Prime ends and Orlicz–Sobolev classes

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.