Boundedness And Compactness Of Cauchy-Type Integral Commutator On Weighted Morrey Spaces

TL;DR

This paper characterizes when the commutator of Cauchy-type integrals is bounded or compact on weighted Morrey spaces over strongly pseudoconvex domains, linking these properties to BMO and VMO spaces.

Contribution

It provides the first characterization of boundedness and compactness of Cauchy-type integral commutators on weighted Morrey spaces in this setting.

Findings

Boundedness of commutator $[b, \, \mathcal C]$ iff $b$ in BMO.

Compactness of commutator $[b, \, \mathcal C]$ iff $b$ in VMO.

Extension of Calderón-Zygmund theory to non-Calderón-Zygmund operators.

Abstract

In this paper we study the boundedness and compactness characterizations of the commutator of Cauchy type integrals $\mathcal C$ on a bounded strongly pseudoconvex domain $D$ in $C^n$ with boundary $bD$ satisfying the minimum regularity condition $C^{2}$ based on the recent result of Lanzani-Stein and Duong-Lacey-Li-Wick-Wu. We point out that in this setting the Cauchy type integral $\mathcal C$ is the sum of the essential part $\mathcal{C}^\sharp$ which is a Calder\'on-Zygmund operator and a remainder $\mathcal R$ which is no longer a Calder\'on-Zygmund operator. We show that the commutator $[b, \mathcal C]$ is bounded on weighted Morrey space $L_{v}^{p,\kappa}(bD)$ ($v\in A_p, 1<p<\infty$) if and only if $b$ is in the BMO space on $bD$. Moreover, the commutator $[b, \mathcal C]$ is compact on weighted Morrey space $L_{v}^{p,\kappa}(bD)$ ($v\in A_p, 1<p<\infty$) if and only if $b$ is in the VMO space on $bD$.