2018
Том 70
№ 5

# Generalization of one Poletskii lemma to classes of space mappings

Sevost'yanov E. A.

Abstract

The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings $f ∈ W^{1,n}_{\text{loc}}$ such that their outer dilatation $K_O (x, f)$ belongs to $L^{n−1}_{\text{loc}}$ and the measure of the set $B_f$ of branching points of $f$ is equal to zero have finite length distortion. In other words, the images of almost all curves $γ$ in the domain $D$ under the considered mappings $f : D → ℝ^n,\;n ≥ 2$, are locally rectifiable, $f$ possesses the $(N)$-property with respect to length on $γ$, and, furthermore, the $(N)$-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii lemma proved for quasiregular mappings.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 7, pp 1151-1157.

Citation Example: Sevost'yanov E. A. Generalization of one Poletskii lemma to classes of space mappings // Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 969-975.

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