Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness

Abstract

In this paper we study the commutator of Cauchy type integrals C on a bounded strongly pseudoconvex domain D in C with boundary bD satisfying the minimum regularity condition C as in the recent result of Lanzani–Stein. We point out that in this setting the Cauchy type integrals C is the sum of the essential part C which is a Calderón–Zygmund operator and a remainder R which is no longer a Calderón–Zygmund operator. We show that the commutator [b,C] is bounded on L(bD) (1 < p < ∞) if and only if b is in the BMO space on bD. Moreover, the commutator [b, C] is compact on L(bD) (1 < p < ∞) if 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 C-linearly convex domain D in C with the boundary bD satisfying the minimum regularity C. Such a Cauchy–Leray integral is a Calderón–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–Szegő operator on a bounded strongly pseudoconvex domain D in C with smooth boundary (first established by Krantz–Li).