Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics [1st ed. 2019] 978-3-030-22590-2, 978-3-030-22591-9

Table of contents :
Front Matter ....Pages i-xvii
Introduction (Vesna Todorčević)....Pages 1-4
Quasiconformal and Quasiregular Harmonic Mappings (Vesna Todorčević)....Pages 5-56
Hyperbolic Type Metrics (Vesna Todorčević)....Pages 57-83
Distance Ratio Metric (Vesna Todorčević)....Pages 85-110
Bi-Lipschitz Property of HQC Mappings (Vesna Todorčević)....Pages 111-129
Quasi-Nearly Subharmonic Functions and QC Mappings (Vesna Todorčević)....Pages 131-146
Possible Research Directions (Vesna Todorčević)....Pages 147-152
Back Matter ....Pages 153-163

Citation preview

Vesna Todorčević

Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics

Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics

Vesna Todorˇcevi´c

Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics

123

Vesna Todorˇcevi´c Faculty of Organizational Sciences University of Belgrade Belgrade, Serbia Mathematical Institute Serbian Academy of Sciences and Arts Belgrade, Serbia

ISBN 978-3-030-22590-2 ISBN 978-3-030-22591-9 (eBook) https://doi.org/10.1007/978-3-030-22591-9 Mathematics Subject Classification: 30C65, 30C62, 31B05, 31B15, 31B25 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The goal of this book is to present a research area in Geometric Function Theory concerned with harmonic quasiconformal mappings and hyperbolic type metrics defined on planar and multidimensional domains. The classes of quasiconformal and quasiregular mappings are well-established areas of study in this field as these classes are natural and fruitful generalizations of the class of analytic functions in the planar case. Harmonic mappings are another natural generalization of conformal mappings and analytic functions and form another well-established class. Injective quasiregular mappings are quasiconformal, and conformal mappings are both harmonic and quasiconformal. On the other hand, harmonic mappings are smooth, and when quasiregular, they are also locally quasiconformal independently of the dimension. So in higher dimensions, the study of the class of mappings that are both harmonic and quasiconformal suggests itself. It turns out that while this seems at first a rather restrictive class, the study of this class uncovers new and unexpected phenomena and is today recognized as an important research area in Geometric Function Theory. The book contains many concrete examples, as well as detailed proofs and explanations of motivations behind given results, gradually bringing the reader to the forefront of current research in the area. The book is written for a wide readership from graduate students of mathematical analysis to researchers working in this or related areas who want to learn the tools or work on the open problems, many of which are listed in various parts of the book, especially in the last chapter. An extensive bibliography of the field is also given for the readers who wish to explore deeper into the results presented in the book or related results that are not covered here. Prerequisite knowledge for reading this book includes the basic knowledge of real and complex analysis, harmonic functions, and the topology of metric spaces. The book is primarily based on research done in the last 12 years, starting with the author’s master and doctoral dissertations and followed by a number of papers that are either single authored or jointly authored with other experts in the field. The author is therefore grateful to all her collaborators

v

vi

Preface

and other mathematicians who have built this research area and have shared their expertise with enthusiasm. Without their help, this book would not have come into its existence. Belgrade, Serbia January 2019

Vesna Todorˇcevi´c

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Quasiconformal and Quasiregular Harmonic Mappings. . . . . . . . . . . . . . . . 2.1 Moduli of Curve Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition of Quasiconformal Mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hölder Continuity of Quasiconformal Mappings. . . . . . . . . . . . . . . . . . . . . . 2.4 Moduli of Continuity in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Moduli of Continuity in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 An Example of Non-Lipschitz HQC Mapping on the Unit Ball . . . . . . 2.7 Hölder Continuity of HQC Mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Subharmonicity of the Modulus of HQR Mappings. . . . . . . . . . . . . . . . . . .

5 5 14 23 31 40 43 46 52

3

Hyperbolic Type Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Möbius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Chordal Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hyperbolic Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Distance Ratio Metric jD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Quasihyperbolic Metric kD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Other Hyperbolic Type Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Quasiconformal Mappings and kD and jD Metrics . . . . . . . . . . . . . . . . . . . 3.8 Quasiconformal Mappings with Identity Boundary Values . . . . . . . . . . .

57 58 60 63 67 70 73 74 77

4

Distance Ratio Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1 Refinements of the Gehring–Osgood Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Lipschitz Continuity and Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5

Bi-Lipschitz Property of HQC Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Bi-Lipschitz Property of HQC Mappings in Plane . . . . . . . . . . . . . . . . . . . . 5.2 When Part of Boundary Is Flat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions. . . . .

111 111 115 119

vii

viii

Contents

6

Quasi-Nearly Subharmonic Functions and QC Mappings . . . . . . . . . . . . . . 6.1 Quasi-Nearly Subharmonic Functions and Conformal Mappings . . . . 6.2 Regularly Oscillating Functions and Conformal Mappings . . . . . . . . . . . 6.3 Quasi-Nearly Subharmonic Functions and QC Mappings . . . . . . . . . . . . 6.4 Regularly Oscillating Functions and QC Mappings . . . . . . . . . . . . . . . . . . . 6.5 Some Generalizations and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 133 134 144 145

7

Possible Research Directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.1 Characterizations of Boundary Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2 QC and HQC Mappings on Non-smooth Domains. . . . . . . . . . . . . . . . . . . . 149

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Notation

Z N R C |z| arg(z) Rn Rn = Rn ∪ {∞} |x| Bn (x, r) Vn S n−1 (x, r) ωn−1 Hn Sρ P (a, t) G M (Rn ) M3 (R) π(x) q(x, y) Q(x, r) |a, b, c, d| a∗ D(a, M) dist(x, A) ∂A diam(A) χA l(γ ) ρ(x, y) jD

Set of integers Set of positive integers Set of real numbers Set of complex numbers Modulus of the complex number z Argument of the complex number z n-dimensional Euclidean space Möbius space Euclidean norm of a vector x ∈ Rn Open ball centered at x ∈ Rn with radius r > 0 Volume of the n-dimensional unit ball Sphere centered at x ∈ Rn with radius r > 0 (n − 1)-dimensional measure of S n−1 (0, 1) Poincare half-space Planar angular domain (n − 1)-dimensional hyperplane Group of Möbius transformations Set of square matrices of order 3 Stereographic projection Spherical (chordal) distance between x and y Spherical ball (Absolute) cross ratio Image of the point a under an inversion on S n−1 Hyperbolic ball with center a and radius M Distance of a point x ∈ Rn to a set A ⊆ Rn Boundary of a set A ⊆ Rn Diameter of a set A ⊆ Rn Characteristic function of a set A ⊆ Rn Length of the curve γ Hyperbolic distance between x and y Distance ratio metric ix

x

kD δG αG sG λG μG Δ(E, F ; G) RG,n (s) RT ,n (s) γ (s), γn (s) τ (s), τn (s) ϕK (r), ϕK,n (r) log Φn (s) log ψn (s) λn mod(R) cap(R) capW (K) p-cap E, cap E Λα (F ) N(f, A, y) Lip(f ) K(f ), KO (f ), KI (f ) H (x, f ) Mp (Γ ), M(Γ ) I dK (∂Bn) C 1 (C 2 ) CC∞ (Ω) ∇f Df Jf (x) αf (z) Hu ||Du(x)|| Δf Lp (Ω) ||f ||Lp p Lloc W 1,p (Ω) 1,p Wloc (Ω) P (x, ξ ) GBn (x, y)

Notation

Quasihyperbolic metric Seittenranta metric Apollonian metric Triangular ratio metric Ferrand metric Modulus metric Family of all closed nonconstant curves joining E, F in G Grötzch ring Teichmüller ring Capacity of RG,n (s) Capacity of RT ,n (s) Special function related to the Schwarz lemma Modulus of the Grötzsch ring Modulus of the Teichmüller ring Grötzsch ring constant Modulus of a ring Capacity of a ring Wiener capacity (p-)capacity of a condenser α-dimensional Hausdorff measure of F Number of preimages of point y in A under f Lipschitz constant of f Maximal, outer, and inner dilatation of f Linear dilatation of a mapping f at x (p-)modulus of the curve Γ K-qc maps with identity boundary values Class of functions with continuous first-order (secondorder) derivatives Class of compactly supported functions with derivatives of all orders Gradient of mapping f : Ω −→ Rn Weak derivative of mapping f Jacobian of mapping f : Ω −→ Rn at x Average derivative Hessian of u Hilbert–Schmidt norm Laplacian of f Lebesgue space Lp -norm of function f Local Lebesgue space Sobolev space Local Sobolev space Poisson kernel for the unit ball Green function on the unit ball

Notation

Is h QNSK (Ω) QNS(Ω) RO(Ω) ||u||QN S ||u||RO ||u||BMO

xi

Riesz potential of order s Class of all quasi-nearly subharmonic functions for a fixed K Class of all quasi-nearly subharmonic function defined in Ω Class of regularly oscillating functions in Ω QNS-norm of function u RO-norm of function u BMO-norm of function u

Preliminaries

The background material necessary for reading this book can be found in standard texts of this area of mathematics such as [23, 138] and [155]. We shall also try to follow the standard terminology and notation as much as possible, but in this chapter, we list the notation and the terminology that are specific for this book. A pair (X, d) is called a metric space if X = ∅ and d : X × X → [0, +∞) satisfies the following four conditions: (M1) (M2) (M3) (M4)

d(x, y)  0, for all x, y ∈ X, d(x, y) = 0 iff x = y, d(x, y) = d(y, x) for all x, y ∈ X, d(x, z)  d(x, y) + d(y, z) for all x, y, z ∈ X.

