Elizabeth Wulcan
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.
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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.
We study the third Cauchy-Fantappie formula, an integral representation formula for holomorphic functions on a domain in affine space, presented by Jean Leray in the third paper of his famous series "Probleme de Cauchy", published in 1959. We show by means of examples that this formula does not hold without some additional conditions, left unmentioned by Leray. We give sufficient conditions and a necessary condition for the formula to hold, and, in the case of a contractible domain, characterize it cohomologically.
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.
The purpose of this paper is to complement the results in [LS-1] by showing the dense definability of the Cauchy-Leray transform for the domains that give the counterexamples of [LS-1], where $L^p$-boundedness is shown to fail when either the "near" $C^2$ boundary regularity, or the strong $\mathbb C$-linear convexity assumption is dropped.
In this paper we generalize Leray's calculus of residues in several complex variables, to the situation of an abstract smooth CR manifold M of general type (n,k).
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。
We prove $L^p(b\mathcal{D})$-regularity of the Cauchy-Leray integral for bounded domains $\mathcal{D}\subset\mathbb C^n$ whose boundary satisfies the minimal regularity condition of class $C^{1,1}$, together with a naturally occurring notion of convexity.
We describe various approaches to Coleff-Herrera products of residue currents $R^j$ (of Cauchy-Fantappiè-Leray type) associated to holomorphic mappings $f_j$. More precisely, we study to which extent (exterior) products of natural regularizations of the individual currents $R^j$ yield regularizations of the corresponding Coleff-Herrera products. Our results hold globally on an arbitrary pure-dimensional complex space.
We prove that the Coleff-Herrera residue current, corresponding to a pair of holomorphic functions defining a complete intersection, can be obtained as the unrestricted weak limit of a natural smooth $(0,2)$-form depending on two parameters. Moreover, we prove that the rate of convergence is Hölder. This result is in contrast to the fact, first discovered by Passare and Tsikh, that the residue integral in general is discontinuous at the origin. We also generalize our regularization results to pairs of so called Bochner-Martinelli, or more generally, Cauchy-Fantappie-Leray blocks in the case of…
Cauchy's method from two centuries ago for computing integrals along the real axis by passing into the complex plane is not rigorous by present-day standards. Yet when properly formulated, his original approach is simpler than modern presentations of the residue calculus.
We survey recent work and announce new results concerning two singular integral operators whose kernels are holomorphic functions of the output variable, specifically the Cauchy-Leray integral and the Cauchy-Szeg\H o projection associated to various classes of bounded domains in $\mathbb C^n$ with $n\geq 2$.
In the introduction part of this paper, first of all, the concept of absolute integral sum of complex function is defined, as more general one with respect to the concept of integral as well as of integral sum of "ordinary'' integral calculus. In the main part of the paper, on the basis of redefined concepts of a spatial derivative as well as of a residue of a complex function, the fundamental results of Cauchy's calculus of residues of analytic functions are generalized and the concept of total value of improper integral of complex function is defined. Finally, the results in the form of fund…
In this paper we study the commutator of Cauchy type integrals $\EuScript C$ on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ 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 integrals $\EuScript C$ is the sum of the essential part $\EuScript C^\sharp$ which is a Calderón--Zygmund operator and a remainder $\EuScript R$ which is no longer a Calderón--Zygmund operator. We show that the commutator $[b, \EuScript C]$ is bounded on $L^p(bD)$ ($1<p<\infty$) if {\color{blac…
We extend the Cauchy residue theorem to a large class of domains including differential chains that represent, via canonical embedding into a space of currents, divergence free vector fields and non-Lipschitz curves. That is, while the classical Cauchy theorems involve integrals over piecewise smooth parameterized curves, these classical theorems actually hold for far more general notions of "curve." We also extend the definition of winding number to these domains and show that it behaves as expected.
Functional Equations for Generalized Collatz Dynamics Using Integral Representations and Residue Calculus
本文利用复分析中的积分表示与留数计算，系统推导广义Collatz型动力学生成函数的递推公式和函数方程，并推广至多维情形。
In this paper, we study the existence of higher order Poisson type systems. In detail, we prove a Residue type phenomenon for the fundamental solution of Laplacian in $\RR^n, n\ge 3$. This is analogous to the Residue theorem for the Cauchy kernel in $\CC$. With the aid of the Residue type formula for the fundamental solution, we derive the higher order derivative formula for the Newtonian potential and obtain its appropriate $\s C^{k, α}$ estimates. The existence of solutions to higher order Poisson type nonlinear systems is concluded as an application of the fixed point theorem.
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