption as the uniform boundedness assumption of Dunkl translations. There have been many results based on this long open assumption and it would be an excellent work if one could prove this assumption. In [11], the authors proved the uniform boundedness of the spherical average of Dunkl translations, and as applications, they found this uniform boundedness assumption can be avoided in the proof of some related results. But it is still inevitable for many other results, such as the $L^{p}$ boundedness of the Dunkl multiplier operator for $s>\mathbf{N}/2$ (see [7, Theorem 8.1]), and the Theorem 1.1 in this paper as well.

In this paper we will define Dunkl–type $BMO$ space and $\mathcal{R}_{j}f$, where $\mathcal{R}_{j}$ is the Riesz transform for Dunkl transforms, as Dunkl–type $BMO$ functions for all $f\in L^{\infty}$. Then under the uniform boundedness assumption of Dunkl translations, we will prove the boundedness of the Riesz transforms from $L^{\infty}$ to Dunkl–type $BMO$ space. This will also mean a half part of duality of the Hardy space $H_{\triangle}^{1}$ and Dunkl–type $BMO$ space.

Theorem 1.1**.**

Under the uniform boundedness assumption of Dunkl translations, the Riesz transform for Dunkl transforms are bounded operators from $L^{\infty}$ to the Dunkl–type $BMO$ space.

The part ii of the following theorem shows that the support of $\tau_{-x}f$ obtained in [7, Theorem 1.7] (part i of the following Theorem) is precise when the multiplicity function $k>0$. The preciseness has been proved for characteristic functions by Gallardo and Rejeb [9] and we extend the result to any nonnegative radial functions on $L^{2}(m_{k})$ in this paper.

Theorem 1.2**.**

*If $f\in L^{2}(m_{k})$ and $suppf\subseteq B(0,r)$, then for any $x\in\mathbb{R}^{N}$

i).(See [7, Theorem 1.7])*

[TABLE]

ii). If the multiplicity function $k>0$ and let $f$ be a nonnegative radial function on $L^{2}(m_{k})$, $suppf=B(0,r)$, then

[TABLE]

The part i of this theorem also means that the Dunkl translation of a function on $L^{2}(m_{k})$ with a compact support is compactly supported. This will be used in the proof of Theorem 1.1. However, different from classical analysis, $suppf\subseteq\bigcup_{g\in G}B(gx,r)$, $x\in\mathbb{R}^{N}$, can not usually imply $supp\tau_{x}f\subseteq B(0,r)$ as will be shown in Section 2. This led to the differences between the proof of the boundedness of Riesz transforms from $L^{\infty}$ to the $BMO$ space in Dunkl setting and that in classical case.

This paper is organized as follows. In Section 2 we present some definitions and fundamental results from Dunkl’s analysis. In Section 3, we will prove Theorem 1.1. ii) and give more information about the support of Dunkl translations on compactly supported functions based on the results of [7]. Section 4 is devoted to Riesz transforms for Dunkl transform. In Section 5, the Dunkl–type $BMO$ space and Riesz transforms for Dunkl transform on $L^{\infty}$ will be defined and we will prove the boundedness of the Riesz transforms from $L^{\infty}$ to Dunkl–type $BMO$ space. We first prove for compactly supported functions and then for all functions on $L^{\infty}$ using a Lemma we will give in the section.

2. Preliminaries

For any $x,y$ in the Euclidean space $\mathbb{R}^{N}$, denote by $\left\langle x,y\right\rangle=\sum\nolimits_{j=1}^{N}x_{j}y_{j}$ the standard inner product associated with norm $\left\|x\right\|$. For any nonzero vector $\alpha\in\mathbb{R}^{N}$, define the reflection $\sigma_{\alpha}$ with respect to the hyperplane $\alpha^{\perp}$ orthogonal to $\alpha$,

[TABLE]

A finite set $R\subset\mathbb{R}^{N}\backslash\left\{0\right\}$ is called a $root$ $system$ if $\sigma_{\alpha}(R)=R$ for any $\alpha\in R$. Given a root system $R$, the finite subgroup $G$ of $O(N)$ generated by the reflections $\sigma_{\alpha}$ is called the $finite$ $relection$ $group$ of the root system. Define a $multiplicity$ $function$ $k:R\rightarrow\mathbb{C}$ such that $k$ is $G$-invariant, that is, $k\left(\alpha\right)=k\left(\beta\right)$ if $\sigma_{\alpha}$ and $\sigma_{\beta}$ are conjugate. We assume $k\geqslant 0$ in this paper. The $Dunkl\;operators\;T_{\xi}$, $\xi\in\mathbb{R}^{N}$, which were introduced in [5], are defined by the following deformations by difference operators of directional derivatives $\partial_{\xi}$:

