Singular integrals in the rational Dunkl setting

TL;DR

This paper establishes boundedness properties of singular integral operators and maximal functions in the Dunkl setting, extending classical harmonic analysis results to a more general algebraic framework involving root systems and multiplicity functions.

Contribution

It proves $L^p$ boundedness and weak-type (1,1) estimates for Dunkl singular integral operators and analyzes associated maximal functions, advancing harmonic analysis in the Dunkl context.

Findings

Dunkl singular integral operators are bounded on $L^p$ for $1<p< \\infty$

The operators are of weak-type (1,1)

Maximal functions related to Dunkl convolutions are also studied

Abstract

On $\mathbb R^N$ equipped with a normalized root system $R$ and a multiplicity function $k\geq 0$ 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 $K$.