The theory of shell-based $Q$-mappings in geometric function theory

Open, discrete $Q$-mappings in ${\mathbb R}^n$, $n\ge2$, $Q\in L^1_{\mathrm{loc}}$, 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.
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