On spatial mappings with integral restrictions on the characteristic

Abstract:

For a given domain $D\subset {\mathbb R}^n$, some families ${\mathfrak F}$ of mappings $f : D\rightarrow \overline {{\mathbb R}^n}$, $n\ge 2$ are studied; such families are more general than the mappings with bounded distortion. It is proved that a family is equicontinuous if $\int _{\delta _0}^{\infty } \frac {d\tau }{\tau [\Phi ^{-1}(\tau )]^{\frac {1}{n-1}}}= \infty$, where the integral depends on each mapping $f\in {\mathfrak F}$, $\Phi$ is a special function, and $\delta _0>0$ is fixed. Under similar restrictions, removability results are obtained for isolated singularities of $f$. Also, analogs of the well-known Sokhotsky–Weierstrass and Liouville theorems are proved.