The regularity of inverses to Sobolev mappings and the theory of $\mathscr Q_{q,p}$-homeomorphisms

We prove that each homeomorphism $\varphi: D\to D'$ of Euclidean domains in $\Bbb R^n$, $n\geq2$, belonging to the Sobolev class $W^1_{p,\operatorname{loc}}(D)$, where $p\in[1,\infty)$, and having finite distortion induces a bounded composition operator from the weighted Sobolev space $L^1_p(D';\omega)$ into $L^1_p(D)$ for some weight function $\omega:D'\to (0,\infty)$. This implies that in the cases $p>n-1$ and $n\geq 3$ as well as $p\geq1$ and $n\geq 2$ the inverse $\varphi^{-1}: D'\to D$ belongs to the Sobolev class $W^1_{1,\operatorname{loc}}(D')$, has finite distortion, and is differentiable ${\Cal H}^{n}$-almost everywhere in $D'$. We apply this result to $\Cal Q_{q,p}$-homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of $\Cal Q_{q,p}$-homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes.