Composition operators on weighted Sobolev spaces and the theory of $\mathscr{Q}_p$-homeomorphisms

We define the scale $\mathscr{Q}_p$, $n-1<p<\infty$, of homeomorphisms of spatial domains in $\mathbb{R}^n$, a geometric description of which is due to the control of the behavior of the p-capacity of condensers in the image through the weighted p-capacity of the condensers in the preimage. For $p=n$ the class $\mathscr{Q}_n$ of mappings contains the class of so-called $\mathscr{Q}_p$-homeomorphisms, which have been actively studied over the past 25 years. An equivalent functional and analytic description of these classes $\mathscr{Q}_p$ is obtained. It is based on the problem of the properties of the composition operator of a weighted Sobolev space into a nonweighted one induced by a map inverse to some of the class $\mathscr{Q}_p$.