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The theory of shell-based $Q$-mappings in geometric function theory
R. R. Salimov,
E. A. Sevost'yanov Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
Abstract:
Open, discrete
$Q$-mappings in
${\mathbb R}^n$,
$n\geqslant2$,
$Q\in L^1_{\mathrm{loc}}$, are proved to be absolutely continuous on lines, to belong to the Sobolev class
$W_{\mathrm{loc}}^{1,1}$, to be differentiable almost everywhere and to have the
$N^{-1}$-property (converse to the Luzin
$N$-property). It is shown
that a family of open, discrete shell-based
$Q$-mappings leaving out a subset of positive capacity is normal, provided that either
$Q$ has finite mean oscillation at each point or
$Q$ has only logarithmic singularities of order at most
$n-1$. Under the same assumptions on
$Q$ it is proved that an isolated singularity
$x_0\in D$ of an open discrete shell-based
$Q$-map $f\colon D\setminus\{x_0\}\to\overline{\mathbb R}{}^n$ is removable; moreover, the extended map is open and discrete. On the basis of these results analogues of the well-known Liouville, Sokhotskii-Weierstrass and Picard theorems are obtained.
Bibliography: 34 titles.
Keywords:
quasiconformal mappings and their generalizations, moduli of families of curves, capacity, removing singularities of maps, theorems of Liouville, Sokhotskii and Picard type.
UDC:
517.548.2+
517.548.9+
517.547.26
MSC: 30C65 Received: 23.01.2009 and 19.01.2010
DOI:
10.4213/sm7529
Fulltext:
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