Equicontinuous classes of ring Q-homeomorphisms

Abstract

We give a description of ring Q-homeomorphisms in ℝn, n ≥ 2, and find a series of conditions for normality of families of ring Q-homeomorphisms. For a family to be normal it is sufficient that the dominant Q(x) have logarithmic-type singularities of order at most n-1. Another sufficient condition for normality is that Q(x) has finite mean oscillation at each point; for example, Q(x) has finite mean value over infinitesimal balls. The definition of ring Q-homeomorphism is motivated by the ring definition of Gehring for quasiconformality. In particular, the mappings with finite length distortion satisfy a capacity inequality that justifies the definition of ring Q-homeomorphism. Therefore, deriving consequences of the theory to be presented, we obtain criteria for normality of families of homeomorphisms f with finite length distortion and homeomorphisms of the Sobolev class W loc 1, in terms of the inner dilation K I (x, f). Moreover, the class of strong ring Q-homeomorphisms for a locally summable Q is closed.