# Absolute continuity on paths of spatial open discrete mappings

Absolute continuity on paths of spatial open discrete mappings We prove that open discrete mappings of Sobolev classes $$W_\mathrm{loc}^{1, p},$$ W loc 1 , p , $$p>n-1,$$ p > n - 1 , with locally integrable inner dilatations admit $$ACP_p^{\,-1}$$ A C 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 Poletskiĭ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Absolute continuity on paths of spatial open discrete mappings

, Volume 8 (1) – Dec 28, 2016
11 pages

/lp/springer-journals/absolute-continuity-on-paths-of-spatial-open-discrete-mappings-LigCFnFs3n
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-016-0159-z
Publisher site
See Article on Publisher Site

### Abstract

We prove that open discrete mappings of Sobolev classes $$W_\mathrm{loc}^{1, p},$$ W loc 1 , p , $$p>n-1,$$ p > n - 1 , with locally integrable inner dilatations admit $$ACP_p^{\,-1}$$ A C 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 Poletskiĭ 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.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Dec 28, 2016