On mappings with finite length distortion on Riemann surfaces

2019 ◽  
Vol 33 ◽  
Author(s):  
Serhii Volkov ◽  
Vladimir Ryazanov
Keyword(s):  
Finite Length ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Main Tool ◽  
Natural Generalization ◽  
Finite Distortion ◽  
Lipschitz Mappings ◽  
Geometric Function ◽  
Mappings With Finite Distortion ◽  
Length Distortion

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

scholarly journals Prime ends in theory of mappings with finite distortion in the plane

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1349-1366 ◽  
Author(s):  
Denis Kovtonyuk ◽  
Igor Petkov ◽  
Vladimir Ryazanov
Keyword(s):  
Boundary Behavior ◽  
Natural Generalization ◽  
Finite Distortion ◽  
Beltrami Equations ◽  
Lipschitz Mappings ◽  
Prime Ends ◽  
The Given ◽  
Mappings With Finite Distortion ◽  
Homeomorphic Extension ◽  
Effective Conditions

In the present paper, it is studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi-Lipschitz mappings that a far-reaching extension of the known classes of isometric and quasiisometric mappings.

scholarly journals On connection of finite distortion mappings with length distortion in $\mathbb{R}^n$

2021 ◽  
Vol 17 ◽  
pp. 112
Author(s):  
Ye.A. Sevostianov
Keyword(s):  
Finite Length ◽  
Measure Zero ◽  
Finite Distortion ◽  
Mappings With Finite Distortion ◽  
Length Distortion

The present paper is devoted to the investigations of mappings with finite distortion in $\mathbb{R}^n$, $n \geqslant 2$. In the work it is proved that every open discrete mapping with finite distortion by Iwaniec such that the branch set of $f$ is of measure zero is a mapping with finite length distortion provided that the corresponding outer dilatation satisfies the inequality $K_O (x, f) \leqslant K(x)$ a.e., where $K(x) \in L_{loc}^{n-1}(D)$.

Mappings with finite length distortion and prime ends on Riemann surfaces

2020 ◽  
Vol 17 (1) ◽  
pp. 60-76
Author(s):  
Vladimir Ryazanov ◽  
Sergei Volkov
Keyword(s):  
Finite Length ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Continuous Extension ◽  
Mean Deviation ◽  
Generalized Derivatives ◽  
Sobolev Classes ◽  
Inverse Mapping ◽  
Prime Ends ◽  
Length Distortion

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 Caratheodory 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 Caratheodory prime ends are obtained.

Caratheodory theorem about prime ends on Riemann surfaces

2021 ◽  
Vol 34 ◽  
pp. 100-110
Author(s):  
Vladimir Ryazanov ◽  
Serhii Volkov
Keyword(s):  
Finite Length ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Bounded Mean Oscillation ◽  
Mean Deviation ◽  
General Criterion ◽  
Boundary Points ◽  
Mean Oscillation ◽  
Prime Ends ◽  
Length Distortion

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class 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 monograph of these authors in the modern mapping theory at 2009. As it 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 quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

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