Levi metrics and the Romanov-Henkin kernel
type: Item
instance of: scholarly article
title: Levi metrics and the Romanov-Henkin kernel
zbMATH Open document ID: 0784.32015
DOI: 10.1007/BF01258061
author: Ingo Lieb
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: There have been two distinct approaches to the study of the tangential Cauchy-Riemann operator \(\overline\partial_ b\) on the boundary \(M\) of a strictly pseudo-convex domain in \(\mathbb{C}^{n+1}\). The first, due originally to A. V. Romanov and G. M. Henkin, is based upon operators \(R_ q\) and \(R_{q+1}\) such that \(f=R_{q+1}\overline\partial_ bf+\overline\partial_ bR_ qf\) for \((0,q)\)-forms \(f\), \(1\leq q\leq n-1\). The second method is due to \textit{G. Folland} and \textit{E. Stein} [Commun. Pure Appl. Math. 27, 429-522 (1974; Zbl 0293.35012)], and involves the adjoint \(A_ b\) of \(\overline\partial_ b\), defined through a Levi metric on \(M\). The present work is devoted to showing for \(M\) with Levi metric that these two methods yield comparable results. The approximation of integral operators depends upon a scale of types that measure the degree of singularity of the kernels. It is shown that the Romanov-Henkin operator is the ``principal part'' in the sense of types of the canonical solution operator \(S\) such that \(\overline\partial_ bSf=f\) and \(A_ bSf=0\). The (kernel of the) principal part of the tangential Neumann operator is also given.
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32A40
Mathematics Subject Classification ID: 35N15
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 446299
zbMATH Keywords: homotopy formula
zbMATH Keywords: integral operator
zbMATH Keywords: principal part
zbMATH Keywords: tangential Cauchy- Riemann operator
zbMATH Keywords: strictly pseudo-convex domain
zbMATH Keywords: Levi metric
reviewed by: Edwin J. Akutowicz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01258061
OpenAlex ID: W2058407734
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description: scientific article; zbMATH DE number 446299