\(p\)-harmonic tensors and quasiregular mappings. (English) Zbl 0785.30009

This paper is a continuation of a joint paper of the author and G. Martin [IM] [Acta Math. 170, 29-81 (1993; reviewed above)]. As in [IM] the author mentions the paper of S. K. Donaldson and D. P. Sullivan [DS] [Acta Math. 163, No. 3/4, 181-252 (1989; Zbl 0704.57008)] as an important source of inspiration and motivation. The basic philosophy here is to combine different methods and techniques to such differential forms in \(L^ p\)-sense, Hodge theory, harmonic tensors, maximal functions and to use these to prove new results for quasiregular maps. These methods are used not only as formal tools but with deep insight and great skill. Together with [IM] this paper opens new avenues to the theory of quasiregular maps. Ideas from Calderón-Zygmund theory, nonlinear potential theory, variational calculus add to the richness of the methods used. Some of the main results are (a) a new regularity result for quasiregular maps, (b) a Caccioppoli type estimate, (c) a removable singularity theorem. The author regards (c) as the primary result of the paper and because it is simple to formulate, we state it here: For each dimension \(n=2,3 \dots\) and \(K \geq 1\) there is an \(\varepsilon=\varepsilon(K,n)>0\) such that every closed set \(E \subset R^ n\) of Hausdorff dimension \(<\varepsilon\) is removable under bounded \(K\)-quasiregular mappings. There are several recent results of this type due to P. Koskela and O. Martio [Ann. Acad. Sci. Fenn., Ser. AI 15, No. 2, 381-399 (1990; Zbl 0717.30015)]. P. Järvi and the reviewer [J. Reine Angew. Math. 424, 31-45 (1992; Zbl 0733.30017)] and S. Rickman (to appear). There is a recent survey of the author [in Lect. Notes Math. 1508, 39-64 (1992; reviewed below)] where also this paper is discussed.


30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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