Riesz transform and commutators in the Dunkl setting

TL;DR

This paper characterizes the size and regularity of the Dunkl Riesz transform kernel using Euclidean and Dunkl metrics, establishing bounds and properties of commutators in these contexts.

Contribution

It provides the first comprehensive analysis of the Dunkl Riesz transform kernel's estimates and commutator bounds involving two non-equivalent metrics.

Findings

Optimal pointwise estimates for the Dunkl Riesz kernel.

Lower bounds for the kernel based on Euclidean metric.

Bounds for commutators in BMO and VMO spaces.

Abstract

In this paper we characterise the optimal pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where these two metrics are not equivalent. We further establish a suitable version of the pointwise kernel lower bound of the Dunkl Riesz transform via the Euclidean metric only. Then we show that the lower bound of commutator of the Dunkl Riesz transform is with respect to the BMO space associated with the Euclidean metric, and that the upper bound is respect to the BMO space associated with the Dunkl metric. Moreover, the compactness and the two types of VMO are also addressed.