Mappings with Direct and Inverse Poletsky Inequalities (Developments in Mathematics, 78)

Preface
Acknowledgements
Contents
1 General Definitions and Notation
	1.1 Introduction
	1.2 Modulus of Families of Paths
	1.3 Chordal (Spherical) Metric and Weak Sphericalization
	1.4 Condensers and Capacities
	1.5 Sobolev and Orlicz-Sobolev Classes
	1.6 Dilatations
	1.7 Path Liftings
	1.8 Liftings in Metric Spaces
	1.9 Quasiconformal and Quasiregular Mappings
	1.10 Mappings with Finite Distortion
	1.11 Poletsky Inequality
	1.12 Inverse Poletsky Inequality
	1.13 Hausdorff Measure and Surface Integrals
	1.14 Moduli of Families of Surfaces
	1.15 Finite Mean Oscillation
	1.16 Integrals Over Spheres and the Main Lemma on Singular Integrals
	1.17 On Restrictions of Integral Type
	1.18 Ziemer and Hesse Equalities
	1.19 Moduli in Metric Spaces
	1.20 Loewner-Type Estimates in Metric Spaces
	1.21 Loewner Spaces
2 Boundary Behavior of Mappings with Poletsky Inequality
	2.1 Introduction
	2.2 Strongly Accessible Boundaries
	2.3 Main Lemma on Boundary Behavior
	2.4 The Boundary Extension of Mappings Under FMO and Integral Conditions
3 Removability of Singularities of Generalized Quasiisometries
	3.1 Introduction
	3.2 Auxiliary Results
	3.3 Main Results
	3.4 A Case of Integrable Majorant
	3.5 Sharpness of Assumptions in Theorem 3.3
	3.6 Illustrative Examples
4 Normal Families of Generalized Quasiisometries
	4.1 Introduction
	4.2 Main Lemmas
	4.3 Main Results
	4.4 Distortion Estimates for Bounded Ring Q-Mappings with Respect to p-Modulus
	4.5 Applications to Orlicz-Sobolev\'s Spaces
5 On Boundary Behavior of Mappings with Poletsky Inequality in Terms of Prime Ends
	5.1 Introduction
	5.2 Statement of the Main Results
	5.3 Extension of Lower Q-Mappings to the Boundary
	5.4 Boundary Behavior for Another Class of Mappings
	5.5 Equicontinuity of Mappings in the Closure of a Domain with Strongly Accessible Boundary
	5.6 Equicontinuity of Lower Q-Mappings in QED-Domains
	5.7 Some Examples
6 Local and Boundary Behavior of Mappings on Riemannian Manifolds
	6.1 Introduction
	6.2 Orlicz–Sobolev Classes and Poletsky Inequalities on Riemannian Manifolds. The Main Results
	6.3 Normal Families in a Class of Mappings with Unbounded Coefficient of Quasiconformality
	6.4 The Removability of Isolated Singularities of Ring Q-Mappings on Riemannian Manifolds
	6.5 The Normality of Orlicz–Sobolev Classes on Riemannian Manifolds
	6.6 Some Examples
7 Local and Boundary Behavior of Maps in Metric Spaces
	7.1 Introduction
	7.2 Equicontinuity of Homeomorphisms in Metric Spaces
	7.3 Equicontinuity of Families of Open Discrete Mappings
	7.4 Removal of Isolated Singularities
	7.5 Boundary Behavior of Ring Q-Mappings in Metric Spaces
	7.6 Examples and Open Problems
8 On Sokhotski-Casorati-Weierstrass Theorem on Metric Spaces
	8.1 Introduction
	8.2 Preliminaries
	8.3 On Capacity Estimates Through Chordal Diameter
	8.4 The Main Lemma
	8.5 Removability of Singularities in Ptolemaic Spaces
9 On Boundary Extension of Mappings in Metric Spaces in Terms of Prime Ends
	9.1 Introduction
	9.2 Main Lemma
	9.3 Proof of the Main Result
	9.4 Homeomorphic Extension to the Boundary
	9.5 Equicontinuity of Families of Homeomorphisms
	9.6 Equicontinuity of Families of Maps with Branching
10 On the Openness and Discreteness of Mappings with the Inverse Poletsky Inequality
	10.1 Introduction
	10.2 Main Lemma
	10.3 Proof of the Main Result
	10.4 Corollaries
	10.5 Examples
11 Equicontinuity and Isolated Singularities of Mappings with the Inverse Poletsky Inequality
	11.1 Introduction
	11.2 Preliminaries
	11.3 Proof of Theorem 11.1
	11.4 On the Behavior of Mappings in the Closure of a Domain
	11.5 On Isolated Singularities of Inverse Mappings
	11.6 Some Examples
12 Equicontinuity of Families of Mappings with the Inverse Poletsky Inequality in Terms of Prime Ends
	12.1 Introduction
	12.2 Preliminaries
	12.3 Proof of Theorem 12.1
	12.4 Examples
13 Logarithmic Hölder Continuous Mappings and Beltrami Equation
	13.1 Introduction
	13.2 The Main Results
	13.3 The Case When a Majorant Is Integrable over Some Set of Spheres
	13.4 Logarithmic Hölder Continuity of Mappings of the Unit Ball
	13.5 On Logarithmic Hölder Type Maps in Arbitrary Domains
	13.6 Applications to the Beltrami Equation
14 On Logarithmic Hölder Continuity of Mappings on the Boundary
	14.1 Introduction
	14.2 Boundary Behavior of Mappings with the Inverse Poletsky Inequality
	14.3 Auxiliary Lemma and Proof of Theorem 14.1
15 The Poletsky and Väisälä Inequalities for the Mappings with (p, q)-Distortion
	15.1 Introduction
	15.2 The Proof of the Main Results
	15.3 The Analog of the Väisälä Inequality
	15.4 Applications
16 An Analog of the Väisälä Inequality for Surfaces
	16.1 Introduction
	16.2 Proofs of Main Results
17 Modulus Inequalities on Riemannian Surfaces
	17.1 Introduction
	17.2 Preliminaries
	17.3 Basic Bound for Distortion of the Modulus of Families of Paths
	17.4 Boundary Extension of Mappings with Lower ModulusDistortion
	17.5 Equicontinuity of Families Homeomorphisms
	17.6 Equicontinuity of Sobolev Classes with Branching
18 On the Local and Boundary Behavior of Mappings of Factor Spaces
	18.1 Introduction
	18.2 Preliminaries
	18.3 Proof of Theorem 18.1
	18.4 On Estimates of the Modulus of Families of Paths in the Preimage Under the Mapping
	18.5 Local and Boundary Behavior of Mappings on Factor Spaces
References
Index