Absolute continuity on paths of spatial open discrete mappings

Abstract

We prove that open discrete mappings of Sobolev classes $W_{\mathrm{loc}}^{1,p},$ $p>n-1,$ with locally integrable inner dilatations admit $AC{P}_p^{-1}$-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\uı 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.

9 pages