Functional and analytical properties of a class of mappings of quasiconformal analysis on Carnot groups

This article addresses the conceptual questions of quasiconformal analysis on Carnot groups. We prove the equivalence of the three classes of homeomorphisms: the mappings of the first class induce bounded composition operators from a weighted Sobolev space into an unweighted one; the mappings of the second class are characterized by way of estimating the capacity of the preimage of a condenser in terms of the weighted capacity of the condenser in the image; the mappings of the third class are described via a pointwise relation between the norm of the matrix of the differential, the Jacobian, and the weight function. We obtain a new proof of the absolute continuity of mappings.