Toward the theory of ring Q-homeomophisms |
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Authors: | V Ryazanov E Sevost’yanov |
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Institution: | (1) Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg Str., Donetsk, 83114, Ukraine |
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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 of all ring Q-homeomorphisms f of a domain D ⊂ ℝ
n
into
, n ≥ 2, with
, forms a normal family, if Q(x) has finite mean oscillation in D. We also prove normality of , 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. |
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Keywords: | |
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