On boundary behavior of generalized quasi-isometries

Kovtonyuk, Denis; Ryazanov, Vladimir

doi:10.1007/s11854-011-0025-8

Cited by 26 publications

(14 citation statements)

References 38 publications

(48 reference statements)

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“…At the same time, this class is a generalization of the known class of mappings with bounded length distortion by Martio-Väisälä from paper [18]. Moreover, this class contains, as a subclass, the so-called finitely bi-Lipschitz mappings introduced for R n , n 2, in paper [10], see also Section 10.6 in book [17], that is, in turn, a natural generalization of the well-known classes of bi-Lipschitz mappings, as well as isometries and quasiisometries.…”

Section: Introductionmentioning

confidence: 88%

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

Ryazanov

Волков

2020

J Math Sci

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The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries.We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Carathéodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Carathéodory prime ends are obtained.

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Section: Introductionmentioning

confidence: 88%

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

Ryazanov

Волков

2020

J Math Sci

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“…Remark 9.1 Note that by Theorem 5.1 and Remark 5.1 in [16] condition (9.2) is not only sufficient but also necessary for a continuous extendibility to the boundary of all mappings f with the integral restriction (9.1).…”

Section: Recall That a Functionmentioning

confidence: 98%

Prime ends on the Riemann surfaces

Ryazanov¹,

Волков²

2017

Dopov. Nac. akad. nauk Ukr.

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The theory of the boundary behavior in the prime ends for the mappings with finite distortion has been developed in [1] and [2] for the plane domains and in [3] for the spatial domains. The pointwise boundary behavior of the mappings with finite distortion in regular domains on Riemann surfaces was recently studied by us in [4]. Moreover, the problem was investigated in regular domains on the Riemann manifolds for 3 n as well as in metric spaces, see, e.g., [5]. For basic definitions and notations, discussions, and historic comments in the mapping theory on Riemann surfaces, see our previous papers [4,6,7].1. Definition of the prime ends and preliminary remarks. We act similarly to Carathéodory under the definition of the prime ends of domains on a Riemann surface S , see Chapter 9 in [8]. First of all, recall that a continuous mapping :We also use the notations , σ σ , and ∂σ for ( ), (

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“…Remark 8.11. Note that condition (8.10) is not only sufficient but also necessary for a continuous extension to the boundary of all direct mappings f with integral restrictions of type (8.9), see, e.g., Theorem 5.1 and Remark 5.1 in [21]. Recall also that condition (8.10) is equivalent to each of conditions (7.13)-(7.17).…”

Section: Boundary Behavior Of Homeomorphic Solutionsmentioning

confidence: 99%

Ring Q-Homeomorphisms at Boundary Points

Gutlyanskiî

Ryazanov

Srebro

et al. 2012

Developments in Mathematics

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We show that every homeomorphic W 1,1 loc solution f of a Beltrami equation ∂f = µ ∂f in a domain D ⊆ C is the so-called ring Q−homeomorphism with Q(z) = K T µ (z, z 0 ) where K T µ (z, z 0 ) is the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point z 0 ∈ D. In this connection, we develop the theory of the boundary behavior of the ring Q−homeomorphisms with respect to prime ends. On this basis, we show that, for wide classes of degenerate Beltrami equations ∂f = µ ∂f , there exist regular solutions of the Dirichlet problem in arbitrary simply connected domains in C and pseudoregular and multivalent solutions in arbitrary finitely connected domains in C with boundary datum ϕ that are continuous with respect to the topology of prime ends.Remark 1.1. This topology can be described in terms of metrics. Namely, as known, every bounded finitely connected domain D in C can be mapped by a conformal mapping g 0 onto the so-called circular domain D 0 whose boundary consists of a finite collection of mutually disjoint circles and isolated points, see, e.g., Theorem V.6.2 in [8]. Moreover, isolated singular points of bounded conformal mappings are removable by Theorem 1.2 in [4] due to Weierstrass. Hence isolated points of ∂D correspond to isolated points of ∂D 0 and inversely.Reducing this case to the Caratheodory theorem, see, e.g., Theorem 9.4 in [4] for simple connected domains, we have a natural one-to-one correspondence between points of ∂D 0 and prime ends of the domain D. Determine in D P the metric ρ 0 (p 1 , p 2 ) = | g 0 (p 1 ) − g 0 (p 2 )| where g 0 is the extension of g 0 to D P just mentioned.If g * is another conformal mapping of the domain D on a circular domain D * , then the corresponding metric ρ * (p 1 , p 2 ) = | g * (p 1 ) − g * (p 2 )| generates the same convergence in D P as the metric ρ 0 because g 0 •g −1 * is a conformal mapping between the domains D * and D 0 that is extended to a homeomorphism between D * and D 0 , see, e.g., Theorem V.6.1 ′ in [8]. Consequently, the given metrics induce the same topology in the space D P .This topology coincides with topology of prime ends described in inner terms of the domain D in Section 9.5 of [4]. Later on, we prefer to apply the description of the topology of prime ends in terms of the given metrics because it is more clear, more convenient and it is important for us just metrizability of D P . Note also that the space D P for every bounded finitely connected domain D in C with the given topology is compact because the closure of the circular domain D 0 is a compact space and by the construction g 0 : D P → D 0 is a homeomorphism.Applying the description of the topology of prime ends in Section 9.5 of [4], we reduce the case of bounded finitely connected domains to Theorem 9.3 in [4] for simple connected domains and obtain the following useful fact.Lemma 1.2. Each prime end P of a bounded finitely connected domain D in C contains a chain of cross-cuts σ m lying on circles S(z 0 , r m ) with z 0 ∈ ∂D and r m → 0 as m →...

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On boundary behavior of generalized quasi-isometries

Cited by 26 publications

References 38 publications

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

Prime ends on the Riemann surfaces

Ring Q-Homeomorphisms at Boundary Points

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