[TABLE]

where $R^{+}$ is any fixed positive subsystem of $R$. They commute pairwise and are skew-symmetric with respect to the $G$-invariant measure $dm_{k}(x)=h_{k}^{2}(x)dx$, where

[TABLE]

and $m_{k}$ is a doubling measure, that is, there is a constant $C>0$ such that

[TABLE]

for $x\in\mathbb{R}^{N},\;r>0$, where $B(x,r)=\{y\in\mathbb{R}^{N}:$ $\left\|x-y\right\|\leq r\}.$ Denote by $\mathbf{N}\boldsymbol{=}N+\sum_{\alpha\in R}k(\alpha)$ the homogeneous dimension of the root system. Let $e_{j},\;j=1,2,...,N,$ be the canonical orthonormal basis in $\mathbb{R}^{N}$ and denote $T_{j}=T_{e_{j}}.$ The $Dunkl$ $Laplacian$ is defined by $\triangle={\textstyle\sum_{j=1}^{N}}T_{j}^{2}$. It commutes with the action of $G$, that is, $g\circ\triangle=\triangle\circ g$ for any $g\in G$, and has the following explicit expression,

[TABLE]

The operator $-\triangle$ is essentially self-adjoint and positive definite and so $\triangle$ is the generator of the contraction semigroup ${\left\{e^{t\triangle}\right\}}_{t\geq 0}$.

The operators $\partial_{\xi}$ and $T_{\xi}$ are intertwined by a Laplace–type operator

[TABLE]

associated to a family of probability measures $\left\{\mu_{x}|\;x\in\mathbb{R}^{N}\right\}$ with compact support, that is,

[TABLE]

Specifically, the support of $\mu_{x}$ is contained in the convex hull $co(G\cdot x)$, where $G\cdot x=\left\{g\cdot x|\;g\in G\right\}$ is the orbit of $x$. For any Borel set $B$ and any $r>0$, $g\in G$, the probability measures satisfy

[TABLE]

The Dunkl kernel $E(x,y)$ is defined by

[TABLE]

It is the generalization of exponential function $e^{\left\langle x,y\right\rangle}$. For any fixed $y\in\mathbb{R}^{N}$, the Dunkl kernel $E(x,y)$ is the unique analytic solution to the differential equation system

[TABLE]

For $f\in L^{1}(m_{k})$ the Dunkl transform is defined by

[TABLE]

Obviously, $F\left(\triangle f\right)\left(\xi\right)=-\left\|\xi\right\|^{2}Ff\left(\xi\right)$ and $F(e^{t\triangle}f)=e^{t\left|\cdot\right|^{2}}F\left(f\right),\;f\in L^{2}\left(m_{k}\right).$ It follows that

[TABLE]

where $k_{t}(x)=c_{k}^{-1}\left(2t\right)^{-\mathbf{N}/2}e^{-\left|x\right|^{2}/\left(4t\right)}$ and the heat kernel $h_{t}(x,y)=\tau_{x}k_{t}(-y).$

Let $x\in\mathbb{R}^{N}$, the Dunkl translation operator $\tau_{x}$ is defined on $L^{1}(m_{k})$ by,

[TABLE]

It can also be defined by

[TABLE]

Here are some basic properties of Dunkl translatons.

[TABLE]

The Dunkl translations can be defined on $L^{p}(m_{k}),\;1\leq p\leq\infty$ in the distributional sense due to the latter formula. Further,

[TABLE]

The following formula for radial functions was first proved by Rösler [13] for Schwartz functions, and was then extended to all continuous radial functions in [4]:

[TABLE]

where $f(x)=\mathaccent 869{f}(\left\|x\right\|)$ and

[TABLE]

It follows from the symmetry of Dunkl translations that (see [10])

[TABLE]

The Dunkl convolution of Schwartz functions is defined by

[TABLE]

or can be written as

[TABLE]

The following are some basic properties of Dunkl convolution,

$1.\;F(f\ast g)=Ff\cdot Fg;$

$2.\;F(f\cdot g)=Ff\ast Fg;$

$3.\;f\ast g=g\ast f;$

$4.\;(f\ast g)\ast h=f\ast(g\ast h);$

$5.\;{\left\|f\ast g\right\|}_{2,\;k}\leq{\left\|f\right\|}_{1,\;k}{\left\|g\right\|}_{2,\;k},\;f\in L^{1}(m_{k}),\;g\in L^{2}(m_{k}).$

