Generalization of one Poletskii lemma to classes of space mappings
The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings f ∈ W^1,n_loc such that their outer dilatation K_O (x, f) belongs to L^n−1_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., number=7, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Sevost'yanov, E. A., year=2009, month=Jul., pages=969–975