Toward the theory of ring Q-homeomophisms

Abstract

We investigate classes of the so-called ring Q-homeomorphisms including, in particular, Q-homeomorphisms, various classes of homeomorphisms with finite length distortion, Sobolev’s classes etc. In terms of the majorant Q(x), we give a series of criteria for normality based on estimates of the distortion of the spherical distance under ring Q-homeomorphisms. In particular, it is shown that the class \( \Re _{Q,\Delta } \) of all ring Q-homeomorphisms f of a domain D ⊂ ℝ n into Open image in new window , n ≥ 2, with Open image in new window , forms a normal family, if Q(x) has finite mean oscillation in D. We also prove normality of \( \Re _{Q,\Delta } \), for instance, if Q(x) has singularities of logarithmic type whose degrees are not greater than n − 1 at every point x ∈ D. The results are applicable, in particular, to mappings with finite length distortion and Sobolev’s classes.