Singular integrals in the rational Dunkl setting

Abstract

On $$\mathbb {R}^N$$ equipped with a normalized root system R and a multiplicity function $$k\ge 0$$ let us consider a (not necessarily radial) kernel $$K({\mathbf {x}})$$ satisfying $$|\partial ^\beta K({\mathbf {x}})|\lesssim \Vert {\mathbf {x}}\Vert ^{-{\mathbf {N}}-|\beta |}$$ for $$|\beta |\le s$$ , where $${\mathbf {N}}$$ is the homogeneous dimension of the system $$({\mathbb {R}}^N,R,k)$$ . We additionally assume that $$\begin{aligned} \sup _{0<a<b<\infty }\Big |\int _{a<\Vert {\mathbf {x}}\Vert<b} K({\mathbf {x}})\, dw({\mathbf {x}})\Big |<\infty , \end{aligned}$$ where dw is the associated measure. We prove that if s large enough then a singular integral Dunkl convolution operator associated with the kernel $$K({\mathbf {x}})$$ is bounded on $$L^p(dw)$$ for $$1<p<\infty $$ and of weak-type (1,1). Furthermore, we study a maximal function related to the Dunkl convolutions with truncation of K.