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Riesz transform and commutators in the Dunkl setting

Yongsheng Han, Ming-Yi Lee, Ji Li, Brett D. Wick

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.

Riesz transform and commutators in the Dunkl setting

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.

Paper Structure

This paper contains 5 sections, 6 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['smooth r']} and \ref{['th pointwise lower']}
  3. Proof of Theorem \ref{['commutator']}: Upper bound of commutator
  4. Proof of Theorem \ref{['commutator']}: Lower bound of commutator
  5. Proof of Theorem \ref{['compact']} : The compactness of $[b,R_j]$

Key Result

Theorem 1.1

There exists a constant $C$ such that for $j=1,2,\ldots,N$ and for every $x,y$ with $d(x,y)\not=0$,

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: ADH
  • proof : Proof of Theorem \ref{['smooth r']}
  • proof : Proof of Theorem \ref{['th pointwise lower']}
  • proof : Proof of Theorem \ref{['commutator']}: upper bound of commutator
  • Definition 4.1
  • proof : Proof of Theorem \ref{['commutator']}: lower bound of commutator
  • ...and 1 more