Absolute continuity on paths of spatial open discrete mappings

TL;DR

This paper proves that certain open discrete Sobolev mappings with integrable inner dilatations are absolutely continuous on almost all preimage paths, extending classical results like Poletskii's lemma for quasiregular mappings.

Contribution

It establishes the $ACP_p^{-1}$-property for Sobolev mappings with $p>n-1$, broadening the understanding of path properties in geometric function theory.

Findings

Mappings are absolutely continuous on almost all preimage paths.

Provides upper bounds for $p$-module in terms of inner dilatations.

Extends Poletskii's lemma to a wider class of mappings.

Abstract

We prove that open discrete mappings of Sobolev classes $W_{\rm loc}^{1, p},$ $p>n-1,$ with locally integrable inner dilatations admit $ACP_p^{\,-1}$-property, which means that these mappings are absolutely continuous on almost all preimage paths with respect to $p$-module. In particular, our results extend the well-known Poletski\u\i\ lemma for quasiregular mappings. We also establish the upper bounds for $p$-module of such mappings in terms of integrals depending on the inner dilatations and arbitrary admissible functions.