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

We define a two-index scale Q(q,p), n - 1 < q <= p < infinity, of home-omorphisms of spatial domains in 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 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 Q(q,p). When q = p = n, the class of mappings Q(n,n) coincides with the set of so-called Q-homeomorphisms which have been studied extensively in the last 25 years.