Prime ends and Orlicz–Sobolev classes

Kovtonyuk, D.; Ryazanov, V.

doi:10.1090/spmj/1416

Prime ends and Orlicz–Sobolev classes
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by D. A. Kovtonyuk and V. I. Ryazanov
Translated by: the authors

St. Petersburg Math. J. 27 (2016), 765-788

DOI: https://doi.org/10.1090/spmj/1416

Published electronically: July 26, 2016

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Abstract:

A canonical representation of prime ends is obtained in the case of regular spatial domains, and the boundary behavior is studied for the so-called lower $Q$-homeomorphisms, which generalize the quasiconformal mappings in a natural way. In particular, a series of efficient conditions on a function $Q$ are found for continuous and homeomorphic extendibility to the boundary along prime ends. On that basis, a theory is developed that describes the boundary behavior of mappings in the Sobolev and Orlicz–Sobolev classes and also of finitely bi-Lipschitz mappings, which are a far-reaching generalization of the well-known classes of isometries and quasiisometries.

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