Sobolev Mappings and Moduli Inequalities on Carnot Groups
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Sobolev mappings and moduli inequalities on Carnot groups Evgenii A. Sevost’yanov, Alexander Ukhlov (Presented by V. Ya. Gutlyanskii) Abstract. We study the mappings that satisfy moduli inequalities on Carnot groups. We prove that the homeomorphisms satisfying the moduli inequalities (Q-homeomorphisms) with a locally integrable function Q are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem, we prove that, on the Carnot groups G, the mappings inverse to Sobolev homeomorphisms of finite distortion of the 1 1 (Ω′ ; Ω). (Ω; Ω′ ) belong to the Sobolev class W1,loc class Wν,loc Keywords. Sobolev spaces, moduli inequalities, Carnot group.
1.
Introduction
It is known that the Sobolev mappings on Carnot groups G cannot be characterized only in terms of its coordinate functions. The basic approach to the theory of Sobolev mappings on Carnot groups is based on the notion of absolutely continuity on almost all horizontal lines, which allows one to define a weak upper gradient of mappings. In the present work, we prove that the homeomorphisms satisfying the moduli inequalities on Carnot groups are Sobolev mappings. On this base, we obtain a weak version of the inverse mapping theorem on Carnot groups. Namely, we prove that the mappings inverse to 1 (Ω; Ω′ ) are Sobolev mappings of the Sobolev homeomorphisms of finite distortion of the class Wν,loc 1 (Ω′ ; Ω). The problem of regularity of the mappings inverse to Sobolev homeomorphisms class W1,loc represents a significant part of the weak inverse mapping theorem and was studied in [50] for a bimeasurable Sobolev homeomorphism φ : Ω → Ω′ , Ω, Ω′ ⊂ Rn of the class Wp1 (Ω; Ω′ ), p > n − 1. In [38], it was proved that the homeomorphism inverse to φ ∈ L1p (Ω; Ω′ ), p > n−1, satisfies φ−1 ∈ BVloc (Ω′ ; Ω). In the last decades, the regularity of mappings inverse to Sobolev homeomorphisms was intensively studied in the frameworks of the non-linear elasticity theory [1], see, for example, [8, 14, 17, 18, 29]. The suggested approach on Carnot groups is based on the moduli inequalities, namely on the notion of Q-mappings introduced in [24] (see also [25–26]). Recall that a homeomorphism φ : Ω → Ω′ of domains Ω, Ω′ ⊂ G is called a Q-homeomorphism with a non-negative measurable function Q, if the inequality ∫ M (φΓ) 6
Q(x) · ρν (x)dx Ω
holds for every family Γ of rectifiable paths in Ω and every admissible function ρ for Γ. 1 For the Euclidean space Rn , it was proved [25] that a homeomorphism φ ∈ Wn,loc (Ω) such that −1 1 φ ∈ Wn,loc is a Q-mapping with Q = KI (x, φ), where KI (x, φ) is the inner quasiconformal dilatation of φ. The systematic applications of the moduli theory to the geometric mapping theory can be found in [27]. The main result of the present work concerns the weak differentiability of mappings satisfying the moduli inequalities on Carnot groups (Theorem 5.1). The proof is based on the capacity estimates Translated from Ukrains’ki˘ı Matematychny˘ı Visnyk, Vol. 17, No. 2, pp. 215–233 April–June, 2020. Original article submitted March 27, 202
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