2017
Том 69
№ 5

All Issues

On some properties of generalized quasiisometries with unbounded characteristic

Sevost'yanov E. A.

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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.

English version (Springer): Ukrainian Mathematical Journal 63 (2011), no. 3, pp 443-460.

Citation Example: Sevost'yanov E. A. On some properties of generalized quasiisometries with unbounded characteristic // Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 385-398.

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