3. Some results on the supports of Dunkl translations

Proof of Theorem 1.2. ii). It suffices to prove that

[TABLE]

Firstly, we will prove for continuous nonnegative radial functions. Suppose there exists a $y\in\bigcup\limits_{g\in G}B(gx,\;r)$, that is, there exists a $g\in G$, $\left\|y-g\cdot x\right\|\leq r$, such that $y\not\in supp\tau_{x}f\left(-\cdot\right)$, that is, there exists $\varepsilon>0$, for any $z\in B(y,\varepsilon)$,

[TABLE]

then

[TABLE]

By a result of Gallardo and Rejeb (see [9]), that the orbit of $x$, $G\cdot x$, is contained in the support of $\mu_{x}$ if $k>0$, for the above $g$ we can select $\eta=g\cdot x$, then $f(z-g\cdot x)=\mathaccent 869{f}(\left\|z-g\cdot x\right\|)=0$. For any $z_{1}\in B(y-g\cdot x,\varepsilon)$, $z_{1}+g\cdot x\in B(y,\varepsilon)$, and so $f\left(z_{1}\right)=f(z_{1}+g\cdot x-g\cdot x)=0$, which means $y-g\cdot x\not\in suppf$, and this leads to a contradiction to that $suppf=B(0,r)$.

Then for any nonnegative radial functions $f$ on $L^{2}(m_{k})$, $suppf=B(0,r)$, by the density of continuous functions with compact support $B(0,r)$ in $L^{2}(B(0,r),m_{k})$, there exists a sequence of continuous nonnegative radial functions $g_{n}$ whose support is $B(0,r)$, such that $f/2$ can be approximated by $g_{n}$ with respect to $L^{2}$-norm. So for any nonnegative smooth function $\varphi$ on $\mathbb{R}^{N}$ with compact support, $\int g_{n}\varphi\rightarrow\int\frac{f}{2}\varphi$. If $(supp\varphi)^{\circ}\cap B(0,r)\neq\varnothing$, then $\int f\varphi>0$, where $A^{\circ}$ stands for the interior of $A$ for any $A\subseteq\mathbb{R}^{N}$. So there exists a sufficiently large natural number $L$ such that $\int g_{L}\varphi<\int f\varphi$. If $(supp\varphi)^{\circ}\cap B(0,r)=\varnothing$, then for any $n\in\mathbb{N}$, $\int g_{n}\varphi=\int f\varphi=0$. So for any nonnegative smooth function $\varphi$ on $\mathbb{R}^{N}$ with compact support, $\int g_{L}\varphi\leq\int f\varphi$. Thus $g_{L}\leq f\;a.e.$ and $\int\tau_{-x}g_{L}\cdot\varphi\leq\int\tau_{-x}f\cdot\varphi$ by positivity of Dunkl translations on radial functions. Let $D=(supp\tau_{-x}f)^{c}$, then $D$ is the largest open set such that $0=\int\tau_{-x}f\cdot\varphi$ for any smooth functions functions $\varphi$ with compact support in $D$. If $\varphi\geq 0$, then $\int\tau_{-x}g_{L}\cdot\varphi=0$. Then by $\tau_{-x}g_{L}\geq 0$,

[TABLE]

Remark 3.1*.*

This theorem does not hold for $k\geqslant 0$. For example, for any nontrival finite reflection group $G$, we can take $k=0$. Then $supp\tau_{x}f(-\cdot)=B(x,r)$ when $suppf=B(0,r)$ and is obviously not $\bigcup_{g\in G}B(gx,\;r)$ since $G$ is nontrival. We refer to [9, Example 3.1] for more counterexamples.

Corollary 3.2**.**

Let $f\in L^{2}(m_{k})$ and $x\in\mathbb{R}^{N}$, $suppf\cap\bigcup\limits_{g\in G}B(gx,r)=\varnothing$, then $supp\tau_{x}f\cap B(0,r)=\varnothing.$

Proof.