A pair (X, d) is called a pseudometric space if X = ∅ and d : X ×X → [0, +∞) satisfies the following four conditions: (M1) d(x, y)  0, for all x, y ∈ X, (M2’) d(x, x) = 0, (M3) d(x, y) = d(y, x) for all x, y ∈ X, (M4) d(x, z)  d(x, y) + d(y, z) for all x, y, z ∈ X. The inner product (a1 , . . . , an ), (b √1 , . . . , bn ) is defined to be equal to the sum a1 b1 + · · · + an bn , and we let |a| = a, a . Let (X, d1 ) and (Y, d2 ) be metric spaces, and let f : X → Y be a continuous mapping. Then, we say that f is uniformly continuous if there exists an increasing continuous function ω : [0, ∞) → [0, ∞) with ω(0) = 0 such that d2 (f (x), f (y)) ≤ ω(d1 (x, y)) for all x, y ∈ X. We call the function ω the modulus of continuity of f. If there exist C, α > 0 such that ω(t) ≤ Ct α for all t ∈ (0, 1) , we say that f is Hölder continuous with xiii

xiv

Preliminaries

Hölder exponent α. If α = 1 , we say that f is Lipschitz with the Lipschitz constant C or simply C-Lipschitz. If f is a homeomorphism and both f and f −1 are CLipschitz, then f is C-bi-Lipschitz or C-quasiisometry, and if C = 1, we say that f is an isometry. These conditions are said to hold locally if they hold for each compact subset of X. A very special case of these mappings are the isometries. Recall that if (X1 , d1 ) and (X2 , d2 ) are metric spaces and f : X1 → X2 a homeomorphism, then we call f an isometry if d2 (f (x), f (y)) = d1 (x, y) for all x, y ∈ X1 . We use the notation B n (x, r) = {y ∈ Rn : |x − y| < r}, S n−1 (x, r) = {y ∈ Rn : |x − y| = r}, Hn = {(x1 , . . . , xn ) ∈ Rn : xn > 0} and abbreviations B n (r) = B n (0, r), Bn = B n (1), S n−1 (r) = S n−1 (0, r), and S n−1 = S n−1 (1). Recall that the volume of the n-dimensional unit ball can be expressed as n

π2 , Vn = Γ ( n2 + 1) where Γ is Euler gamma function. Recall also that if by ωn−1 , we denote the (n−1)dimensional measure of the sphere S n−1 , then ωn−1 = nVn . A Möbius transformation in complex plane is a mapping of the form z →

az + b , cz + d

where a, b, c, and d are complex constants such that ad = bc. The homeomorphism f : D −→ D  is called conformal, where D and D  are domains in Rn , if f is C 1 (D), if Jf (x) = 0 for all x ∈ D, and if |f  (x)h| = |f  (x)| |h| for all x ∈ D and h ∈ Rn . An anticonformal is the complex conjugate of a conformal mapping. If D and D  are domains in Rn , a homeomorphism f : D −→ D  is conformal if its restriction to D \ {∞, f −1 (∞)} is conformal. This definition of conformal homeomorphism is preferable to others, since if f is conformal, then f −1 is conformal as well because Jf (x) = 0. n Let D and D  be domains in R . We call a C 1 -homeomorphism f : D −→ D  sense-preserving (orientation-preserving) if Jf (x) > 0 for all x ∈ D \

Preliminaries

xv

{∞, f −1 (∞)}. If Jf (x) < 0 for all x ∈ D \ {∞, f −1 (∞)}, then we call f sensereversing (orientation-reversing). The mapping f : D −→ D  is regular at x ∈ D when f is differentiable at x and Jf (x) = 0. A bijective mapping f : G1 → G2 between domains G1 and G2 in Rn is a diffeomorphism if f and f −1 are continuously differentiable. Note that if f is a diffeomorphism, then Jf (x) = 0 for all x ∈ G1 . Let D and D  be domains in Rn . The Hessian of a function f : D −→ Rn is defined to be   n ∂ 2f det . ∂xi ∂xj i,j =1 A function f : D −→ R is called harmonic if f ∈ C 2 (D) and Δf :=

n  ∂ 2f i=1

∂xi2

= 0.

The operator Δ is called Laplacian. The set of harmonic functions forms a linear space. A complex valued function f : Ω −→ C, where Ω is domain in Rn , is called harmonic if both of its real and imaginary parts are harmonic functions in Ω. Let X, Y be domains in Rn . If a function f : X −→ Y belongs to C 2 (X) and if each of its coordinate functions is harmonic, then we say that f is harmonic. Note that the real and imaginary parts of the complex analytic function are harmonic functions. Note that if f is harmonic and g analytic, then the composition f ◦g is a harmonic function, but the composition g ◦ f need not be harmonic in general. Note also that all conformal mappings are analytic and that, in the plane, all analytic functions are harmonic. Additionally, in the plane, each harmonic function is locally the real (imaginary) part of some analytic function. The Harnack inequality for positive harmonic function u : G −→ (0, ∞) states that for every s ∈ (0, 1), there is C ≥ 1 such that max u(z) ≤ C min u(z),

z∈Bx

z∈Bx

n

holds, whenever B n (x, r) ⊆ G and Bx = B (x, sr). Green’s function on the unit ball is defined by G(x, y) = Φ(y − x) − Φ(|x|(y − x ∗ )),

x, y ∈ Bn ,

x = y.

Here, ωn−1 is the (n − 1)-dimensional measure of S n−1 , x ∗ = |x|−2 x, and the function

xvi

Preliminaries

 Φ(x) =

1 − 2π log |x|,

n = 2,

1 1 n(n−2)ωn−1 |x|n−2 ,

n ≥ 3,

defined for x ∈ Rn , x = 0 is the fundamental solution of the Laplace’s equation. Recall that P (x, ξ ) =

1 − |x|2 |x − ξ |n

is the Poisson kernel for the unit ball in Rn . Recall also that a continuous function u : G −→ R defined on a domain G ⊂ C is subharmonic if for all z0 ∈ G, there exists ε > 0 such that u(z0 ) ≤

1 2π



2π

u(z0 + reit ) dt for 0 < r < ε.

(*)

0

A continuous function u : G −→ R defined on a domain G ⊂ C is superharmonic if −u is subharmonic. Let a and b are reals such that a < b. For path α : [a, b] −→ Rn , we define its length l(α) as the supremum of values of the form n 

|α(ti ) − α(ti−1 )|

i=1

where n ∈ N and a = t0 < · · · < tn = b. If l(α) < ∞, then we say that α is a rectifiable path. If α is a rectifiable path, then there is a unique path α 0 : [0, l(α)] −→ Rn such that there is a continuous increasing function h : [a, b] −→ [0, l(α)] such that α = α 0 ◦ h and for any t ∈ [0, l(α)], l(α 0 |[0,t] ) = t. We call the path α 0 a normal representation of α. Let I ⊆ R is interval and f : I −→ Rn . We say that f is absolutely continuous if for every ε > 0, there is δ > 0 such that for every m ∈ N and for every sequence (a1 , b1 ), . . . , (am , bm ) of pairwise disjoint subintervals of I, we have that n  i=1

|ai − bi | < δ ⇒

n 

|f (ai ) − f (bi )| < ε.

i=1

A Jordan curve or simple closed path in metric space (X, d) is a continuous (with respect to the metric d) mapping γ : [0, 1] −→ X such that for all x, y ∈ [0, 1] with x < y,

Preliminaries

xvii

γ (x) = γ (y) ⇔ x = 0 ∧ y = 1. A plane domain is Jordan if its boundary is a Jordan curve. For higher dimensions n, we say that a domain in Rn is a Jordan domain if its boundary is homeomorphic to the unit sphere. We denote the α-dimensional Hausdorff measure of a set F ⊂ Rn by Λα (F ). Recall that Λδα (F ) = inf{

∞ 

d(Ui )α },

i=1

where the infimum is taken over all countable coverings of F by sets Ui with d(Ui ) < δ, then set Λα (F ) = limδ→0 Λδα (F ). The Hausdorff dimension of a set F is defined as follows dimH (F ) = inf{α : Λα (F ) < ∞}. The β-dimensional Hausdorff content is defined to be  Λ (E) = inf β

∞ 

β ri

,

i=1

where the infimum is taken over all coverings of E ⊆ Rn with countably many (Euclidean) balls of radii ri . Let G ⊂ Rn be a domain, and let w : G −→ (0, ∞) be continuous. For given x, y ∈ G, let  dw (x, y) = inf{lw (γ ) : γ ∈ Γxy , l(γ ) < ∞},

lw (γ ) =

w(γ (z))|dz|. γ

It turns out that this is a metric on G. If a length-minimizing curve exists, it is called a geodesic. For a proper subdomain G of Rn , the quasihyperbolic length of a rectifiable curve γ in G is given by  lk (γ ) = γ

|dz| . d(z, ∂G)

The quasihyperbolic distance between points x and y from G is the infimum of quasihyperbolic lengths over all rectifiable curves in G joining x and y. For an easy reference, we also record the Bernoulli inequality, log(1 + as)  a log(1 + s),

a  1, s > 0.

Chapter 1

Introduction

Geometric Function Theory began as a branch of Complex Analysis dealing with geometric aspects of analytic functions, but has since grown considerably, both in scope and in methodology. It considers, for example, the class of quasiregular mappings proven to be a natural and especially fruitful generalization of analytic functions in the planar case. Another class considered is the class of quasiconformal mappings characterized by the property1 that there is a constant C ≥ 1 such that infinitesimal spheres are mapped onto infinitesimal ellipsoids in such a manner that the ratio of the longest axis to the shortest axis is bounded from above by C. Injective quasiregular mappings are quasiconformal and conformal mappings in the plane are both harmonic and quasiconformal. Moreover, harmonic mappings are smooth and if they are also quasiregular they are locally quasiconformal in higher dimensions. This gives us a motivation to study harmonic quasiconformal mappings in higher dimensions. Today the study of these classes of mappings is recognized as an important research area of Geometric Function Theory. In more precise analytic terms, a quasiconformal map f : X → Y is a homeomorphism of two domains in Rn that is differentiable almost everywhere, n such that f belongs to Sobolev space W1,loc and there is a uniform bound K on the ratio of the largest and smallest absolute value of eigenvalue of a differential of f , valid almost everywhere in X. There are many other alternative definitions of quasiconformal mappings that use, for example, moduli of families of curves or linear dilatation which are more geometric in nature showing that quasiconformality is a fruitful notion. The equivalence of geometric and analytic definitions of quasiconformal mappings has been established for quite some time. The two-dimensional quasiconformal theory was developed by mathematicians including Lars Ahlfors, Lipman Bers, Oswald Teichmüller, Frederick Gehring, and William Thurston in the 20th century. Quasiconformal mappings have compactness properties similar to conformal mappings. For instance, they can be used to form 1 In

Chap. 2, we will formally define the notions of quasiconformal and quasiregular mappings.

