# Generalization of one Poletskii lemma to classes of space mappings

### 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.
Published

25.07.2009

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 61, no. 7, July 2009, pp. 969-75, https://umj.imath.kiev.ua/index.php/umj/article/view/3071.

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Section

Research articles