The third Cauchy-Fantappie formula of Leray

External area integral inequality for the Cauchy-Leray-Fantappié integral

In this paper we extend Luzin inequality for functions deined by the Cauchy-Leray-Fantappié integral on the complement of a convex domain in $\mathbb{C}^n$. The work is supported by Russian Science Foundation Grant 14-41-00010.

Products of residue currents of Cauchy-Fantappiè-Leray type

With a given holomorphic section of a Hermitian vector bundle, one can associate a residue current by means of Cauchy-Fantappiè-Leray type formulas. In this paper we define products of such residue currents. We prove that, in the case of a complete intersection, the product of the residue currents of a tuple of sections coincides with the residue current of the direct sum of the sections.

一个使得复椭球上的Cauchy-Leray-Fantappiè算子无界的变指数空间$L^{p(\cdot)}$的例子

An example of a space $L^{p(\cdot)}$ on which the Cauchy-Leray-Fantappiè operator for complex ellipsoid is not bounded

本文构造了一个变指数Lebesgue空间，证明复椭球上的Cauchy-Leray-Fantappiè算子在该空间上无界，从而将对数连续性条件的尖锐性推广到非严格凸区域。

Cauchy-Fantappie Type Operators And Duality On Poletsky-Stessin Hardy Spaces of Complex Ellipsoids

In the first part of this study we consider the boundedness and compactness properties of Cauchy-Fantappie type operators on Poletsky-Stessin Hardy spaces $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ of complex ellipsoids. We show that boundedness and compactness criteria are given by the Carleson conditions. In addition we give a basic compactness property for the subsets of $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ spaces and the characterization of weakly convergent sequences in $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$. In the second part we will discuss the dual complement of the complex ellipsoid and we w…

Cauchy-type integrals in several complex variables

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).

The Cauchy-Leray integral: counter-examples to the $L^p$-theory

We prove the optimality of the hypotheses guaranteeing the $L^p$-boundedness for the Cauchy-Leray integral in $\mathbb C^n$, $n\geq 2$, obtained in [LS-4]. Two domains, both elementary in nature, show that the geometric requirement of strong $\mathbb C$-linear convexity, together with regularity of order 2, are both necessary.