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 (non-radial) kernel $K(\mathbf{x})$ which has properties similar to those from the classical theory. We prove that a singular integral Dunkl convolution operator associated with the kernel $K$ is bounded on $L^p$ for $1<p<\infty$ and of weak-type (1,1). Further we study a maximal function related to the Dunkl convolutions with truncation of $K$.