For any function $g\in L^{2}(m_{k})$, $suppg\subseteq B(0,r)$, from Theorem 1.2. i),

[TABLE]

By the skew-symmetry of Dunkl translations,

[TABLE]

Then $\tau_{x}f(y)=0$, $y\in B(0,r).$ ∎

Define the distance of the orbits $G\cdot x$ and $G\cdot y$ (see [6]),

[TABLE]

For any fixed point $x$ and a ball $B(x,r)$ with center $x$, let $B^{\ast}=B(x,2r)$ and $Q^{\ast}=\bigcup_{g\in G}gB^{\ast}$. For any $y\in B(x,r)$, if $z\in\mathbb{R}^{N}\backslash Q^{\ast}$, then (see [2])

[TABLE]

Theorem 3.3**.**

Let $f\in L^{p}(m_{k}),\;1\leq p<\infty$ be a radial function, $suppf\cap B(0,r)=\varnothing,$ then for any $x\in\mathbb{R}^{N}$,

[TABLE]

Proof.

Let us prove for continuous radial functions first. It is easy to see that

[TABLE]

for any $x,\;y\in\mathbb{R}^{N}$ and $\eta\in co(G\cdot x)$. For any continuous radial functions $f$ with support contained in $B(0,r)^{c}$, if

[TABLE]

then $\underset{g\in G}{max}\left\|g\cdot x-y\right\|\geq r$. Therefore, $supp\tau_{x}f(-\cdot)\cap\bigcap\limits_{g\in G}B(gx,\;r)=\varnothing$. By the density of continuous functions on $L^{p}(m_{k})$ and the continuity of Dunkl translations on $L^{p}(m_{k})$ for radial functions, (3.2) can be extended to any radial functions in $L^{p}(m_{k})$. ∎

Remark 3.4*.*

One may expect that $supp\tau_{x}f(-\cdot)\cap\bigcup_{g\in G}B(gx,r)=\varnothing$, but this is not correct even for a characteristic function for a general finite reflection group $G$. From a similar argument as in Corollay 3.2, this also means that $suppf\subseteq\bigcup_{g\in G}B(gx,r)$ can not usually imply $supp\tau_{x}f\subseteq B(0,r)$ as in classical case.

As an immediate consequence of the theorem, the condition of the Corollary 4.1 in [7] can be weakened for radial functions.

Corollary 3.5**.**

Suppose for all $g\in G$ and $x,\;y\in\mathbb{R}^{N}$, $\left\|g\cdot x-y\right\|<1$. Let $f$ be a radial function in $L^{p}(m_{k}),\;1\leq p<\infty$, $f(z)=0$ for all $z\in B(0,1)$, then $\tau_{x}f(y)=0$.

Remark 3.6*.*

This theorem cannot be extended to functions not necessarily radial because the Stone-Weierstrass theorem does not hold on $B(0,r)^{c}$, and there is no more precise result on the support of the distribution associated to $\tau_{x}f(y)$ other than [1].

4. Riesz transforms for Dunkl transform

The Riesz transforms ${\mathcal{R}}_{j}\;$in the Dunkl setting are defined by

[TABLE]

where $j=1,\cdots,N$ and $d_{k}=2^{\mathbf{N}/2}\Gamma((\mathbf{N}+1)/2)/\sqrt{\pi},\;$ $p_{k}=\mathbf{N}+1$. It has been proved in [14] that

[TABLE]

and $\mathcal{R}_{j}$ is a bounded operator on $L^{2}(m_{k})$. Clearly,

[TABLE]

and the integral converges for $f\in L^{2}(m_{k})$. It is obvious that the Riesz transforms commute with the Dunkl translations. If $f\in L^{2}(m_{k})$ and has a compact support, it was shown in [2] that for all $x\in\mathbb{R}^{N}$ such that $g\cdot x\in\mathbb{R}^{N}\backslash supp(f)$ for any $g\in G$,

[TABLE]

where

$\mathcal{K}_{j}(x,y)=d_{k}\left\{\mathcal{K}_{j}^{(1)}(x,y)+\displaystyle\sum\limits_{\alpha\in R^{+}}\frac{k\left(\alpha\right)\alpha_{j}}{p_{k}-2}\mathcal{K}_{j}^{(\alpha)}(x,y)\right\},$

$\mathcal{K}_{j}^{(1)}(x,y)=\displaystyle\int_{\mathbb{R}^{N}}\frac{\eta_{j}-y_{j}}{A^{p_{k}}(x,y,\eta)}d\mu_{x}(\eta),$

$\mathcal{K}_{j}^{(\alpha)}(x,y)=\displaystyle\frac{1}{\left\langle y,\alpha\right\rangle}\int_{\mathbb{R}^{N}}\left[\frac{1}{A^{p_{k}-2}\left(x,y,\eta\right)}-\frac{1}{A^{p_{k}-2}\left(x,\sigma_{\alpha}\left(y\right),\eta\right)}\right]d\mu_{x}(\eta),\;\alpha\in R^{+}$,

and $\mathcal{K}_{j}(x,y)$ satisfies the condition

[TABLE]

And the authors proved that $\mathcal{R}_{j}$ is a bounded operator on $L^{p}(m_{k})$, $1<p<\infty$ in [2] using this Calderón–Zygmund condition in Dunkl setting.

