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Equicontinuous classes of ring Q-homeomorphisms

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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 1,nloc 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.

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Correspondence to V. I. Ryazanov.

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Original Russian Text Copyright © 2007 Ryazanov V. I. and Sevost’yanov E. A.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1361–1376, November–December, 2007.

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Ryazanov, V.I., Sevost’yanov, E.A. Equicontinuous classes of ring Q-homeomorphisms. Sib Math J 48, 1093–1105 (2007). https://doi.org/10.1007/s11202-007-0111-4

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  • DOI: https://doi.org/10.1007/s11202-007-0111-4

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