Geodesic
Coincidence of set functions in quasiconformal analysis
Sbornik. Mathematics, Tome 213 (2022) no. 9, pp. 1157-1186

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It is known that mappings occurring in quasiconformal analysis can be defined in several equivalent ways: 1) as homeomorphisms inducing bounded composition operators between Sobolev spaces; 2) as Sobolev-class homeomorphisms with bounded distortion whose operator distortion function is integrable; 3) as homeomorphism changing the capacity of the image of a condenser in a controllable way in terms of the weighted capacity of the condenser in the source space; 4) as homeomorphism changing the modulus of the image of a family of curves in a controllable way in terms of the weighted modulus of the family of curves in the source space. A certain set function, defined on open subsets, can be associated with each of these definitions. The main result consists in the fact that all these set functions coincide. Bibliography: 48 titles.
Keywords: Sobolev space, composition operator, condenser capacity, outer operator distortion function, set function.
Mots-clés : quasiconformal analysis 
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S. K. Vodopyanov. Coincidence of set functions in quasiconformal analysis. Sbornik. Mathematics, Tome 213 (2022) no. 9, pp. 1157-1186. http://geodesic.mathdoc.fr/item/SM_2022_213_9_a0/