Generalization of one Poletskii lemma to classes of space mappings
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- Sevost’yanov, E.A. Ukr Math J (2009) 61: 1151. doi:10.1007/s11253-009-0267-0
The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings f ∈ W1,nloc such that their outer dilatation KO(x, f) belongs to Ln−1loc and the measure of the set Bf 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 → ℝn, 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.