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The fractional Riesz transform and their commutator in Dunkl setting

Yanping Chen, Xueting Han, Liangchuan Wu

TL;DR

The paper addresses the boundedness and compactness of fractional Dunkl Riesz transforms and their commutators. It derives kernel size and smoothness estimates, and establishes $(L^p,L^q)$-boundedness of $R_j^{\alpha}$ for $0<\alpha<N$, along with a full characterization of the commutator $[b,R_j^{\alpha}]$ in terms of Dunkl BMO spaces: $\mathrm{BMO}_d$ for boundedness and $\mathrm{CBMO}_{\text{Dunkl}}$ for sharp lower bounds, as well as a complete compactness theory via $\mathrm{VMO}_d$ and $\operatorname{CVMO}_{\text{Dunkl}}$. The work leverages heat-kernel representations, kernel estimates, and Dunkl metric geometry to extend Euclidean harmonic analysis tools to the Dunkl setting, providing precise criteria for when commutators are bounded or compact. These results deepen the understanding of fractional operators in Dunkl analysis and have potential implications for related PDEs and function-space theory in reflection-invariant contexts.

Abstract

In this paper, we study the boundedness of the fractional Riesz transforms in the Dunkl setting. Moreover, we establish the necessary and sufficient conditions for the boundedness of their commutator with respect to the central BMO space associated with Euclidean metric and the BMO space associated with Dunkl metric, respectively. Based on this, we further characterize the compactness of the commutator in terms of the corresponding types of VMO spaces.

The fractional Riesz transform and their commutator in Dunkl setting

TL;DR

The paper addresses the boundedness and compactness of fractional Dunkl Riesz transforms and their commutators. It derives kernel size and smoothness estimates, and establishes -boundedness of for , along with a full characterization of the commutator in terms of Dunkl BMO spaces: for boundedness and for sharp lower bounds, as well as a complete compactness theory via and . The work leverages heat-kernel representations, kernel estimates, and Dunkl metric geometry to extend Euclidean harmonic analysis tools to the Dunkl setting, providing precise criteria for when commutators are bounded or compact. These results deepen the understanding of fractional operators in Dunkl analysis and have potential implications for related PDEs and function-space theory in reflection-invariant contexts.

Abstract

In this paper, we study the boundedness of the fractional Riesz transforms in the Dunkl setting. Moreover, we establish the necessary and sufficient conditions for the boundedness of their commutator with respect to the central BMO space associated with Euclidean metric and the BMO space associated with Dunkl metric, respectively. Based on this, we further characterize the compactness of the commutator in terms of the corresponding types of VMO spaces.

Paper Structure

This paper contains 9 sections, 8 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['operatorbound']}
  4. Proof of Theorem \ref{['commutatorbound']}
  5. Upper bound of $[b,R_j^{\alpha}]$
  6. Lower bound of $[ b, R_j^\alpha]$
  7. Proof of Theorem \ref{['commutatorcompact']}
  8. Sufficiency
  9. Necessity

Key Result

Theorem 1.1

Given $0<\alpha<N$ and $1<p<\frac{\mathbf{N}}{\alpha}$. Let $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{\mathbf{N}}$. Then $R_j^{\alpha}$ is bounded from $L^p(\mathbb{R}^N, d \omega)$ to $L^q(\mathbb{R}^N, d \omega)$ with

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Remark 3.2
  • proof : Proof of Lemma \ref{['smoothness']}
  • proof : Proof of Theorem \ref{['operatorbound']}
  • Lemma 4.1
  • ...and 2 more