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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 3, pp. 385–398, March, 2011.
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Sevost’yanov, E.A. On some properties of generalized quasiisometries with unbounded characteristic. Ukr Math J 63, 443–460 (2011). https://doi.org/10.1007/s11253-011-0514-z
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DOI: https://doi.org/10.1007/s11253-011-0514-z