The failure of \(L^ p\) estimates for harmonic measure in chord-arc domains
type: Item
instance of: scholarly article
title: The failure of \(L^ p\) estimates for harmonic measure in chord-arc domains
zbMATH Open document ID: 0536.30007
published in: Michigan Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Suppose that D is a bounded plane domain whose boundary is a rectifiable Jordan curve. D is said to be a chord-arc domain if there is a constant C such that for any w,w'\(\in \partial D\) we have \(\sigma(w,w')\leq C| w- w'|,\) where \(\sigma\) denotes the length of the shorter arc of \(\partial D\) connecting w and w'. These domains were introduced by Lavrent'ev in 1936, who proved that conformal mappings from the unit disk onto chord-arc domains satisfy \(1/f'\in H^ p\) for some \(p>0,\) where \(H^ p\) denotes the Hardy class. Chord-arc domains have attracted considerable attention recently because of their connections with functions of bounded mean oscillation, the theory of \(A_ p\) weights, and the theory of Cauchy integrals along curves. If \(\partial D\) is Lipschitz then \(1/f'\in H^ 1.\) The question arose whether this would be true for every chord-arc domain. In this paper the author decisively disproves this by showing that given any \(p>0\) there exists a chord-arc domain D for which \(1/f'\not\in H^ p.\) The counterexamples are obtained by clever construction starting from the non-Smirnov domain of Keldysh and Lavrent'ev. The same result has been proved non-constructively and independently by \textit{P. W. Jones} and \textit{M. Zinsmeister} [C. R. Acad. Sci., Paris Sér. I 295, 563-566 (1982; Zbl 0536.30008)].
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 3851446
zbMATH Keywords: chord-arc domain
zbMATH Keywords: functions of bounded mean oscillation
author: David Jerison
reviewed by: Albert Baernstein II
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029002849
OpenAlex ID: W2081395690
DOI: 10.1307/MMJ/1029002849
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description: scientific article; zbMATH DE number 3851446