Quasiregular mappings in even dimensions
type: Item
instance of: scholarly article
title: Quasiregular mappings in even dimensions
zbMATH Open document ID: 0785.30008
DOI: 10.1007/BF02392454
author: Tadeusz Iwaniec
author: Gaven J. Martin
published in: Acta Mathematica
publication date: 1994-04-11T00:00:00Z
review text: In their introduction to this long important paper the authors say that the paper has grown from their study of a paper of \textit{S. K. Donaldson} and \textit{D. P. Sullivan} [DS] [Acta Math. 163, No. 3/4, 181-252 (1989; Zbl 0704.57008)]. Using and developing the ideas of [DS] the authors study differential forms with integrable coefficients and Hodge theory on \(L^ p\)-spaces and with great success apply these methods to investigate several topics and open problems in the theory of quasiconformal and quasiregular maps in \(R^ n\), \(n \geq 2\). Among such topics are (a) a definition of a new Beurling-Ahlfors type singular integral operator \(S\) suited for higher dimensional studies, (b) regularity questions in Liouville's theorem in even dimensions, (c) discovery of new differential equations satisfied by quasiregular maps in even dimensions, (d) new definition of quasiregular maps, (e) derivation of Caccioppoli type estimates. The authors give various examples showing the sharpness of their results and also present various conjectures. It is not possible to adequately describe even the main results of the paper in this review. Therefore we refer the reader to a recent survey of the first author [in Lect. Notes Math. 1508, 39-64 (1992; reviewed below)] where also this paper is discussed and conclude by what the authors regard as their most important result. This result deals with the study of exceptional sets of bounded quasiregular maps. Here the main problem is the following one. Let \(E\) be a compact subset of the unit ball \(B^ n\) of measure zero and let \(f:B^ n \backslash E \to R^ n\) be a bounded quasiregular map. Under which conditions on \(E\) and \(f\) the map \(f\) has an extension to \(E\)? \textit{Yu. G. Reshetnyak} [Sib. Mat. Zh. 10, 1300-1310 (1969; Zbl 0201.098)] and \textit{O. Martio}, \textit{S. Rickman} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser. A I 465, 1-13 (1970; Zbl 0197.057)] settled the case when \(E\) is of zero \(n\)-capacit…
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 205188
zbMATH Keywords: quasiregular maps
zbMATH Keywords: Hausdorff dimension
reviewed by: M. Vuorinen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02392454
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description: scientific article; zbMATH DE number 205188