Commutators of Cauchy-Szegő type integrals for domains in Cn with minimal smoothness

Xuan Thinh Duong, Michael Lacey, Ji Li, Brett D. Wick, Qingyan Wu

    Research output: Contribution to journalArticlepeer-review

    Abstract

    In this paper we study the commutator of the Cauchytype integral C on a bounded strongly pseudoconvex domain D in Cn with boundary bD satisfying the minimum regularity condition C2 as in the recent result of Lanzani-Stein. We point out that in this setting the Cauchy-type integral 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 Lp(bD) (1 < p < ∞) if and only if b is in the BMO space on bD. Moreover, the commutator [b,C] is compact on Lp(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 Cn with the boundary bD satisfying the minimum regularity C1,1. 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 the commutator of the Cauchy-Szegő operator on a bounded strongly pseudoconvex domain D in Cn with smooth boundary (first established by Krantz-Li in [18]).
    Original languageEnglish
    Pages (from-to)1505-1541
    Number of pages37
    JournalIndiana University Mathematics Journal
    Volume70
    Issue number4
    DOIs
    Publication statusPublished - 2021

    Keywords

    • Cauchy-type integrals
    • domains in Cn
    • BMO space
    • VMO space
    • commutator

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