Обобщение одной леммы Е. А. Полецкого на классы пространственных отображений

The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings
f ∈ W1,n
loc such that their outer dilatation KO(x, f) belongs to Ln−1
loc 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.