© Springer Nature Switzerland AG 2019 V. Todorˇcevi´c, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, https://doi.org/10.1007/978-3-030-22591-9_1

1

2

1 Introduction

normal families of functions under quite general conditions, which gives them a special place in Geometric Function Theory. The theory of two-dimensional quasiconformal mappings has applications in numerous areas of mathematics such as Teichmüller theory of Riemann surfaces, Complex Dynamics [30] (the famous No Wandering Domains Theorem of Dennis Sullivan), low dimensional Topology, as well as in Physics (String Theory). Higher dimensional applications have been less versatile, due to the rigidity of quasiconformal mappings, discovered by George Mostow in 1968, [119]. Harmonic mappings are another natural generalization of conformal mappings and analytic functions. These are used extensively in the study of Teichmüller space. The first significant application was given by M. Wolf [161]. Later R. Schoen [139] posed a well-known conjecture stating that every quasisymmetric homeomorphism u : ∂H2 → ∂H2 admits a unique harmonic quasiconformal extension f : H2 → H2 . This conjecture was resolved recently by V. Markovi´c in [108] and his result can be used to find a canonical HQC representative for each class in the Universal Teichmüller space. Since the theories of harmonic mappings and quasiconformal mappings are both well developed, it is of interest to consider how the corresponding results can be strengthened in the presence of both harmonicity and quasiconformality [1, 74, 75, 85, 102, 117]. While these definitions impose strong limitations, some of the results are unexpected and elegant. One such result is, for example, the preservation of boundary modulus of continuity in the case of the unit ball given in [15]. Harmonic quasiconformal (abbreviated as HQC) mappings in the plane were first introduced by Olli Martio in [110]. Today they are investigated both in the planar and in the multidimensional setting from several different points of view. Unfortunately, the powerful machinery developed for the plane is not available in the space, and so our approach is to combine the analytic and geometric aspects of the theory of quasiconformal mappings together with a number of tools from Harmonic Analysis. Among the topics considered in this book are the boundary behavior, including Hölder and Lipschitz continuity, and the more general moduli of continuity, behavior with respect to natural metrics, especially quasihyperbolic metric, distortion estimates, bi-Lipschitz properties with respect to different metrics, and characterization of boundary mappings. We shall also explain an array of tools used in this study such as the conformal moduli of families of curves, Poisson kernels, estimates from the theory of second order elliptic operators, notions of capacity, subharmonic functions, and Riesz potentials. In Chap. 2 we introduce basic notions and examples on which the rest of the book relies. Conformal invariants, such as harmonic measure, hyperbolic distance, condenser capacity, and modulus of a curve family are fundamental tools of Geometric Function Theory. The theory of quasiconformal mappings in Rn , n ≥ 2 makes effective use of all these tools. For example, even the definition of quasiconformal mappings can be expressed in terms of moduli of families of curves. (For the planar case n = 2 the reader can see this in the L. Ahlfors book [5] and in the higher dimensional case in the J. Väisälä book [155].) The relationship between moduli of continuity of a harmonic quasiregular mapping on the boundary and

1 Introduction

3

inside the ball in dimension n = 2 is given in [88] and in higher dimensions in [15]. Chapter 2 also contains an example of a non-Lipschitz harmonic quasiconformal mapping on the unit ball. In [16] it is shown that for a wide range of domains, including those with uniformly perfect boundary, Hölder continuity on the boundary implies Hölder continuity (with the same exponent) inside the domain for the class of HQC mappings, a result which does not hold for the class of qc mappings. In Chap. 3 we introduce some hyperbolic type metrics and explain their connection with the theory of qc mappings. This turns out to be a fruitful theme explored by a number of authors. The geometric properties of domains have been another central theme of research, especially in relation to the boundary phenomena. The geometric nature of the boundary is reflected by the quasihyperbolic metric introduced by Gehring and Palka [54] as a tool for the study of quasiconformal homogeneity. It turns out that the quasihyperbolic metric is invariant under Euclidean similarities, but it is not invariant under conformal mappings, not even under Möbius transformations. From Gehring and Osgood’s result [53] it follows that for each domain Ω ⊆ Rn and points x, y ∈ Ω, there exists a quasihyperbolic geodesic and that the quasihyperbolic metric is quasiinvariant under quasiconformal mappings. Another hyperbolic type metric of interest is the distance ratio metric jG . The hyperbolic metric in the unit ball or half space is Möbius invariant. However, the distance ratio metric is not invariant under Möbius transformations. Therefore it is natural to ask what is its Lipschitz constant under conformal mappings or Möbius transformations in higher dimensions. Gehring and Osgood proved that this metric is not changed by more than a factor 2 under Möbius transformations. In Chap. 4 we present a refinement of this result given by Simi´c and Vuorinen [147]. It turns out that the factor 2 can be improved in the cases of the unit ball and the punctured unit ball in Rn and that, in fact, the best possible factors can be identified. The Gehring–Osgood theorem provides a Hölder-type estimate for the modulus of continuity of quasiconformal mappings with respect to the quasihyperbolic metrics of the domain and the target domain of the mapping. As shown by Vuorinen in [157, Example 3.10], there is no counterpart of this result for analytic functions in the plane. Recall that the famous Schwarz–Pick lemma provides a Lipschitz-type modulus of continuity estimate for analytic functions of the unit disk into itself with respect to the hyperbolic metrics of the domain and target disks. In recent years there has been a large amount of activity in the study of hyperbolic type metrics [39, 58, 60, 81, 82, 97]. Chapter 5 includes bi-Lipschitz properties of harmonic quasiconformal mappings in the planar and the higher dimensional case. The author has shown in [99] that HQC mappings between any two proper domains in the plane are bi-Lipschitz with respect to the corresponding quasihyperbolic metrics. In the course of proving this, the author also showed that a sense preserving harmonic mapping between two planar domains has a superharmonic logarithm of the Jacobian. This theorem has found some applications on its own. For instance, Tadeusz Iwaniec [70] used this result in establishing the minimum principle for the Jacobian determinant, a remarkable novelty which leads us to the new analytic proof of the celebrated Radó– Kneser–Choquet theorem. The result from author’s paper [99] was also used for

4

1 Introduction

higher dimensional generalizations of the Pavlovi´c’s bi-Lipschitz condition. In the joint paper [21] of K. Astala and the author the bi-Lipschitz property is proved for gradient harmonic quasiconformal mappings in the unit ball B3 . However, in higher dimensions, Pavlovi´c’s approach seems difficult to apply. The Lipschitz property follows from the regularity theory of elliptic PDEs established by Kalaj [76] and by a simple and self-contained argument that works for all dimensions given in [21]. The co-Lipschitz condition is much more difficult to tackle, as it is not even known to hold in higher dimensions when the HQC mappings have nonvanishing Jacobian. Indeed, a famous example by J. C. Wood [162] shows that Jacobian can vanish for harmonic injective mappings in dimensions higher than two. In Chap. 6 we present a result of the author from [87] which solves a problem posed by Pavlovi´c about the functions that are quasi-nearly subharmonic (QNS) in the plane by showing that this class is conformally invariant. An analogous result for regularly oscillating (RO) functions is also proved in the same paper. These results motivated Riihentaus (who introduced the QNS class) and Dovgoshey to partially extend these results to the class of bi-Lipschitz mappings [37]. Since biLipschitz mappings are quasiconformal, the general problem of invariance of QNS and RO classes remained open. In cooperation with P. Koskela, the author solved this problem in [90] by showing that both classes remain invariant under quasiregular mappings with bounded multiplicity, which includes the quasiconformal case. In the paper [90] the problem of composition u ◦ φ is solved, where u is QNS and φ is QC, not only in the plane but also in the space. This generalizes results from [87], as well as results of Riihentaus and Dovgoshey which were based on [87]. In Chap. 7 we introduce problems related to characterization of boundary values of harmonic quasiconformal mappings. From many aspects for harmonic quasiconformal mappings, the quantity log J (z, f ) seems the natural counterpart of log f  (z). In particular, the question arises if the counterparts of Pommerenke’s and Kellogs’ theorems hold for HQC mappings and log J (z, f ) instead of conformal mappings and log f  (z). In the last chapter we also pose some problems in this direction.

Chapter 2

Quasiconformal and Quasiregular Harmonic Mappings

In this chapter we build the foundation for the work that comes in the rest of the book. We begin with the definition of two conformal invariants, the modulus of a curve family and the capacity of a condenser, which are two closely related notions. These tools enable us to define quasiconformal and quasiregular mappings which are the basic classes of mappings to be studied. Several examples of quasiconformal mappings are given illustrating the importance of this class of functions and their role in Geometric Function Theory. Moduli of continuity of harmonic mappings, which are either quasiconformal or quasiregular at the same time, are considered and some sharp estimates are given for all dimensions n ≥ 2. In particular, we study the case of Lipschitz continuity of mappings defined in the unit ball.

2.1 Moduli of Curve Families Conformal invariance has played a predominant role in the Geometric Function Theory during the past century. The set of landmark works include the pioneering contributions of Grötzsch and Teichmüller prior to the Second World War and the paper of Ahlfors and Beurling [6] in 1950. These results led to far-reaching applications and have stimulated many later studies [92]. Gehring and Väisälä [51, 155] built the theory of quasiconformal mappings in Rn based on the notion of the modulus of a curve family introduced by Ahlfors and Beurling [6] in the plane and extended to Rn by Fuglede [45], which is an essential tool in the investigation of quasiconformal mappings and the theory of quasiregular mappings (see [133]). For the notions of path integral and modulus M(Γ ) of a family Γ of curves in Rn , we refer the reader to [155] and [158] where the following result is found. Theorem 2.1 ([155, Theorem 1.3, p. 2]) The length function s : [a, b] −→ R of rectifiable path α : [a, b] −→ Rn has the following properties:

© Springer Nature Switzerland AG 2019 V. Todorˇcevi´c, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, https://doi.org/10.1007/978-3-030-22591-9_2

5

6

1. 2. 3. 4. 5. 6.

2 Quasiconformal and Quasiregular Harmonic Mappings

a ≤ t1 ≤ t1 ≤ b implies l(α|[t1 ,t2 ] ) = s(t2 ) − s(t1 ) ≥ |α(t2 ) − α(t1 )|, s is increasing, s is continuous, s is absolutely continuous iff α is absolutely continuous, s  (t) and α  (t) exist a.e. and s  (t) = |α  (t)| a.e.,

b

b l(α) ≥ a s  (t) dt = a |α  (t)| dt,

where the equality holds iff s (or α) is absolutely continuous. For locally rectifiable path γ : Δ −→ X and a continuous function f : γ Δ −→ [0, ∞], the path integral is defined in two steps. Recall that γ o is the normal representation of a rectifiable path γ . 1. If γ is rectifiable, we set 

 f ds =

l(γ )

f (γ o (t))|(γ o ) (t)| dt.

0

γ

2. If γ is locally rectifiable, we set 

 f ds = sup γ

f ds : l(β) < ∞, β is subpath of γ .

β

Let us consider a family Γ of curves in Rn . We say that a nonnegative Borel measurable ρ : Rn → R is an admissible metric for Γ if  lρ (γ ) = ρ ds  1 for each locally rectifiable γ ∈ Γ. γ

Let F (Γ ) be the set of all admissible metrics for Γ . Finally, for each p  1 we define the p-modulus of Γ by  Mp (Γ ) =

inf

ρ∈F (Γ ) Rn

ρ p dm.

If F (Γ ) = ∅, we define Mp (Γ ) = ∞. This happens only if Γ contains a constant path1 because otherwise the constant function ρ(x) = ∞ belongs to F (Γ ). Clearly, 0 ≤ Mp (Γ ) ≤ ∞. The case p = n is the most important. In that case we simply write M(Γ ) for the modulus. Note: Let Γlr = {γ ∈ Γ | γ is locally rectifiable} and Γr = {γ ∈ Γ | γ is rectifiable}. By definition, we see that Mp (Γ ) = Mp (Γlr ). It can be shown that M(Γ ) = M(Γr ).

1A

case which will never be considered in this book.

