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 ${\mathcal{C}}^{♯}$ which is a Calderón-Zygmund operator and a remainder $\mathcal{R}$ which is no longer a Calderón-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 $bD$.