To a theory of ring Q-homeomorphisms with respect to a p-modulus

Abstract

We consider the ring Q-homeomorphisms with respect to a p-modulus. A criterion of the membership to this class is established. On this basis, we solve a Lavrentiev-type problem [17] of estimation of the measures for images of balls under such mappings and study their asymptotic behavior at a point. The theory developed by us is applicable to the description of mappings that are either quasiconformal on the average (see [8, 16]) and to the so-called (p, q)-quasiconformal mappings (see [32]), as well as to the mappings of the Orlicz—Sobolev class \( W_{loc}^{{1,\varphi }} \) under the Calderon condition and, in particular, to Sobolev’s classes \( W_{loc}^{1,p } \) for p > n − 1 (see [1, 15]).