2.1 Moduli of Curve Families

7

Theorem 2.2 ([155, Theorem 6.2, p. 16]) Mp is an outer measure on the set of all n curves in R : 1. Mp (∅) = 0, Γ2 ⇒ M 2. Γ1 ⊂  1 )  Mp (Γ2 ),  p (Γ ∞ ∞ 3. Mp i=1 Γi ≤ i=1 Mp (Γi ). Proof 1. Since the zero function belongs to F (∅), Mp (∅) = 0. 2. If Γ1 ⊂ Γ2 then F (Γ1 ) ⊃ F (Γ2 ), whence Mp (Γ1 )  Mp (Γ2 ). 3. We may assume that Mp (Γi ) < ∞ for each i. For ε > 0 pick ρi ∈ F (Γi ) such that  ε p ρi dm < Mp (Γi ) + i . 2

 p 1/p ρi Then the function ρ =  ρi is admissible for each family Γi , and  therefore for i Γi . Thus  Mp (Γ ) 

 ρ dm =



p

Rn

Rn

p ρi dm

=



i

i

p

Rn

ρi dm 



Mp (Γi ) + ε.

i

Letting ε → 0 we obtain  Mp

 i

 Γi





Mp (Γi ).

i

 

Definition 2.3 ([155, Definition 6.3, p. 17]) Let Γ1 and Γ2 be curve families in Rn . We say that Γ2 is minorized by Γ1 and denote Γ2 > Γ1 if every γ ∈ Γ2 has a subcurve which belongs to Γ1 . Theorem 2.4 ([155, Theorem 6.4, p. 17]) If Γ1 < Γ2 , then Mp (Γ1 )  Mp (Γ2 ). Proof If ρ is admissible for Γ1 , then it is also admissible for Γ2 , and hence F (Γ1 ) ⊂ F (Γ2 ). Therefore Mp (Γ1 )  Mp (Γ2 ).   Note that if Γ1 ⊃ Γ2 , then Γ1 < Γ2 . So, Theorem 2.2, part 2, is a special case of Theorem 2.4. Roughly speaking, Mp (Γ ) is large if the family Γ is large or if curves in Γ are short. Definition 2.5 ([155, Definition 6.6, p. 17]) The curve families Γ1 , Γ2 , . . . are called separate if there exist disjoint Borel sets Ei in Rn such that if γ ∈ Γi is locally rectifiable, then γ gi ds = 0, where gi is the characteristic function of Eic .

8

2 Quasiconformal and Quasiregular Harmonic Mappings

This condition says that each locally rectifiable γ ∈ Γi lies almost entirely (in terms of arc length) in Ei . Theorem 2.6 ([155, Theorem 6.7, p. 18]) If Γ1 , Γ2 , . . . are separate and if Γ < Γi for all i, then Mp (Γ ) 



Mp (Γi ).

i

Proof For ρ ∈ F (Γ ) set ρi (x) = (1 − gi (x)) ρ(x). Then ρi ∈ F (Γi ), whence 

Mp (Γi ) 



i

p

ρi dm =



i

i

 ρ p dm 

ρ p dm.

Ei



Mp (Γi )  Mp (Γ ).    Note that if Γ = Γi , where Γi are separated, then by combining this theorem with subadditivity (Theorem 2.2, part 3), one gets i Mp (Γi ) = Mp (Γ ). If we apply this to a single Γ1 , we obtain the previous theorem. It is in general very difficult to compute Mp (Γ ) for a given family Γ . However, choosing a suitable ρ ∈ F (Γ ) one can often get a good upper bound for Mp (Γ ), for if we take any ρ ∈ F (Γ ), then Mp (Γ )  ρ p dm. Thus

i

Theorem 2.7 ([155, Theorem 7.1, p. 20]) Suppose that the curves of a family Γ lie in a Borel set G ⊂ Rn , and that l(γ )  r > 0 for every locally rectifiable γ ∈ Γ . Then m(G) . rp

Mp (Γ )  Proof Define ρ : Rn → R by

⎧1 ⎨ , x ∈ G, ρ(x) = r ⎩ 0, x ∈ G. Then



1 γ r

ds  r ·

1 r

= 1, so ρ ∈ F (γ ) and  Rn

ρ p dm =

1 rp

 dm = G

m(G) . rp  

To get a good lower bound for Mp (Γ ) is much more difficult. Now one has to prove an estimate for each ρ ∈ F (Γ ), in contrast to the opposite inequality where a choice of a single ρ is sufficient. Usually that estimate is done by combining the Hölder’s inequality and the Fubini’s theorem.

2.1 Moduli of Curve Families

9

Fig. 2.1 Cylinder

We introduce notation: n For E, F, G ⊂ R let Δ(E, F, G) be the family of all closed curves joining E n to F within G. More precisely, a path γ : [a, b] → R belongs Δ(E, F, G) iff γ (a) ∈ E, γ (b) ∈ F and γ (t) ∈ G for a < t < b. Example 2.8 ([155, Example 7.2, p. 21] (Fig. 2.1) (The Cylinder Example)) Let E be a Borel set in Rn−1 and let h > 0. Set G = {x ∈ Rn | (x1 , . . . , xn−1 ) ∈ E and 0 < xn < h}. Then G is a cylinder with bases E and F = E + h en and with height h. Set Γ = Δ(E, F, G). We show that Mp (Γ ) =

mn−1 (E) m(G) = . hp hp−1

Since l(γ )  h for every γ ∈ Γ , Theorem 2.7 implies Mp (Γ )  m(G) hp . Let ρ ∈ F (Γ ). For each y ∈ E let Yy : [0, h] → Rn be the vertical segment Yy (t) = y + t en . Then Yy ∈ Γ . Assume that p > 1. Then we have 

p

1

1 · ρ ds



h



1/q  1q

0

Yy



h

= hp/q

h

1/p p ρ p ds

0

 ρ p ds = hp−1

0

h

ρ p (y + t en ) dt.

0

Here we used admissibility of ρ in the first and Hölder inequality in the second inequality. Now we integrate over E and use Fubini’s theorem: 





1 dm  E

h

hp−1 dmn−1 E

0

ρ p (y + t en ) dt,

10

2 Quasiconformal and Quasiregular Harmonic Mappings



in other words mn−1 (E)  hp−1 G ρ p dm  hp−1 ρ p dm. Since this holds for (E) . every ρ ∈ F (Γ ) we get Mp (Γ )  mn−1 hp−1 In fact we proved that Mp (Γ ) = Mp (Γ0 ), where Γ0 is the subfamily of all vertical segments. Example 2.9 ([155, Example 7.5, p. 22] (The Spherical Ring)) If 0 < a < b < ∞, n the domain A = B n (b) \ B (a) is called a spherical ring. Let E = S n−1 (a), F = S n−1 (b), and ΓA = Δ(E, F, A). We are going to prove that b M(ΓA ) = ωn−1 (log )1−n , a where ωn−1 is the surface measure of the unit sphere S n−1 . Pick ρ ∈ F (ΓA ). For each unit vector y ∈ S n−1 we let Yy : [a, b] → Rn be the radial segment, defined n and q = n) we obtain by Yy (t) = t · y. By Hölder’s inequality (for p = n−1 n

 1

 =

ρ ds



b

(ρ(ty) · t

1/n 

1− n1 n

) dt

 a

b

b

(t

a

=

(ρ(ty) · t

1− n1

)t

1 n −1

n dt

a

Yy



b

n 1 n −1 n−1

)

(n−1)/n n dt

a

 ρ n (ty)t n−1 dt ·

b

t −1 dt

n−1

a

b = (log )n−1 a



b

ρ n (ty) t n−1 dt.

a

Integrating over y ∈ S n−1 yields ωn−1

b  (log )n−1 a

 ρ n dm.

Since it is true for each ρ ∈ F (Γ ), we get M(ΓA )  ωn−1 (log ab )1−n . Now we choose a concrete ρ ∈ F (Γ ):

ρ(x) =

⎧ ⎪ ⎨ ⎪ ⎩

1 |x| log ab

, x ∈ A, x ∈ A.

0,

It is geometrically obvious that lρ (γ ) 

1 log ab

b

1 a r

dr = 1, so ρ ∈ F (Γ ).

Note that this can be generalized to cone domains. Let Y be a Borel set in S n−1 and let C be the cone {x ∈ Rn \ {0} | x/|x| ∈ Y }. Set Γ = {γ ∈ ΓA | |γ | ⊂ C} where A is as above. Then we have

2.1 Moduli of Curve Families

11

b M(Γ ) = mn−1 (Y )(log )1−n . a Example 2.10 ([155, Example 7.8, p. 23] (The degenerated ring)) Let Γ = Δ(E, F, G) where E = {0}, F = S n−1 (b), and G = B n (b) \ {0}. Since Γ > ΓA n for every spherical ring A = B n (b) \ B (a), we obtain (from Theorem 2.4 and Example 2.9) M(Γ )  M(Γ1 ) = ωn−1 (log ab )1−n . Since this is true for every a > 0 when a → 0, we get M(Γ ) = 0. Example 2.11 ([155, Example 7.9, p. 23] (Paths Through a Point)) Let Γ be the family of all nonconstant paths γ passing through a fixed point a ∈ Rn . We prove that M(Γ ) = 0. Without loss of generality  assume a = 0 and set  Γk = ΓA , where A = {x ∈ Rn | 0 < |x| < k1 }. Since Γ > k Γk , then M(Γ )  k M(Γk ) = 0. Suppose that A is a subset of Rn and that f : A → Rn is continuous. If Γ is a family of curves in A, then the family Γ  = {f ◦ Γ | γ ∈ Γ } is called the image of Γ under f . Theorem 2.12 ([155, Theorem 8.1, p. 25]) If f : D → D  is conformal, then M(Γ  ) = M(Γ ) for every family of curves Γ in D. Proof We can assume that Γ and Γ  do not pass through ∞. Let ρ ∈ F (Γ ), and set ρ1 (f (x)) = ρ(x) ·

1 |f  (x)|

for x ∈ D and 0 otherwise.

,

Since, ρ ∈ F (Γ ), we have  ρ |dx|  1,

for γ ∈ Γ.

γ

A change of variables gives (see [155, Theorem 5.6, p. 14]) |dy| = |f  (x)| · |dx|

y = f (x), and hence 

 ρ1 (y) |dy| =

f (γ )

ρ1 (f (x)) |f  (x)| |dx| =

γ

 ρ(x) |dx|  1, γ

so ρ1 ∈ F (Γ  ). We have M(Γ  ) = inf ρ1



 ρ1n (y) dy = inf ρ

ρ n (x)

1 Jf (x) dx = M(Γ ). |f  (x)|n  

12

2 Quasiconformal and Quasiregular Harmonic Mappings

1 Note that this works only for p = n (cancellation of |f  (x)| p with Jf (x) occurs only for p = n). However, if f (x) = k x, (k > 0), then by a similar calculation one gets the following result.

