Hölder and \(L^ p\)-estimates for the \({\bar\partial}\)-equation in some convex domains with real-analytic boundary
type: Item
instance of: scholarly article
title: Hölder and \(L^ p\)-estimates for the \({\bar\partial}\)-equation in some convex domains with real-analytic boundary
zbMATH Open document ID: 0533.32010
DOI: 10.1007/BF01450763
author: Joaquim Bruna
author: Joan del Castillo
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163946
review text: It is well known that for convex domains \(D=\{z:r(z)<0\}\) in \({\mathbb{C}}^ n\) and using the ''canonical'' support function \(F(\zeta,z)=\sum^{n}_{i=1}\partial r/\partial \zeta_ i(\zeta)(\zeta_ i-z_ i)\) one can define a reproducing kernel \(H(\zeta\),z), \(\zeta\in bD\), \(z\in D\), for holomorphic functions in \(D\) and also an explicit integral operator \(T\) which satisfies \({\bar \partial}Tf = f\) for \((0,1)\)-forms, \({\bar \partial}f=0.\) In this paper it is shown that: if \(r\) is a convex, real-analytic function of the type \(r(z)=\sum^{n}_{i=1}s_ i(| z_ i|^ 2)-1\), then the kernel \(H(\zeta,z)\) and the operator \(T\) have analogous properties as those in the strictly pseudoconvex case. Namely, \(\int_{bD}|\zeta-z|\;| H(\zeta,z)| d\sigma(\zeta)\leq c\), \(\zeta\in bD\), \(z\in D\), and for a regular, \({\bar\partial}\)-closed (0,1)-form \(\| Tf\|_{L^ p(D)}\leq c\| f\|_{L^ p(D)},\) 1\(\leq p\leq \infty\) and \(\| Tf\|_{Lip(D,1/m)}\leq c\| f\|_{L^{\infty}(D)}.\) The latter property generalizes previous work of \textit{R. M. Range} [cf. Several complex variables, Proc. Int. Conf., Cortona/Italy 1976/77, 247-267 (1978; Zbl 0421.32021)]. The proofs depend on the estimate, for \(\zeta\in bD\), \(z\in D\) and \(| \zeta-z|\) small \(2Re\;F(\zeta,z)\geq -r(z)+c L^r(\zeta)(\zeta -z)^ 2+c| \zeta -z|^ m\) where \(L^r(\zeta)\) is the Levi form of \(r\) at \(\zeta\), saying that the order of contact between \(bD\) and the tangent space at \(\zeta\) is controlled even at points of degeneracy by \(L^r(\zeta)\).
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32F99
Mathematics Subject Classification ID: 32A35
zbMATH DE Number: 3845955
zbMATH Keywords: convex domains
zbMATH Keywords: \(\bar\partial\)-equation
zbMATH Keywords: Hölder estimates
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3845955