Composition Operators on Weighted Sobolev Spaces and the Theory of ${{\mathcal{Q}}_{p}}$-Homeomorphisms

Abstract

We define the scale ${{\mathcal{Q}}_{p}}$, $n - 1 < p < \infty$, 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 ${{\mathcal{Q}}_{n}}$ of mappings contains the class of so-called Q-homeomorphisms, which have been actively studied over the past 25 years. An equivalent functional and analytic description of these classes ${{\mathcal{Q}}_{p}}$ 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 ${{\mathcal{Q}}_{p}}$.