Theorem 2.13 ([155, Theorem 8.2, p. 25]) Mp (k Γ ) = k n−p Mp (Γ ). Theorem 2.14 ([155, Theorem 10.12, p. 31]) Suppose that 0 < a < b and that E and F are disjoint sets such that every sphere S n−1 (t), a < t < b, meets both E and F . If G contains the spherical ring A = B n (b) \ B n (a) and if Γ = Δ(E, F, G), then b M(Γ )  cn log , a

(2.1)

where cn =

1 ωn−2 2



∞

1

n−2

t − n−1 (1 + t 2 )− n−1 dt

1−n ,

c2 =

0

1 . π

There is equality in (2.1) if G = A and if E and F are the components of L ∩ A, where L is a line through the origin. It is important to observe that the constants Ωn and ωn−1 depend strongly on the dimension, indeed Ωn → 0 and ωn−1 → 0 when n → ∞ by [10, 2.28]. Various estimates for the constant cn in 2.1 are given [10, pp. 41–44, 458]. Definition 2.15 ([155, p. 32]) A domain A ⊂ Rn is a ring if C(A) has exactly two components, where C(A) denotes the complement of A ⊂ Rn . If the components of C(A) are C0 and C1 , we denote A = R(C0 , C1 ), B0 = C0 ∩ A and B1 = C1 ∩ A. To each ring A = R(C0 , C1 ), we associate the curve family ΓA = Δ(B0 , B1 , A). Definition 2.16 ([158]) The modulus mod(R) of the ring R(C0 , C1 ) is defined by  mod(R) = mod(R(C0 , C1 )) =

ωn−1 M(Δ(C0 , C1 ))

1/(n−1) .

The capacity cap(R) of R(C0 , C1 ) is M(Δ(C0 , C1 )). n

The complementary components of the Grötzsch ring RG,n (s) in Rn are B and [s · e1 , ∞], s > 1, while those of the Teichmüller ring RT ,n (t) are [−e1 , 0] and [t e1 , ∞], t > 0. We shall need two special functions γn (s), s > 1 and τn (t), t > 0 to designate the moduli of the families of all those curves which connect the complementary components of the Grötzsch and Teichmüller rings in Rn respectively (Fig. 2.2). γn (s) = M(Γs ) = γ (s),

Γs = ΓRG,n (s),

τn (t) = M(Δt ) = τ (t),

Δt = ΓRT ,n (t).

2.1 Moduli of Curve Families

13

Fig. 2.2 The Grötzsch and Teichmüller rings

These functions are related by a functional identity [48, Lemma 6] γn (s) = 2n−1 τn (s 2 − 1).

(2.2)

We define the functions Φ = Φn and Ψ = Ψn by ωn−1 (log(Φ(s)))1−n = γn (s),

s>1

ωn−1 (log(Ψ (t)))1−n = τn (t),

t > 0.

Lemma 2.17 ([158, √ 7.20, p. 88]) The function Φ(t)/t is increasing for t > 1 and Ψ (t − 1) = Φ( t)2 for t > 1. Moreover, the functions γn and τn are strictly decreasing. By the previous lemma, the function log Φ(t) − log t is increasing and therefore has a limit as t → ∞. We define the number λn by log λn = lim (log Φ(t) − log t). t→∞

This number is usually called the Grötzsch (ring) constant. Only for n = 2 is the exact value of the Grötzsch constant known, λ2 = 4. Various estimates for λn , n ≥ 3, are given in [48, p. 518], [33, pp. 239–241], [8]. Lemma 2.18 ([158, 7.22, p. 89]) For each n  2 there exists a number λn ∈ [4, 2 · en−1 ), λ2 = 4 such that 1. t  Φ(t)  λn t, t > 1, 2. t + 1  Ψ (t)  λ2n (t + 1), t > 0. 1/n

Furthermore, λn → e as n → ∞ and in particular, λn → ∞ as n → ∞. The inequality Φ(t)  λn t is equivalent, by the definition of the function Φ, to γn (t)  ωn−1 (log λn t)1−n ,

t > 1.

(2.3)

14

2 Quasiconformal and Quasiregular Harmonic Mappings

Definition 2.19 Given r > 0, we let RΨn (r) be the set of all rings A = R(C0 , C1 ) in Rn with the following properties: 1. C0 contains the origin and a point a such that |a| = 1. 2. C1 contains ∞ and a point b such that |b| = r. Teichmüller was the first to consider the following infimum in the planar case (n = 2), τn (r) = inf M(ΓA ), where infimum is taken over all rings A ∈ RΨn (r). For n  3 it was studied in [48]. Theorem 2.20 ([155, Theorem 11.7, pp. 34–35], [48]) The function τn (0, ∞) → (0, ∞) has the following properties: 1. 2. 3. 4.

:

τn is decreasing, limr→∞ τn (r) = 0 , limr→0 τn (r) = ∞ , τn (r) > 0 for every r > 0.

Moreover, τn : (0, ∞) → (0, ∞) and γn : (1, ∞) → (0, ∞) are homeomorphisms. From the conformal invariance of the modulus and from the definition of τn , we obtain the following estimate: Theorem 2.21 ([155, Theorem 11.9, p. 36], [48]) Suppose that A = R(C0 , C1 ) is a ring and that a, b ∈ C0 and c, ∞ ∈ C1 . Then  M(ΓA )  τn

 |c − a| . |b − a|

2.2 Definition of Quasiconformal Mappings Let f : D → D  be a homeomorphism. If Γ is a family of curves in D, then Γ  denotes the family {f ◦ γ | γ ∈ Γ } of curves in D  . We set KI (f ) = sup

M(Γ  ) , M(Γ )

KO (f ) = sup

M(Γ ) , M(Γ  )

where the suprema are taken over all families of curves Γ ⊂ D such that M(Γ ) and M(Γ  ) are not simultaneously 0 or ∞. Note that the two quantities are equal if f is a conformal mapping (see Theorem 2.12). Definition 2.22 ([155, Definition 13.1, p. 42]) If f : D → D  is a homeomorphism, KI (f ) is the inner dilatation and KO (f ) is the outer dilatation of f . The

2.2 Definition of Quasiconformal Mappings

15

maximal dilatation of f is K(f ) = max{KI (f ), KO (f )}. If K(f )  K < ∞, f is K-quasiconformal. f is quasiconformal (qc) if K(f ) < ∞. Equivalently, f is K-quasiconformal iff M(Γ )  M(Γ  )  K M(Γ ), K for every family of curves Γ in D. This is the geometric definition of quasiconformal mappings. Definition 2.23 ([155, Definition 26.2, p. 88]) Let Q = {x ∈ Rn : ai ≤ xi ≤ bi } be a closed n-interval. A mapping f : Q −→ Rm is said to be ACL (absolutely continuous on lines) if f is continuous and if f is absolutely continuous on almost every line segment in Q, parallel to the coordinate axes. Definition 2.24 [155, Definition 26.5, p. 89] An ACL-mapping f : U −→ Rm is said to be ACLp , p ≥ 1, if the partial derivatives of f are locally Lp -integrable. A homeomorphism f : D −→ D  is ACLp if the restriction of f to D\{∞, f −1 (∞)} is ACLp . Definition 2.25 ([158, Definition 10.1, pp. 127–128], [113]) Let G ⊆ Rn be a domain. A mapping f : G −→ Rn is said to be quasiregular (QR) if f is ACLn and if there exists a constant K ≥ 1 such that |f  (x)|n ≤ K Jf (x),

|f  (x)| = max |f  (x)h|, |h|=1

a.e. in G.

Here f  (x) denotes the formal derivative of f at x. The smallest K ≥ 1 for which this inequality is true is called the outer dilatation of f and denoted by KO (f ). If f is quasiregular, then the smallest K ≥ 1 for which the inequality Jf (x) ≤ K l(f  (x))n ,

l(f  (x)) = min |f  (x)h|, |h|=1

holds a.e. in G is called the inner dilatation of f and denoted by KI (f ). The maximal dilatation of f is the number K(f ) = max{KI (f ), KO (f )}. If K(f ) ≤ K, f is said to be K-quasiregular (K-QR). If f is not quasiregular, we set KO (f ) = KI (f ) = K(f ) = ∞. It turns out that injective quasiregular mappings are quasiconformal and that all quasiregular mappings are orientation-preserving (Jf ≥ 0 a.e). Definition 2.22 gives that a quasiconformal mapping is orientation reversing. Therefore, in particular, all Möbius transformations and all sense-reversing transformations are 1-quasiconformal, but they are not 1-quasiregular because quasiregular mappings are required to have Jacobian greater than zero, hence they are sense-preserving. Theorem 2.26 ([130, Theorem 1.1]) For a given n ≥ 3 there exists a constant e(n) depending on n such that if f : Ω → Rn is a quasiregular mapping of a domain Ω ⊂ Rn and if K(f ) ≤ 1 + e(n) , then f is a local homeomorphism.

16

2 Quasiconformal and Quasiregular Harmonic Mappings

Note that from this theorem we have that for small K, harmonic K-quasiregular mappings are diffeomorphism and, therefore, locally quasiconformal (see [155, p. 46]). The following theorem states simple properties of the dilatations just introduced above relative the compositions and inverses and its proof is left to the reader. Theorem 2.27 ([155, Theorem 13.2, p. 42]) 1. 2. 3. 4. 5. 6.

KI (f −1 ) = KO (f ), KO (f −1 ) = KI (f ), K(f −1 ) = K(f ), KI (f ◦ g)  KI (f )KI (g), KO (f ◦ g)  KO (f )KO (g), K(f ◦ g)  K(f )K(g).

Corollary 2.28 ([155, Corollary 13.3, p. 42]) If f is K-quasiconformal, then f −1 is K-quasiconformal.  

Proof This follows from Theorem 2.27, part 3.

Corollary 2.29 If h = f ◦ g, where f is K1 -quasiconformal and g is K2 quasiconformal, then h is K1 K2 -quasiconformal.  

Proof This is a consequence of Theorem 2.27, part 6. Definition 2.30 ([155, Definition 14.1, p. 43]) Let A : bijection. The numbers HI (A) =

|det (A)| , l n (A)

HO (A) =

|A|n , |det (A)|

Rn

→

H (A) =

Rn

be a linear

|A| , l(A)

where l(A) = inf||x||=1 ||Ax||, are called the inner, outer, and linear dilatation of A, respectively. They have the following geometric interpretation: The image of the unit ball Bn under A is an ellipsoid E(A). Let BI (A) and BO (A) be the inscribed and circumscribed balls of E(A), respectively. Then HI (A) =

HO (A) =

a1 · · · an−1 m(E(A)) = , m(BI (A)) ann−1

a1n−1 m(BO (A)) = , m(E(A)) a2 · · · an

H (A) =

a1 , an

where a1  a2  · · ·  an are the semi-axes of E(A). Next we turn to a more interesting task: finding conditions on quasiconformality in the case of a C 1 mapping in terms of its derivative. This is an analytic approach to quasiconformal mappings.

