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We will consider classes of homeomorphisms of Euclidean domains subject to certain natural conditions. The new concept in quasiconformal analysis is based on the fundamental connection between bounded composition operators from a weighted Sobolev space to a weightless one, estimates for the capacity in the image in terms of the weighted capacity in the preimage, and homeomorphisms of the Sobolev class with finite distortion, for which the outer operator distortion function is summable.
The proposed approach includes, as a special case, almost all classes of homeomorphisms previously studied in the framework of the quasiconformal theory.
Both new results obtained in the framework of the new paradigm and new proofs of known results with a wider field of application will be discussed.