Holomorphic approximation and estimates for the \({\bar \partial}\)-equation on strictly pseudoconvex nonsmooth domains
type: Item
instance of: scholarly article
title: Holomorphic approximation and estimates for the \({\bar \partial}\)-equation on strictly pseudoconvex nonsmooth domains
zbMATH Open document ID: 0645.32009
DOI: 10.1215/S0012-7094-87-05528-1
author: Joaquim Bruna
author: Josep M. Burgués
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Denote by X a (non-smooth) compact set in \({\mathbb{C}}^ n\) of the form \(X=\{z:\) r(z)\(\leq 0\}\) where r is a function of class \(C^ 2\), strictly plurisubharmonic in a neighborhood of \(\{\) \(z: r(z)=0\}\) and by H(X) the space of functions holomorphic in some neighborhood of X. For \(m=0,1,..\). and \(o\leq s<1\), we consider the Whitney space \(C^{m,s}(X)\), i.e. the space of jets \(F=(u_{\alpha,\beta})\) where \(\alpha\), \(\beta\) are n- multiindexes with \(| \alpha | +| \beta | \leq m\) and \(u_{\alpha,\beta}\) are continuous functions on X such that the formal Taylor rest \(R_ z^{m-| \alpha | -| \beta |}(u_{\alpha,\beta}(w))\) is \(o(| w-z|^{m+s-| \alpha | -| \beta |})\) for z,w\(\in X\), \(| z-w| \to 0.\) Main results: a) It is described the closure of H(X) in \(C^{m,s}(X).\) For \(p\geq 1\), let ary and initial conditions). The approach rests on a monotone iterative method, which yields the existence of maximal and minimal solutions along with upper and lower estimates of the exact solution of the equation. An application of the theory to the study of a nonlinear integral equation occuring in epidemiology is also included.
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 4051999
zbMATH Keywords: estimates for the delta bar equations
zbMATH Keywords: non-smooth strictly pseudoconvex domains
zbMATH Keywords: holomorphic approximation
zbMATH Keywords: nonisotropic regularization of holomorphic functions
zbMATH Keywords: monotone iterative method
zbMATH Keywords: upper and lower estimates
zbMATH Keywords: epidemiology
reviewed by: Sergiu Aizicovici
MaRDI profile type: Publication
cites work: Mesures de Carleson d'ordre $\alpha$ et solutions au bord de l'équation $\overline\partial$
cites work: Henkin-Ramirez kernels with weight factors
cites work: Holomorphic approximation in \(C^ m\)-norms on totally real compact sets in \({\mathbb{C}}^ n\)
cites work: Cauchy kernels in strictly pseudoconvex domains and an application to a Mergolyan type approximation problem
cites work: Interpolation by holomorphic functions smooth to the boundary in the unit ball of \({\mathbb{C}}^ n\)
cites work: Q5621558
cites work: Zero sets of non-negative strictly plurisubharmonic functions
cites work: Holomorphic approximation and hyperfunction theory on a C\(^1\) totally real submanifold of a complex manifold
cites work: INTEGRAL REPRESENTATIONS OF FUNCTIONS HOLOMORPHIC IN STRICTLY PSEUDO-CONVEX DOMAINS AND SOME APPLICATIONS
cites work: INTEGRAL REPRESENTATION OF FUNCTIONS IN STRICTLY PSEUDOCONVEX DOMAINS AND APPLICATIONS TO THE $ \overline{\partial}$-PROBLEM
cites work: EXACT HÖLDER ESTIMATES FOR THE SOLUTIONS OF THE $ \bar\partial$-EQUATION
cites work: Q3692032
cites work: Uniform Approximation on Compact Sets in $\mathsf{C}^n$.
cites work: Hölder and Lp estimates for solutions of ∂u = f in strongly pseudoconvex domains
cites work: Ein Approximationssatz auf streng pseudokonvexen Gebieten
cites work: Die Cauchy-Riemannschen Differentialgleichungen auf streng pseudokonvexen Gebieten. Beschränkte Lösungen
cites work: The Cauchy-Riemann differential equations on strongly pseudo-convex domains. Continuous boundary values.
cites work: Q5566770
cites work: Approximation Theorems on Differentiable Submanifolds of a Complex Manifold
cites work: Integral Representation Formulas and $L^p$-Estimates for the $\bar{\partial}$-Equation.
cites work: C\(^k\) approximation by holomorphic functions and \(\bar \partial\)-closed forms on C\(^k\) submanifolds of a complex manifold
cites work: Valeurs au bord pour les solutions de l'opérateur $d^n$, et caractérisation des zéros des fonctions de la classe de Nevanlinna
cites work: Singular integrals and estimates for the Cauchy-Riemann equations
description: scientific article; zbMATH DE number 4051999