2.2 Definition of Quasiconformal Mappings

17

Module estimates will play crucial role here. In order to do this, we define HO (f  (x)) =

|(f  (x))|n , |Jf (x)|

|Jf (x)| , l(f  (x))n

HI (f  (x)) =

where Jf (x) = 0 is Jacobian of f . Theorem 2.31 ([155, Theorem 15.1, p. 46]) Suppose that f : D → D  is a diffeomorphism. Then KI (f ) = sup HI (f  (x)),

KO (f ) = sup HO (f  (x)).

x∈D

x∈D

Theorem 2.32 ([155, Theorem 15.3, p. 47]) Let f : D → D  be a homeomorphism. If f is differentiable at a point a ∈ D and if KO (f ) < ∞, then |f  (a)|n  KO (f ) · |Jf (a)|. Corollary 2.33 ([155, Corollary 15.5, p. 48]) A diffeomorphism f : D → D  is K-quasiconformal iff the double inequality |f  (x)|n  |Jf (x)|  K · l(f  (x))n K holds for every x ∈ D. Example 2.34 ([155, Example 16.1, pp. 48–49] (A Linear Mapping)) Let A : Rn → Rn be a linear bijection. Then A (x) = A for all x ∈ Rn . From Theorem 2.31 we obtain KI (A) = HI (A)

KO (A) = HO (A).

Thus A is quasiconformal. Example 2.35 ([155, Example 16.2, p. 49] (A Radial Mapping)) Let a = 0 be a real number, and set f (x) = |x|a−1 x. Then f is a diffeomorphism of Rn \{0} onto itself. n n We can extend f to a homeomorphism f ∗ : R −→ R by defining f ∗ (0) = 0, f ∗ (∞) = ∞ for a > 0 and f ∗ (0) = ∞, f ∗ (∞) = 0 for a < 0. Then f is a quasiconformal with KI (f ) = |a|,

KO (f ) = |a|n−1

if |a|  1,

KI (f ) = |a|1−n ,

KO (f ) = |a|−1

if |a|  1.

Example 2.36 ([86], (General Radial Mappings)) Now we consider radial mappings of a more general type: f (x) = ϕ(|x|) ·

x , |x|

18

2 Quasiconformal and Quasiregular Harmonic Mappings

Fig. 2.3 Radial mappings

where ϕ is continuously differentiable on [0, +∞), ϕ  (r) > 0 and ϕ(0) = 0 ([86]). Now we want to calculate the Jacobi matrix of f at a given point x. In fact, we are interested in l(f  (x)) and |f  (x)|. It is easier to work in a new rectangular coordinate system, one coordinate is along the vector x and the other coordinates are in the tangent plane ((n − 1)-dimensional hyperplane) to the sphere through x (Fig. 2.3). Then we have ⎤ ⎡  ϕ (r) 0  n ϕ(r) ⎥ ⎢ ∂f ⎥ ⎢ r =⎢ ⎥. .. ⎦ ⎣ ∂x i,j =1 . 0

ϕ(r) r

Indeed, the rate of stretching along the first coordinate axis is ϕ  (r) by definition of f . In the tangent hyperplane we have a similarity transformation by a coefficient ϕ(r) r , where Δr is “infinitesimal” displacement (Fig. 2.4). Hence, we have the following equations:   ϕ(r) n−1 Jf (x) = ϕ  (r) · = det f  (x), r    ∂f ϕ(r)   = min ϕ (r), , l(f (x)) = l ∂x r  ϕ(r) , |f  (x)| = max ϕ  (r), r !n max ϕ  (r), ϕ(r) r = HO (f  (x)) = #n−1 . " Jf (x) ϕ  (r) ϕ(r) r |f  (x)|n

2.2 Definition of Quasiconformal Mappings

19

Fig. 2.4 The stretching rate

r

Note that this is bounded iff sup r>0

ϕ  (r) · r < +∞ ϕ(r)

and sup r>0

HI (f  (x)) =

ϕ(r) < +∞. r · ϕ  (r)

ϕ  (r)

"

ϕ(r) r

#n−1

Jf (x) = !n l(f  (x)n min ϕ  (r), ϕ(r) r 

= ϕ (r)



ϕ(r) r

n−1



r 1 , · max  ϕ (r) ϕ(r)

n .

ϕ(r) On the other hand, this is bounded iff supr>0 r·ϕ  (r) < +∞ and supr>0 +∞, which is the same condition we obtained for HO .

ϕ  (r)·r ϕ(r)


0

ϕ  (r) · r < +∞ ϕ(r)

and

sup r>0

ϕ(r) < +∞. r · ϕ  (r)

One can write this characterization in a different form: Let α : (0, 1) → (u, v), where 0 < u < v < 1. For f (x) = |x|α(|x|)−1 x we have ϕ(r) = r α(r) and r ϕ  (r) = r · α  (r) · ln(r) + α(r) ϕ(r) and the condition for quasiconformality of f is that r · α  (r) · ln(r) + α(r) is bounded above and bounded away from zero. This is of course true if α(r) = α ∈ (0, 1). In that case ⎡ ⎢ ⎢ Jf = ⎢ ⎣

α · r α−1

0 r α−1 ..

.

⎤ ⎥ ⎥ ⎥. ⎦

r α−1

0

Since 0 < α < 1, l = α · r α−1 , |f  (x)| = r α−1 , Jf (x) = α · (r α−1 )n , and HO (f  (x)) =

(r α−1 )n 1 = α α · (r α−1 )n

and HI (f  (x)) =

α · (r α−1 )n 1 = n−1 . α−1 n (α · r ) α

Note that HI  HO , and we see that the constant of quasiconformality is K = . From here we have α = K 1/(1−n) . The constant α = K 1/(1−n) is the best α n−1 possible exponent. 1

Example 2.37 (Hölder Continuity of Radial Mappings [86, Appendix]) In this example we use some results from [31, Lemma 3.3]. Let f (x) = |x|α−1 x, x ∈ Rn , and α ∈ (0, 1). We prove that f is Hölder continuous with exponent α, i.e., ∀x, y ∈ Rn

|f (x) − f (y)|  C |x − y|α

with C = 21−α . Note that this Hölder estimate is sharp, since the equality holds for x = −y.

2.2 Definition of Quasiconformal Mappings

21

We write the function f in the following form: f (x) = |x|α

x . |x|

(2.4)

Without loss of generality we can assume that |x|  |y| > 0. Put k = |x| |y| and y x x˜ = |x| , y˜ = |y| . Then k  1 and |x| ˜ = |y| ˜ = 1. By the equality (2.4), we need to prove $ $ $ α x $ α y $ $ |x|  C |x − y|α . − |y| $ |x| |y| $ By dividing the previous inequality by |y|α , it follows that ˜  C |k x˜ − y| ˜ α. |k α x˜ − y| This inequality is equivalent to ˜ 2  C 2 |k x˜ − y| ˜ 2α . |k α x˜ − y| By definition of the inner product, this is equivalent to C2 

˜ y ˜ +1 k 2α − 2 k α x, . (k 2 − 2 kx, ˜ y ˜ + 1)α

(2.5)

The inequality C 2  max

˜ y ˜ +1 k 2α − 2 k α x, (k 2 − 2 kx, ˜ y ˜ + 1)α

ensures (2.5). Because |x, ˜ y | ˜  1, it is sufficient to prove that f (k, t) =

k 2α − 2 k α t + 1  22−2α (k 2 − 2k t + 1)α

for each t ∈ [−1, 1]. The function f (k, t) can be written in the form f (k, t) =

(k α − 1)2 + 2 k α (1 − t) . ((k − 1)2 + 2 k (1 − t))α

By differentiating f (k, t) with respect to t we obtain ∂f −2 k ((k α+1 − 1)(1 − k α−1 ) + (1 − α) ((k α − 1)2 + 2 k α (1 − t))) = ∂t (k 2 − 2 k t + 1)α+1 < 0.

22

2 Quasiconformal and Quasiregular Harmonic Mappings

Hence, f (k, t) is decreasing as a function of t when −1  t  1. From this we get   α k +1 2 k 2α + 2 k α + 1 = . f (k, t)  f (k, −1) = 2 (k + 1)α (k + 2 k + 1)α By concavity of the function uα , kα + 1 = (k + 1)α

k α +1 2 (k+1)α 2

" 

#α k+1 2 (k+1)α 2

= 21−α .

Note that in this example we proved the sharp Hölder estimate for f . Consider a homeomorphism f : D → D  . Suppose that x ∈ D, x = ∞ and f (x) = ∞. For each r > 0 such that S n−1 (x, r) ⊂ D we set L(x, f, r) = max |f (y) − f (x)|, |y−x|=r

l(x, f, r) = min |f (y) − f (x)|. |y−x|=r

(2.6) Definition 2.38 ([155, Definition 22.2, p. 78]) The linear dilatation of f at x is the number L(x, f, r) H (x, f ) = lim sup . r→0 l(x, f, r) If x = ∞, f (x) = ∞, we define H (x, f ) = H (0, f ◦ u) where u is the inversion u(x) = |x|x 2 . If f (x) = ∞, we define H (x, f ) = H (x, u ◦ f ) (Fig. 2.5). Example 2.39 The mapping f : Bn → Bn , f (x) = |x|α−1 x has H (0, f ) = 1. This is true for all radial mappings. Theorem 2.40 ([155, Theorem 22.3, pp. 78–79]) Suppose that f : D → D  is a homeomorphism such that one of the following conditions are satisfied for some finite K: 1. M(ΓA )  K · M(ΓA ) for all rings A such that A ⊂ D, 2. KO (f )  K, 3. KI (f )  K. Then H (x, f ) is bounded by a constant which depends only on n and K. Fig. 2.5 A linear dilatation

2.3 Hölder Continuity of Quasiconformal Mappings

23

Remark 2.41 Let us explain some history behind the bounds of linear dilatation for n ≥ 2. First, there was a bound due to Gehring and improved by Vuorinen, see [158, Remark 10.29, p. 136]. A bound which tends to 1 when K tends to 1 was proven by Vuorinen [159] for mappings defined in the whole space in Rn and this was generalized to the case of mappings defined in subdomains G of Rn by P. Seittenranta [140], see also [10, 14.37].

2.3 Hölder Continuity of Quasiconformal Mappings We need more delicate estimates for our further study of Hölder continuity of quasiconformal mappings. Theorem 2.42 ([158, Theorem 7.47, p. 98]) For n  2, K > 0, and 0  r  1 let ϕK (r) = ϕK,n (r) =

1 γn−1 (Kγn (1/r))

.

(2.7)

We set ϕK (0) = 0 and ϕK (1) = 1. Then ϕK,n : [0, 1] → [0, 1] is a homeomorphism and for K  1 α ϕK (r)  λ1−α n r ,

α = K 1/(1−n) ,

(2.8)

β ϕ1/K (r)  λ1−β n r ,

β = K 1/(n−1) .

