On mappings with the inverse Poletsky inequality on Riemannian manifolds

Abstract

We study some problems related to the boundary behavior of maps of domains of Riemannian surfaces. In particular, for mappings satisfying the inverse Poletsky type modulus inequality, we establish the possibility of their continuous extension to the boundary in terms of prime ends. We also study the local behavior of such mappings at boundary points.

On the boundary behavior of weak (p,q)-quasiconformal mappings

“…Let q = n and Q q ∈ L 1 (D). Then we have the class of mappings with capacitory inverse Poletsky inequality which was intensively studied recently [26], [24] and [23].…”

“…Proof of Theorem 1.1. In general, we follow the logic of the proof of Theorem 1.2 in [24], see also Theorem 1.2 in [26] and Theorems 1-2 in [23]…”

“…We will carry out a proof from the opposite (cf. [26,Theorem 1.2], [23,Theorem 5]). Assume that, there is x 0 ∈ ∂D, a number ε 0 > 0, a sequence x m ∈ D, which converges to x 0 as m → ∞ and a sequence of mappings…”

“…These mappings are called weak (p, q)-quasiconformal mappings [3,30] and can be characterized by the inverse capacitory (moduli) Poletsky inequality [27] cap 1/q q (f −1 (E), f −1 (F ); D) K p,q (ϕ; Ω)cap 1/p p (E, F ; D ′ ), 1 < q p < ∞. The detailed study of the mappings with the inverse conformal Poletsky inequality for modulus of paths was given in [26], [24] and [23]. In this case p = q = n the Hölder continuity, the continuous boundary extension, and the behavior on the closure of domains were obtained.…”

“…holds for any continua E ⊂ f −1 (B(y 0 , r 1 )), F ⊂ f −1 (f (D) \ B(y 0 , r 2 )), 0 < r 1 < r 2 < r 0 = sup The case q = n was studied in details in [26], cf. [24] and [23]. The present article is dedicated to the case q = n.…”