Functional and analytic properties of a class of mappings in quasi-conformal analysis

We define a two-index scale $\mathcal Q_{q,p}$, $n-1<q\leq p<\infty$, of homeomorphisms of spatial domains in $\mathbb R^n$, the geometric description of which is determined by the control of the behaviour of the $q$-capacity of condensers in the target space in terms of the weighted $p$-capacity of condensers in the source space. We obtain an equivalent functional and analytic description of $\mathcal Q_{q,p}$ based on the properties of the composition operator (from weighted Sobolev spaces to non-weighted ones) induced by the inverses of the mappings in $\mathcal Q_{q,p}$.
When $q=p=n$, the class of mappings $\mathcal Q_{n,n}$ coincides with the set of so-called $Q$-homeomorphisms which have been studied extensively in the last 25 years.