Реферат: The Sobolev space L_p^1 (D),p∈[1,∞), on the domain D⊂R^n,n≥2, consists of locally integrable functions D have first generalized derivatives that are summable to the power p. The seminorm of a function v∈L_p^1 (D) equals the norm in L_p (D) of its generalized gradient ∇v. If φ:D→D^' is a homeomorphism of two domains D,D^'⊂R^n, then a natural question arises: under what conditions the composition operator φ^*:L_p^1 (D^' )→L_q^1 (D),1≤q≤p<∞, where u=φ^* (v)=v∘φ, will be bounded. We get a more general problem if instead of the space L_p^1 (D^' ) we consider the weighted Sobolev space L_p^1 (D^',ω), where ω is a locally summable weight function. We present a solution to the problem in a generalized setting and show that for some particular summability exponents q and p, and particular weight function ω the resulting classes of mappings coincide with the mappings studied in earlier papers. Within the framework of the generalized theory, results are obtained that are new even for the classical theory of quasiconformal mappings. For example, the norm of the composition operator φ^*:L_n^1 (D^' )→L_n^1 (D) is equal to K^(1/n), where K=(ess sup)┬xϵD⁡〖(|Dφ(x)|)/〖|det⁡〖Dφ(x)〗 |〗^(1/n) 〗 is the quasi-conformity coefficient. It will also be shown new questions appearing in the study of the connection between the Sobolev space and the geometry of mappings on a special class of nilpotent Lee groups: Carnot groups. Here we define a new object on Carnot groups: mappings on Carnot groups of the Sobolev class Соболева W_(ν,loc )^1 (D) with finite distortion. These are mappings D→G of the Sobolev class 〖φ:W〗_(ν,loc )^1 (D), where D is a domain on the Carnot group G, with non-negative Jacobian such that Dφ(x)=0 a.e. on the set of zeros of the Jacobian (here ν is the Hausdorff dimension of the group G). It is established that such mappings are continuous, P-differentiable almost everywhere, and have the Luzin property N.

Библиографическая ссылка: Vodopyanov S.K.
FUNCTION SPACES and GEOMETRY of MAPPINGS
Conference on the Theory of Functions of Several Real Variables, dedicated to the 90th anniversary of O. V. Besov 29 May - 6 Jun 2023