Geodesic
Functional and analytic properties of~a~class of~mappings in quasi-conformal analysis
Izvestiya. Mathematics , Tome 85 (2021) no. 5, pp. 883-931.

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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.
Keywords: quasi-conformal analysis, Sobolev space, composition operator, capacity and modulus of a condenser. 
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S. K. Vodopyanov; A. O. Tomilov. Functional and analytic properties of~a~class of~mappings in quasi-conformal analysis. Izvestiya. Mathematics , Tome 85 (2021) no. 5, pp. 883-931. http://geodesic.mathdoc.fr/item/IM2_2021_85_5_a3/

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