(2.9)

It should be noticed that in the case n = 2, all the functions τn (t), γn (t), ϕK,n (r) can be expressed in terms of classical special functions such as complete elliptic integrals while this is not the case for n ≥ 3 [10]. Therefore, for n = 2 one can expect sharper results than for the case of a general dimension n ≥ 3. We give here some well-known identities between them that can be found in [10]. First, the function ηK,n (t) =

τn−1 (τn (t)/K)

√ 1 − ϕ1/K,n (1/ 1 + t)2 = ,K>0 √ ϕ1/K,n (1/ 1 + t)2

(2.10)

defines an increasing homeomorphism ηK,n : (0, ∞) → (0, ∞) (see [10, p.193]). Later,√in Theorem 3.24 we will need the constant (1 − a)/a, where a = ϕ1/K,n (1/ 2)2 , in (3.31) which can be expressed as follows for K > 1 (1 − a)/a = ηK,n (1) = τn−1 (τn (1)/K) .

(2.11)

The following lemma establishes the extremality property of the Grötszch ring and is based on a symmetrization theorem from [48, Theorem 1].

24

2 Quasiconformal and Quasiregular Harmonic Mappings

Lemma 2.43 Let C be a connected compact set contained in the unit disk that contains points 0 and x. Then the capacity of a ring domain with complementary 1 components C0 = C and C1 = {y : |y|  1} is at least γn ( |x| ). The next theorem is a counterpart of the Schwarz lemma for quasiconformal mappings. Theorem 2.44 ([115, Theorem 3.1]) If f : Bn → Bn is a K-quasiconformal mapping such that f (0) = 0, then |f (x)|  ϕK,n (|x|) for all x ∈ Bn , where ϕK,n is as in Theorem 2.42. Proof We fix x ∈ Bn (i.e., |x| < 1) and consider the ring R(C0 , C1 ) where C0 = {t x : 0  t  1} and C1 = {x : |x|  1}. If Γ is the family of curves joining the boundary components of the ring R(C0 , C1 ), then M(f (Γ ))  K M(Γ ) = Kγn (1/|x|), by definition of K-quasiconformality and the definition of γn (using an inversion with respect to S n−1 to transform R(C0 , C1 ) onto RG,n (1/|x|) (Fig. 2.6). 1 Lemma 2.43 gives M(f (Γ ))  γn ( |f (x)| ). So, 1 γn ( )  K γn |f (x)| i.e., 1  γn−1 |f (x)|





 1 , |x|

 K γn

1 |x|

 ,

because γn is decreasing on (1, +∞). So, |f (x)| 

1 1 γn−1 (K γn ( |x| ))

.  

Fig. 2.6 Schwarz lemma

2.3 Hölder Continuity of Quasiconformal Mappings

25

For K = 1 we have that ϕK,n (r) = r. We note that Theorem 2.44 was proved in [115, Theorem 3.1] for quasiregular mappings. We will need a more general (Möbius invariant) form of this result. Below we will make use of the hyperbolic metric ρBn of the unit ball and a variant of the Schwarz lemma formulated in terms of this metric. We have the formula [158, (2.18), 2.52] |x − y|2 1 th2 ρ(x, y) = 2 2 |x − y| + (1 − |x|2 )(1 − |y|2 )

(2.12)

for x, y ∈ Bn . Theorem 2.45 ([158, Theorem 11.2 (1)]) If f : D → G, D, G ∈ {Bn , Hn } is K-quasiregular and x, y ∈ D, then 1 1 th ρG (f (x), f (y))  ϕK,n ( th ρD (x, y)). 2 2 Proof In [158, Theorem 11.2 (1)] the result is formulated only in the case when D = G = Bn . By Möbious invariance of the hyperbolic metric the same proof also holds in the present case.   Now we are ready to prove the local Hölder continuity of a quasiconformal mapping. Theorem 2.46 ([115]) Suppose that f is a bounded and quasiconformal mapping in a domain G ⊆ Rn and that F is a compact subset of G. Set α = KI (f )1/(1−n) and −α C = λ1−α n d(F, ∂G) d(f G), where λn is the Grötzsch constant. Then f satisfies the Hölder condition |f (x) − f (y)|  C|x − y|α ,

x ∈ F, y ∈ G.

(2.13)

Proof Set r = d(F, ∂G). The main case is |x − y| < r. Define g : Bn → Bn by g(z) =

f (x + rz) − f (x) , d(f G)

|z| < 1.

Then g(0) = 0, |g(z)|  1, and KI (g)  KI (f ). Note that g is not necessarily onto Bn . However, we can still apply the Schwarz lemma, Theorem 2.44, and use the estimate (2.8) to get α |g(z)|  λ1−α n |z| .

Set z =

y−x r

∈ Bn . This gives |f (y) − f (x)| |y − x|α  λ1−α n d(f G) rα

26

2 Quasiconformal and Quasiregular Harmonic Mappings

i.e., · |f (y) − f (x)|  λ1−α n

d(f G) · |y − x|α . d(F, ∂G)α

Now we consider the easier case |x − y|  r. We have |f (x) − f (y)|  d(f G) 

|x − y|α d(f G) d(f G)  λ1−α · |x − y|α , n α r d(F, ∂G)α

because λn  1 (see Lemma 2.18).

 

Note 2.47 Theorem 2.46 is also true if we replace KI (f ) by K(f ). Theorem 2.48 ([49, Theorem 11] and [155, Theorem 18.1, p. 63]) (The distortion theorem) For every K  1 and n ∈ N, n  2, there exists a function θK,n : (0, 1) → R1 with the following properties: 1. 2. 3. 4.

θK,n is increasing, limr→0 θK,n (r) = 0, limr→1 θK,n (r) = ∞. Let D and D  be proper subdomains of Rn and let f : D → D  be Kquasiconformal. If x, y ∈ D such that 0 < |y − x| < d(x, ∂D), then |f (y) − f (x)| ≤ θK,n d(f (x), ∂D  ) |f (y) − f (x)| ≤ θK,n d(f (y), ∂D  )



 |y − x| , d(x, ∂D)



 |y − x| . d(x, ∂D)

Moreover, we can choose (A) θK,n (r) =

τn−1



1 " #1−n  , 1 K ωn−1 log r

where 0 < r < 1

and also (B) θK,n (r) =

1 " " #α # Ψ −1 Φ 1r

for 0 < r < 1.

We give two proofs of this result, based on the same idea, leading to two different functions θK,n . Our motivation for giving two proofs is coming from the fact that the function obtained in the second proof gives a better estimate.

2.3 Hölder Continuity of Quasiconformal Mappings

27

Proof (A) Suppose that f : D → D  and x, y are as in (4). We abbreviate d = d(x, ∂D), d  = d(f (x), ∂D  ), and d  = d(f (y), ∂D  ). Let A be the spherical ring {z | |y − x| < |z − x| < d}. Then A ⊂ D. Setting C0 = f B n (x, |y − x|) and C1 = D  \ f B n (x, d), we have f A = R(C0 , C1 ). Here C0 contains f (x) and f (y) while C1 contains ∞ and points b , b ∈ ∂D  such that |f (x) − b | = d  , |f (y) − b | = d  . By 2.21   d M(Γf A )  τn , (2.14) |f (x) − f (y)|   d  . (2.15) M(Γf A )  τn |f (x) − f (y)| But, K-quasiconformality of f gives M(Γf A )  K · M(ΓA ) (Fig. 2.7). " #1−n d Since M(ΓA ) = ωn−1 log |y−x| ,  K ωn−1 log

d |y − x|



1−n  τn

 d . |f (x) − f (y)|

Since τn is a homeomorphism, we have  τn−1 K ωn−1



d log |y − x|

1−n  

d , |f (x) − f (y)|

i.e., |f (x) − f (y)|  d

1  " #1−n  . d τn−1 K ωn−1 log |y−x|

Defining θK,n (r) =

τn−1



1 " #1−n  , 1 K ωn−1 log r

where 0 < r < 1

we obtain the first inequality. The other inequality can be proved analogously, using (2.15) instead of (2.14). Properties (1)–(3) follow from the corresponding properties of τn in 2.20.   Proof ((B) [33, p. 248]) Now we use, instead of a spherical ring A, a bounded Grötzsch ring R = RG (C0 , C1 ), where C1 = S n−1 (x, a), where |y − x| < a < d = d(x, ∂D) and C0 is a line segment from x to y. In this case,

28

2 Quasiconformal and Quasiregular Harmonic Mappings

Fig. 2.7 The distortion theorem

 mod(R) = log Φ

 a . |x − y|

Now, f (R) = R  is a ring with components C0 and C1 such that C0 contains f (x) and f (y) and C1 contains a point whose distance from f (x) is smaller than d  . Let α = K 1/(1−n) . By the extremal property of Teichmüller ring (see Theorem 2.21) we have 

mod(R )  log Ψ



 d . |f (x) − f (y)|

2.3 Hölder Continuity of Quasiconformal Mappings

29

Since mod(R  )  α mod(R), we obtain  Ψ hence

d |f (x) − f (y)|





a |x − y|

Φ

α ,

  α  d a ,  Ψ −1 Φ |f (x) − f (y)| |x − y|

i.e., |f (x) − f (y)| 1 " " #α # ,   a d Ψ −1 Φ |x−y| letting a → d, we get |f (x) − f (y)| 1 " " #α #   d d Ψ −1 Φ |x−y| and we obtained the desired estimate with θK,n (r) =

"

1

Ψ −1 Φ

" #α # 1 r

for 0 < r < 1.  

We reserve the notation θK,n for the function obtained in proof B. We see that ⎛ " #α ⎞ " #α 1 1 Φ r r θK,n (r) ⎜ ⎟ " " #α # · " #α ⎠ = = lim ⎝ lim r→0 r→0 1 1 rα Ψ −1 Φ r Φ r  lim

t→+∞

t Φ(t)

α · lim

s→+∞

Ψ (s) , s

but both limits are finite (Lemma 2.17), so θK,n (r) = O(r α ). Hence, we derived behavior of θK,n (r) for small r. We can find another expression for θK,n , in terms of γ , τ . Namely, θK,n (t) =

1 " " ## , τ −1 K γ 1t

for 0 < t < 1.

(2.16)

30

2 Quasiconformal and Quasiregular Harmonic Mappings

This follows from the equation   α     1 1 Ψ −1 Φ = τ −1 K γ r r and by the definition of Ψ, we have 

τ (u) Ψ (u) = exp ωn−1

1/(1−n)

and now we obtain " # ⎞1/(1−n) ⎛  α K γ 1r 1 ⎠ Φ = exp ⎝ , r ωn−1 i.e., " # ⎞1/(1−n) ⎛ 1   K γ r 1 ⎝ ⎠ = α log Φ , r ωn−1 which is equivalent to ⎛ " # ⎞1/(1−n) ⎛ " # ⎞1/(1−n) γ 1r γ 1r ⎠ ⎠ = K 1/(1−n) ⎝ α⎝ ωn−1 ωn−1 by the definition of Φ. Next, we relate the function ϕT ,n to θK,n : θK,n (t) =

ϕT ,n (t)2 , 1 − ϕT ,n (t)2

0 < t < 1, T = 2n−1 K.

