Limits on an extension of Carleson's \(\bar\partial\)-theorem
type: Item
instance of: scholarly article
title: Limits on an extension of Carleson's \(\bar\partial\)-theorem
zbMATH Open document ID: 0913.30027
DOI: 10.1007/BF02921564
author: Jeffrey Diller
published in: The Journal of Geometric Analysis
publication date: 1999-05-25T00:00:00Z
review text: The author studies a theorem due essentially to Carleson about solving the \(\overline\partial\)-equation on the unit disk. He shows that this theorem generalizes to bordered Riemann surfaces with finitely generated fundamental groups. However, his main result is that the constant appearing in the generalized theorem cannot be taken to be independent of the bordered Riemann surface in question. He exhibits a sequence of (topologically equivalent) Riemann surfaces on which the constant tends to \(\infty\). Since Carleson's \(\overline\partial\)-theorem depends on the notion of a Carleson measure, the author also discusses Carleson measures at some length in order to define them appropriately on arbitrary Riemann surfaces.
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 30C25
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 930668
zbMATH Keywords: Cauchy-Riemann equations
zbMATH Keywords: Poincaré metric
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: Carleson measures
reviewed by: Justus Diller
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 930668