Cauchy-type integrals in several complex variables

Abstract

We present the theory of Cauchy-Fantappié integral operators, with emphasis on the situation when the domain of integration, $D$, has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter $z\in D$. The goal is to prove $L^p$ estimates for these operators and, as a consequence, to obtain $L^p$ estimates for the canonical Cauchy-Szegö and Bergman projection operators (which are not of Cauchy-Fantappié type).

This is an electronic reprint of the original article published in the Bulletin of the Mathematical Sciences, vol. 3 (2) (2013) 241-285. This reprint differs from the original in pagination and typographical detail

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• The Cauchy-Szeg\H o projection for domains in $\mathbb C^n% with minimal smoothness
• Regularity of a $\bar{\partial }$-solution operator for strongly $\mathbf{C}$-linearly convex domains with minimal smoothness
• The Leray transform: factorization, dual $CR$ structures and model hypersurfaces in $\mathbb{C}{\mathbb{P}}^2$
• Boundary Regularity of Bergman Kernel in Hölder space