Refined geometric characterizations of weak $p$-quasiconformal mappings

In this paper we consider refined geometric characterizations of weak $p$-quasiconformal mappings $\phi :\mathrm{\Omega }\to \tilde{\mathrm{\Omega }}$, where $\mathrm{\Omega }$ and $\tilde{\mathrm{\Omega }}$ are domains in ${\mathbb{R}}^n$. We prove that mappings with the bounded on the set $\mathrm{\Omega }\setminus S$, where a set $S$ has $\sigma$-finite $(n-1)$-measure, geometric $p$-dilatation, are $W_{p,\text{loc}}^1$– mappings and generate bounded composition operators on Sobolev spaces.