# On some properties of generalized quasiisometries with unbounded characteristic

### Abstract

We consider a family of the open discrete mappings $f:\; D \rightarrow \overline{\mathbb{R}^n}$ that distort in a special way the $p$ -modulus of families of curves connecting the components of spherical condenser in a domain $D$ in $\mathbb{R}^n$, $p > n — 1,\;\; p < n$, and omitting a set of positive $p$-capacity. We establish that this family is normal provided that some function realizing the control of the considered distortion of curve family has a finite mean oscillation at every point or only logarithmic singularities of the order, which is not larger than $n − 1$. We prove that, under these conditions, an isolated singularity $x_0 \in D$ of the mapping $f : D \ \{x_0\} \rightarrow \overline{\mathbb{R}^n}$ is removable and, moreover, the extended mapping is open and discrete. As applications we obtain analogs of the known Liouville and Sokhotski – Weierstrass theorems.
Published

25.03.2011

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 63, no. 3, Mar. 2011, pp. 385-98, https://umj.imath.kiev.ua/index.php/umj/article/view/2724.

Issue

Section

Research articles