An inverse Sobolev lemma
type: Item
instance of: scholarly article
title: An inverse Sobolev lemma
zbMATH Open document ID: 0795.30020
DOI: 10.4171/RMI/147
author: Pekka Koskela
published in: Revista Matemática Iberoamericana
publication date: 1994-05-24T00:00:00Z
full work available at URL: https://eudml.org/doc/39458
review text: The inverse Sobolev lemma in this context means the following: Let \(0< p\leq n\), and suppose that \(f\) is a quasiconformal mapping of the unit ball \(B\) of \(\mathbb{R}^ n\) into \(\mathbb{R}^ n\). Then the derivative \(f'\) is in \(L^ q(B)\) for all \(0< q<p\) if and only if \(f\) is in \(L^ s(B)\) for all \(0< s<pn/(n- p)\). The proof employs the average derivative of a quasiconformal mapping [\textit{K. Astala} and \textit{W. F. Gehring}, J. Anal. Math. 46, 16-57 (1986; Zbl 0628.30026)]. This is used to relate the integrability of \(f\) to the growth of the \(f\), and thus to \(f'\).
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 572265
zbMATH Keywords: integrability of the derivative of quasiconformal mapping
reviewed by: O. Martio
MaRDI profile type: Publication
Wikidata QID: Q124822457
OpenAlex ID: W2071434995
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description: scientific article; zbMATH DE number 572265