Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations

Abstract

We study the distortion features of ho\-meo\-mor\-phisms of Sobolev class $W^{1,1}_{\rm loc}$ admitting integrability for $p$-outer dilatation. We show that such mappings belong to $W^{1,n-1}_{\rm loc},$ are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower $Q$-homeomorphisms with appropriate measurable functions $Q.$ This allows us to derive various distortion results like Lipschitz, H\"older, logarithmic H\"older continuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms.