A note on commutators of singular integrals with ${\rm BMO}$ and ${\rm VMO}$ functions in the Dunkl setting

TL;DR

This paper studies commutators of singular integral operators associated with Dunkl analysis, proving boundedness on L^p spaces for BMO functions and compactness for VMO functions, extending classical harmonic analysis results to the Dunkl setting.

Contribution

It extends classical commutator boundedness and compactness results to Dunkl operators with non-radial kernels on spaces of homogeneous type.

Findings

Commutators are bounded on L^p for BMO functions.

Commutators are compact on L^p for VMO functions.

Results extend classical harmonic analysis to Dunkl setting.

Abstract

On $\mathbb R^N$ equipped with a root system $R$, multiplicity function $k \geq 0$, and the associated measure $dw(\mathbf{x})=\prod_{\alpha \in R}|\langle \mathbf{x},\alpha\rangle|^{k(\alpha)}\,d\mathbf{x}$, we consider a (non-radial) kernel ${K}(\mathbf{x})$ which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator $\mathbf{T}f=f*K$ associated with ${K}$. Assuming that $b$ belongs to the ${\rm BMO}$ space on the space of homogeneous type $X=(\mathbb{R}^N,\|\cdot\|,dw)$, we prove that the commutator $[b,\mathbf{T}]f(\mathbf{x})=b(\mathbf{x})\mathbf{T}f(\mathbf{x})-\mathbf{T}(bf)(\mathbf{x})$ is a bounded operator on $L^p(dw)$ for all $1<p<\infty$. Moreover, $[b,\mathbf T]$ is compact on $L^p(dw)$, provided $b\in {\rm VMO} (X)$. The paper extents results of Han, Lee, Li and Wick.