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

S. K. Vodopyanov ; A. O. Tomilov

Izvestiya. Mathematics , Tome 85 (2021) no. 5, pp. 883-931

Résumé

We define a two-index scale $\mathcal Q_{q,p}$, $n-1$, 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.

Détail

Keywords: quasi-conformal analysis, Sobolev space, composition operator, capacity and modulus of a condenser.