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

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 $bD$.