Functional and analytic properties of a class of mappings in quasi-conformal analysis

Two-Weighted Composition Operators on Sobolev Spaces and Quasiconformal Analysis

“…The initial version of Theorem 1 appeared in [5,7] where instead of cubical condensers there were considered more general condensers of the form E = (F, U) in D ′ ( U ⋐ D ′ is an open set and F ⊂ U is a continuum). At papers [6,7], a reader may find also a detailed history of the question and a bibliography.…”

“…In the case ≡ 1 , simple proof of Lemma 6 based on the original method in [, Lemma 1] is given in [7,43]. The same arguments work for an arbitrary A q -weight .…”

“…where Z � = (Σ) and Σ � = (Z) play the same roles for the inverse homeomorphism −1 ∶ D � → D as Z and Σ do for ; for the details, see [, Section 1]. 7 Moreover, if ∶ D → D � belongs to W 1 1,loc (D) or ∈ ACL(D) , then [7, Section 1] shows that Hence, for Sobolev-class mappings, we may assume from the outset that…”

“…This article naturally fits into the cycle of publications [1][2][3][4][5][6][7]. Originating in the series [8][9][10][11][12][13][14][15], it resides at the junction of the theory of function spaces [16][17][18] and geometric function theory [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”

“…In the Euclidean space ℝ n the n-dimensional Hausdorff measure H n (A) of A ⊂ ℝ n coincides with the n-dimensional Lebesgue measure; see [, Theorem 2.3.4], for instance. 38 Recalling the main result of [6], we can say that it amounts to a "weighted" generalization of the results of [7][8][9][10][11] and [12][13][14][15] as regards 1 < q = p < ∞ and 1 < q < p < ∞.…”