Commutators of Cauchy--Szeg\H{o} type integrals for domains in $\mathbb C^n$ with minimal smoothness

Abstract

In this paper we study the commutator of Cauchy type integrals $\EuScript C$ on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with boundary $bD$ satisfying the minimum regularity condition $C^{2}$ as in the recent result of Lanzani--Stein. We point out that in this setting the Cauchy type integrals $\EuScript C$ is the sum of the essential part $\EuScript C^\sharp$ which is a Calder\'on--Zygmund operator and a remainder $\EuScript R$ which is no longer a Calder\'on--Zygmund operator. We show that the commutator $[b, \EuScript C]$ is bounded on $L^p(bD)$ ($1<p<\infty$) if {\color{black}and only if}\ $b$ is in the BMO space on $bD$. Moreover, the commutator $[b, \EuScript C]$ is compact on $L^p(bD)$ ($1<p<\infty$) if {\color{black}and only if}\ $b$ is in the VMO space on $bD$. Our method can also be applied to the commutator of Cauchy--Leray integral in a bounded, strongly $\mathbb C$-linearly convex domain $D$ in $\mathbb C^n$ with the boundary $bD$ satisfying the minimum regularity $C^{1,1}$. Such a Cauchy--Leray integral is a Calder\'on--Zygmund operator as proved in the recent result of Lanzani--Stein. We also point out that our method provides another proof of the boundedness and compactness of commutator of Cauchy--Szeg\H o operator on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with smooth boundary (first established by Krantz--Li).