On the Analytic and Geometric Properties of Mappings in the Theory of $$\mathscr Q_{q,p}$$ -Homeomorphisms
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On the Analytic and Geometric Properties of Mappings in the Theory of Qq,p -Homeomorphisms S. K. Vodop’yanov1* 1
Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, 630090 Russia Received June 30, 2020; in final form, June 30, 2020; accepted July 1, 2020
DOI: 10.1134/S0001434620110309 Keywords: quasiconformal analysis, Sobolev space, capacity of condenser, regularity of mapping.
This note is devoted to the discussion of properties of Qq,p -homeomorphisms introduced in [1, Definition 2] for q = p. The generalized version is defined as follows. Definition 1 (see [2]). We say that a homeomorphism f : D → D, where D, D ⊂ Rn , n ≥ 2, belongs to the class Qq,p (D , ω), where 1 < q ≤ p < ∞ and ω ∈ L1,loc (D ) is a weight function, if there exists (1) a constant Kp (if q = p) or (2) a bounded quasi-additive function Ψq,p defined on open sets in D (if q < p) such that, given any condenser E = (F, U ) in D with image f (E) = (f (F ), f (U )) in D, we have if q = p, cap1/p (f (E); L1p (D)) ≤ Kp cap1/p (E; L1p (D ; ω)) 1/q 1 1/σ 1/p 1 cap (f (E); Lq (D)) ≤ Ψq,p (U \ F ) cap (E; Lp (D ; ω)) if q < p.
(1)
A condenser E = (F, U ) is said to be annular if the boundary of U \ F is formed by concentric spheres, that is, U = B(x, R) D and F = B(x, r), r ∈ (0, R). Requiring condition (1) to hold only for annular condensers, we obtain a larger class of homeomorphisms f : D → D, which we denote by RQq,p (D , ω), where 1 < q ≤ p < ∞ and ω ∈ L1,loc (D ) is a weight function. Obviously, Qq,p (D , ω) ⊂ RQq,p (D , ω). This class includes • quasiconformal mappings, which correspond to q = p = n and ω ≡ 1 (see, e.g. [3]); • Qp -homeomorphisms defined in [4], which correspond to n − 1 < q < p = n and ω ≡ 1; • (P, Q)-quasiconformal mappings defined in [5]–[7], which correspond to 1 < q ≤ p < ∞ and ω ≡ 1; • Q-homeomorphisms defined in the paper [8] and the monograph [9], which correspond to q = p = n. *
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VODOP’YANOV
Thus, Definition 1 suggests a generalizing concept covering, as special cases, many classes of mappings that have been studied during the past seven decades. In this paper, we supplement the examples of [1] with several more subclasses of the family Qp,p (D , ω). Any locally integrable function ω : D → (0, ∞), where D ⊂ Rn is an open set, determines the space L1 (D ; ω) of measurable functions u : D → R integrable with weight ω in the domain D , i.e., satisfying the condition u | L1 (D ; ω) = u · ω | L1 (D ) < ∞, where L1 (D ) is space of Lebesgue integrable functions on D . We also define the weighted measure of a measurable set A ⊂ D by ω(A) = ω | L1 (A; ω) . Recall that a function u : D → R defined on an open set D ⊂ Rn belongs to the Sobolev weight class L1p (D; ω), 1 ≤ p < ∞, if u ∈ L1,loc (D), its generalized partial derivatives are integrable on D (i.e., ∂u/dxj ∈ L1 (D) for all j = 1, . . . , n), and its seminorm u | L1p (D; ω) = ∇u | Lp (D; ω) is finite. In the case ω ≡ 1, we write
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