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- Title
Equicontinuous classes of ring Q-homeomorphisms.
- Authors
Ryazanov, V. I.; Sevost'yanov, E. A.
- 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 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.
- Subjects
RING theory; HOMEOMORPHISMS; MATHEMATICAL mappings; QUASICONFORMAL mappings; ALGEBRA; MATHEMATICAL analysis
- Publication
Siberian Mathematical Journal, 2007, Vol 48, Issue 6, p1093
- ISSN
0037-4466
- Publication type
Article
- DOI
10.1007/s11202-007-0111-4