## Abstract

In this paper we study the commutator of the Cauchytype integral C on a bounded strongly pseudoconvex domain D in 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 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 L^{p}(bD) (1 < p < ∞) if and only if b is in the BMO space on bD. Moreover, the commutator [b,C] is compact on L^{p}(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^{n}with the boundary bD satisfying the minimum regularity C^{1,1}. 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 C^{n}with smooth boundary (first established by Krantz-Li in [18]).Original language | English |
---|---|

Pages (from-to) | 1505-1541 |

Number of pages | 37 |

Journal | Indiana University Mathematics Journal |

Volume | 70 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2021 |

## Keywords

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

## Fingerprint

Dive into the research topics of 'Commutators of Cauchy-Szegő type integrals for domains in C^{n}with minimal smoothness'. Together they form a unique fingerprint.