Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations

"We study the distortion features of homeomorphisms 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, both ring lower $Q$-homeomorphisms with appropriate measurable functions $Q.$ This allows us derive various results like Lipschitz, H\""older, logarithmic H\""older continuity, etc. also establish a weak bounded variation property homeomorphisms."