Consider $\mathaccent 869{\varphi}_{n,\varepsilon}$ a $C^{\infty}-$function on $\mathbb{R}$, such that:

• •

$\mathaccent 869{\varphi}_{n,\varepsilon}$ is odd .

• •

$\mathaccent 869{\varphi}_{n,\varepsilon}$ is supported in $\left\{t\in\mathbb{R};\;|t|\geq\varepsilon\right\}$.

• •

$\mathaccent 869{\varphi}_{n,\varepsilon}=1$ in $\left\{t\in\mathbb{R};\;t\geq\varepsilon+\frac{1}{n}\right\}$.

• •

$\left|\mathaccent 869{\varphi}_{n,\varepsilon}\right|\leq 1$.

Let

[TABLE]

Clearly, $\phi_{n,\varepsilon}$ is a $C^{\infty}$ radial function and

[TABLE]

Denote by $\mathcal{K}_{j}^{(\varepsilon,n)}(x,y)=d_{k}T_{j}\tau_{x}(\phi_{n,\varepsilon})(-y)$. Here the action of $\tau_{x}$ on $\phi_{n,\varepsilon}$ is defined in the sense of distribution. Then from the proof of [2, Proposition 3.2],

[TABLE]

where

[TABLE]

and $\int_{\mathbb{R}^{N}}\mathcal{K}_{j}^{(\varepsilon,n)}(x,y)f(y)dm_{k}(y)$ could be an integrable singular integral for $f\in L^{2}(m_{k})$ because $suppT_{j}\phi_{n,\varepsilon}\subseteq(B{(0,\varepsilon)})^{c}$ does not necessarily imply $suppT_{j}\tau_{x}(\phi_{n,\varepsilon})(-\cdot)$ $\subseteq(\bigcup_{g\in G}B(gx,\varepsilon))^{c}$ as is shown in Section 2.

For any $f\in L^{2}(m_{k})$ with compact support, if $\mathcal{R}_{j}^{\ast}$ is the adjoint operator of $\mathcal{R}_{j}$, then

[TABLE]

By $\mathcal{R}_{j}=-\mathcal{R}_{j}^{\ast}$,

[TABLE]

If $y\in\mathbb{R}^{N}$ satisfies $Gy\cap suppf=\varnothing$, then $d_{G}(x,y)>0$ for all $x\in suppf$. For any $0<\varepsilon<d_{G}(x,y)$, from (3.4),

[TABLE]

Then from the same argument as in the proof of [2, Proposition 3.2],

[TABLE]

and

[TABLE]

with the aid of dominated convergence theorem.

5. The Dunkl–type $BMO$ Space and Proof of Theorem 1.1

The study of Dunkl–type $BMO$ space dates back to [12], where the space was defined for the one dimensional case. Here we will define the Dunkl–type $BMO$ space for multidimensional cases.

Given a function $f\in L_{loc}^{1}(m_{k})$, and a ball $B(x,r)$. Denote $B_{r}\equiv B(0,r)$. Let $f_{B_{r}}(x)$ be the average of $\tau_{x}f$ on $B_{r}$:

[TABLE]

Definition 5.1**.**

The Dunkl–type $BMO$ space is the space of all those functions in $L_{loc}^{1}(m_{k})$ satisfying \begin{array}[]{l}{\left\|f\right\|}_{\ast,\;k}<\infty\end{array}, where

[TABLE]

We can consider $BMO$ as the quotient of the above space by the space of constant functions to let ${\left\|\cdot\right\|}_{\ast,k}$ be a norm.

Proof of Theorem 1.1.

Given a function $f$ in $L^{\infty}$ compactly supported, thanks to Theorem 1.1. i), $\tau_{x}f$ is compactly supported. Write $\tau_{x}f=g_{1}+g_{2}$, where $g_{1}=(\tau_{x}f)\chi_{B_{2r}}$, and $g_{2}=(\tau_{x}f)\chi_{(B_{2r})^{c}}$. This is the only way of decomposition because as is shown in Section 2, $suppf\subseteq\bigcup_{g\in G}B(gx,r)$ can not usually imply $supp\tau_{x}f\subseteq B(0,r)$. For any $y\in B_{r}$, $Gy\cap suppg_{2}=\varnothing$. Then by (3.2), (4.1), (4.2) and the uniform boundedness assumption of Dunkl translations,

[TABLE]

Then using again the uniform boundedness assumption of Dunkl translations, and by the $L^{2}$ boundedness of the Riesz transform and (2.1),

[TABLE]

Therefore,

[TABLE]

We will then extend the definition of Riesz transforms for Dunkl transform to all of $L^{\infty}$. For any function $f\in L^{\infty}$, define

[TABLE]

For any $\varepsilon>0$ and any $y\in\mathbb{R}^{N}$, there exists a sufficiently large $r>2\varepsilon$ such that $y\in B_{r}$. Write $f=f_{1}+f_{2}$, where $f_{1}=f\chi_{B_{2r}}$, and $f_{2}=f\chi_{(B_{2r})^{c}}$. Then $f_{1}$ belongs to $L^{2}(m_{k})$ and so $\mathcal{R}_{j}f_{1}(y)$ converges almost everywhere. For any natural number $n$ larger than $\frac{1}{\varepsilon}$ and any $z\in suppf_{2}$,

[TABLE]

So for the above $\varepsilon$ and $n$,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

This integral converges since

[TABLE]

So the above definition (5.1) for Riesz transforms for Dunkl transform on $L^{\infty}$ makes sense for any $y\in\mathbb{R}^{N}$and coincides with formula (4.2) for compactly supported functions on $L^{2}(m_{k})$ as Dunkl–type $BMO$ functions since the two formulae differ by a constant.

Under the uniform boundedness assumption of Dunkl translations, for any $x\in\mathbb{R}^{n}$, $r>0$ and all $f\in L^{\infty}$, $\tau_{x}f\in L^{\infty}$. Write $\tau_{x}f=g_{1}+g_{2}$, where $g_{1}=(\tau_{x}f)\chi_{B_{2r}}$, and $g_{2}=(\tau_{x}f)\chi_{(B_{2r})^{c}}$. Then

[TABLE]

and for any $y\in B_{r}$,

[TABLE]

By the same argument as for compactly supported functions, we have

[TABLE]

The following Lemma will then imply the boundedness of Riesz transforms for Dunkl transform from $L^{\infty}$ to Dunkl–type $BMO$ space.

Lemma 5.2**.**

Under the uniform boundedness assumption of Dunkl translations, for any $f\in L^{\infty}$ and any fixed $x\in\mathbb{R}^{N}$, $r>0$, $\mathcal{R}_{j}\tau_{x}f(y)$ and $\tau_{x}\mathcal{R}_{j}f(y)$ differ by a constant independent of $y$ for $y\in B_{r}$.

Proof.

This statement is obvious for functions compactly supported on $L^{\infty}$, implying that for any function $f\in L^{\infty}$ compactly supported,

[TABLE]

where $\left[X,Y\right]:=XY-YX.$

For all $f\in L^{\infty}$, if $\left\{D_{i}\right\}$ is a countable open cover of $\mathbb{R}^{N}$, where each $D_{i}$ is bounded, then by partition of unity, $f$ can be written as $\textstyle\overset{\infty}{\underset{i=1}{\sum f_{i}}}$, where $suppf_{i}\subset D_{i}$. Denote $g:=\left[\mathcal{R}_{j},\tau_{x}\right]f$. Then

[TABLE]

and by (5.1),

[TABLE]

And so $g(y)=\left[\mathcal{R}_{j},\tau_{x}\right]f(y)$ is a constant independent of $y$ on $y\in B_{r}$. ∎

Denote by $C_{x}$ the constant $\tau_{x}\mathcal{R}_{j}f(y)$ and $\mathcal{R}_{j}\tau_{x}f(y)$ differ plus ${\mathcal{R}}_{j}(g_{1})(0)$. Then

[TABLE]

$\hfill\Box$

Acknowledgments

The author would like to thank Margit Rösler very much for her correction of Theorem 1.2 ii) and some valuable comments, and thank the reviewer and his former adviser Heping Wang for valuable suggestions. The paper is based on the master thesis of the author at Capital Normal University