Estimates for the Szegö and Henkin kernels in Hardy classes on certain unbounded weakly pseudoconvex domains
type: Item
instance of: scholarly article
title: Estimates for the Szegö and Henkin kernels in Hardy classes on certain unbounded weakly pseudoconvex domains
zbMATH Open document ID: 0761.32002
DOI: 10.1016/0022-247X(91)90154-R
author: Der-Chen Chang
author: Tristan Nguyen
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-26T00:00:00Z
review text: The smooth, unbounded, pseudoconvex domains in question are of the form \[ \Omega=\{(z_ 1,z_ 2)\in C^ 2:\;\text{Im} z_ 2>\varphi(z_ 2)\}, \] where \(\varphi\) is a subharmonic and non-harmonic polynomial of degree \(m\). The first task is the definition of Hardy spaces \({\mathcal H}^ p\) on \(\partial\Omega\). Two definitions (first in terms of atoms and non- isotropic balls in the sense of \textit{A. Nagel}, \textit{E. M. Stein} and \textit{S. Wainger} [Acta Math. 155, 103-147 (1985; Zbl 0578.32044)] and secondly through an appropriate notion of maximal function as in \textit{R. A. Macias} and \textit{C. Segovia} [Adv. Math. 33, 271-309 (1979; Zbl 0431.46019)] and \textit{A. Uchiyama} [Trans. Am. Math. Soc. 262, 579-592 (1980; Zbl 0503.46020)]) are shown to be equivalent. This is not easy and it occupies 10 pages of \S2, Local \({\mathcal H}^ p\) spaces on \(\partial\Omega\). \S3, Estimates for the Szegö kernel in Hardy spaces, is devoted to a lengthy proof that the Szegö projection extends to a bounded operator from \({\mathcal H}^ p(\partial\Omega)\) to the local Hardy spaces \({\mathcal H}_{\text{loc}}^ p(\partial\Omega)\) for \(p_ m<p\leq 1\), where the index \(p_ m\) depends upon the degree \(m\) of \(\varphi\). \S4, Estimates for the Henkin kernel in Hardy spaces, deals with weakly pseudoconvex domains \(H_ m=\{(z_ 1,z_ 2)\in C^ 2\): \(\text{Im} z_ 2>| z_ 1|^{2m}\}\) and proves that the Henkin projection extends to a bounded operator from \(L^ p(\partial H_ m)\) to \(L^ p(\partial H_ m)\) for \(1<p<\infty\), using methods in the book of \textit{E. M. Stein}, Singular integrals and nilpotent Lie groups (in preparation). The reviewer has not checked the delicate estimates involved in the present work.
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 55689
zbMATH Keywords: pseudoconvex domain
zbMATH Keywords: Hardy class
zbMATH Keywords: Szegö projection
zbMATH Keywords: Henkin projection
reviewed by: Edwin J. Akutowicz
MaRDI profile type: Publication
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full work available at URL: 0022-247x(91)90154-r
OpenAlex ID: W2085287452
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description: scientific article; zbMATH DE number 55689