On some properties of generalized quasiisometries with unbounded characteristic

Abstract

We consider a family of open discrete mappings $$ f:D \to \overline {{{\mathbb R}^n}} $$ that distort, in a special way, the p-modulus of a family of curves that connect the plates of a spherical condenser in a domain D in $$ {{\mathbb R}^n} $$ ; p > n-1; p < n; and bypass a set of positive p-capacity. We establish that this family is normal if a certain real-valued function that controls the considered distortion of the family of curves has finite mean oscillation at every point or only logarithmic singularities of order not higher than n - 1: We show that, under these conditions, an isolated singularity x 0 ∈ D of a mapping $$ f:D\backslash \left\{ {{x_0}} \right\} \to \overline {{{\mathbb R}^n}} $$ is removable, and, moreover, the extended mapping is open and discrete. As applications, we obtain analogs of the known Liouville and Sokhotskii–Weierstrass theorems.