This follows easily from the following equation (see [10, 8.70 (7) (a)]) τn−1 where r  =

√

   ψ1/T ,n (r  )2 1 K γn = , r ϕT ,n (r)2

1 − r 2 and ψK,n (r) =

+ 1 − ϕ1/K,n (r  )2 .

 

2.4 Moduli of Continuity in the Plane

31

2.4 Moduli of Continuity in the Plane This section considers a natural topic, the relationship between moduli of continuity of mappings that are continuous on the boundary of the ball and quasiregular inside the ball and it is based on the paper [88]. This was also considered by Heinonen et al. in [64, Theorem 14.45, p.273]. In [88], it was proven that |f |p is subharmonic whenever f is QR, and moreover the optimal exponent p < 1 was explicitly determined. It is well known that if f is a complex-valued harmonic function defined in a domain G of the complex plane C, then |f |p is subharmonic for p ≥ 1, and that in the general case it is not subharmonic for p < 1. However, if f is holomorphic, then |f |p is subharmonic for every p > 0 (see [40, Example 1 and Example 2, pp. 7–8]). Here we consider quasiregular harmonic functions in the plane. We shall follow the practice of the planar case to define k-quasiregular mapping for 0 < k < 1 as follows: A mapping f : G −→ C is k-quasiregular if it is absolutely continuous on lines in G and ¯ (z)| ≤ k|∂f (z)|, |∂f

for almost all z ∈ G,

(2.17)

where ¯ (z) = 1 ∂f 2



∂f ∂f +i ∂x ∂y



1 ∂f (z) = 2

and



∂f ∂f −i ∂x ∂y

 ,

z = x + iy.

As shown in [94], k-quasiregularity in this sense is the same thing as Kquasiregularity with K = (1 + k)/(1 − k) in the sense of Definition 2.25. For a continuous function f : D → C harmonic in D, we define two moduli of continuity: ω(f, δ) = sup{|f (eiθ ) − f (eit )| : |eiθ − eit | ≤ δ, t, θ ∈ R},

δ ≥ 0,

and ω(f, ˜ δ) = sup{|f (z) − f (w)| : |z − w| ≤ δ, z, w ∈ D},

δ ≥ 0.

We prove that |f |p is subharmonic for p ≥ 4k/(1 + k)2 =: q as well as that the exponent q (< 1) is the best possible (see Theorem 2.53). The fact that q < 1 enables us to prove that if f is quasiregular in the unit disk D and continuous on D, then ω(f, ˜ δ) ≤ C ω(f, δ) for some constant C; see Theorem 2.60. If u : G −→ R, u > 0 is a real valued function of the class C 2 defined on a domain G in C, then  |∇u| = 2

∂u ∂x

2

 +

∂u ∂y

2

$ $ $ ∂u ∂u $$2 $ =$ − i $ = 4|∂u|2 , ∂x ∂y

32

2 Quasiconformal and Quasiregular Harmonic Mappings

 |∇u|2 =

∂u ∂x

 Δ=

2

 +

∂u ∂y

∂ ∂ −i ∂x ∂y

2



$ $ $ ∂u ∂u $2 ¯ 2, + i $$ = 4|∂u| = $$ ∂x ∂y

∂ ∂ +i ∂x ∂y

 =4

∂2 , ∂z∂ z¯

or in the more compact form ¯ 2 |∇u|2 = 4|∂u|2 = 4|∂u|

(2.18)

¯ Δu = 4∂ ∂u.

(2.19)

and

Lemma 2.49 If G is a domain in C and if u : G −→ R is a function of class C 2 such that u > 0, then for every α ∈ R, Δ(uα ) = αuα−1 Δu + α(α − 1)uα−2 |∇u|2 ,

(2.20)

Proof    (uα )xi xi = ((uα )xi )xi = (α uα−1 uxi )xi i i i =α ((α − 1)uα−2 (uxi )2 + uα−1 uxi xi )

Δ(uα ) =

i

= αuα−1 Δu + α(α − 1)uα−2 |∇u|2 .   ¯ where g and h are complex valued holomorphic Lemma 2.50 If f = g + h, functions defined on domain G in C, then Δ(|f |2 ) = 4(|g  |2 + |h |2 ). ¯ g¯ + h), we have Proof Since |f |2 = (g + h)( ¯ ) Δ(|f |2 ) = 4∂(h (g¯ + h) + (g + h)g   = 4(h h + gg ) = 4(|g  |2 + |h |2 ).   ¯ where g and h are complex valued holomorphic Lemma 2.51 If f = g + h, functions defined on domain G in C, then |∇(|f |2 )|2 = 4(|g  |2 + |h |2 )|f |2 + 8 Re (g  h f 2 ).

2.4 Moduli of Continuity in the Plane

33

Proof We have |∇(|f |2 )|2 = = = =

4|∂(|f |2 )|2 ¯ g¯ + h))|2 4|∂((g + h)( 4|g  f¯ + f h |2 4(|g  |2 + |h |2 )|f |2 + 8 Re (g  h f 2 ).  

¯ where g and h are complex valued holomorphic Lemma 2.52 If f = g + h, functions defined in domain G in C, then Δ(|f |p ) = p2 (|g  |2 + |h |2 )|f |p−2 + 2p(p − 2)|f |p−4 Re (g  h f 2 ), whenever f = 0. Proof We take α = p/2, u = |f |2 and then use (2.20), Lemma 2.50, and Lemma 2.51 to get the result.   Theorem 2.53 ([88]) If f is a complex-valued k-quasiregular harmonic function defined on a domain G ⊂ C, and q = 4k/(k + 1)2 , then |f |q is subharmonic. The exponent q is optimal. Proof If u(z0 ) = |f (z0 )|2 = 0, then (*) from the Preliminaries holds. If u(z0 ) > 0, then there exists a neighborhood U of z0 such that u is of class C 2 (U ) (because the zeroes of u are isolated), and then we can prove that Δu ≥ 0 on U . Thus the proof reduces to proving that Δu(z) ≥ 0 whenever u(z) > 0. We have to prove that Δ(|f |p )  0, where p = 4k/(1 + k)2 . Since p − 2 < 0, we get from Lemma 2.52 that Δ(|f |p )  p2 (|g  |2 + |h |2 )|f |p−2 + 2p(p − 2)|f |p−4 |g  | · |h | · |f |2 = p2 |g  |2 (m2 + 1)|f |p−1 + 2p(p − 2)|g  |2 |f |p−2 m = p|g  |2 |f |p−2 [p(1 + m) + 2(p − 2)m], where m = |h |/|g  |  k. The function m → p(1 + m2 ) + 2(p − 2)m has a negative derivative (because p < 1 and m < 1), and this implies that (1 + m2 )p + 2(p − 2)m  (1 + k 2 )p + 2(p − 2)k. Observe now that (1 + k 2 )p + 2(p − 2)k  0 if and only if p  4k/(1 + k)2 . This proves that |f |q is subharmonic. To prove that q is optimal consider f (z) = z + k z¯ . Note that ¯ = 1 + k. |f (1)| = |1 + k · 1|

34

2 Quasiconformal and Quasiregular Harmonic Mappings

By Lemma 2.52 for g(z) = z and h(z) = kz, we get Δ(|f |p )(1) = p2 (1 + k 2 )(1 + k)p−2 + 2p(p − 2)(1 + k)p−2 k. It follows that Δ(|f |p )(1)  0 if and only if p(1 + k 2 ) + 2(p − 2)k  0, which, as indicated above, is equivalent to p  q.

 

Lemma 2.54 (Jordan Inequality) For 0 < x ≤ π/2 there holds 2 sin(x) ≤ < 1. π x Lemma 2.55 For t1 , t2 ∈ R such that |t1 − t2 | ≤ π there holds |t1 − t2 | ≤

π it1 |e − eit2 |, 2

|eit1 − eit2 | ≤ |t1 − t2 |. Proof For t1 = t2 the conclusion of the lemma holds. Assume that t1 = t2 . Then, we have |eit1 − eit2 | = 2 sin(|t1 − t2 |/2) |eit1 − eit2 |

sin(|t1 − t2 |/2) |t1 − t2 | ≤ |t1 − t2 |, |t1 − t2 |/2

|eit1 − eit2 | π |eit1 − eit2 | # ≤ |eit1 − eit2 | . |t1 − t2 | = " it it # = " sin(|t1 −t2 |/2) |e 1 −e 2 | 2 |t1 −t2 |

|t1 −t2 |/2

  Lemma 2.56 For all δ1 , δ2 ≥ 0, we have ω0 (f, δ1 + δ2 ) ≤ ω0 (f, δ1 ) + ω0 (f, δ2 ). Proof Let δ1 , δ2 > 0. Let us define the sets S(δ) = {|f (eit1 ) − f (eit2 )| : |t1 − t2 | ≤ δ}. Choose a a ∈ S(δ1 + δ2 ). Then there are t1 , t2 ∈ R such that |t1 − t2 | ≤ δ1 + δ2 and |f (eit1 ) − f (eit2 )| = a. There is a t ∈ R such that |t1 − t| ≤ δ1 and |t − t2 | ≤ δ2 . For

2.4 Moduli of Continuity in the Plane

35

b = |f (eit1 ) − f (eit )|,

c = |f (eit ) − f (eit2 )|,

we have that b ∈ S(δ1 ) and c ∈ S(δ2 ). Therefore, b ≤ sup S(δ1 ) = ω0 (f, δ1 ), c ≤ sup S(δ2 ) = ω0 (f, δ2 ), a ≤ b + c ≤ ω0 (f, δ1 ) + ω0 (f, δ2 ). Because all elements of S(δ1 + δ2 ) are less or equal to ω0 (f, δ1 ) + ω0 (f, δ2 ), we have that ω0 (f, δ1 + δ2 ) = sup S(δ1 + δ2 ) ≤ ω0 (f, δ1 ) + ω0 (f, δ2 ).   Corollary 2.57 For every δ ≥ 0, we have the following inequalities 1. ω0 (f, mδ) ≤ mω( f, δ) for every m ∈ N. 2. ω0 (f, λδ) ≤ n ω0 (f, δ) for every λ ∈ R and n ∈ N such that 0 < λ ≤ n. 3. ω0 (f, λδ) ≤ (λ + 1)ω0 (f, δ) λ ∈ R such that λ > 0. Lemma 2.58 We have the following inequalities: ω0 (f, δ) ≤ ω(f, δ) ≤ 2ω0 (f, δ). Proof Let S(δ) = {|f (eit1 − f (eit2 )| : |eit1 − eit2 | ≤ δ}, S0 (δ) = {|f (eit1 − f (eit2 )| : |t1 − t2 | ≤ δ}. If a ∈ S, then there are t1 , t2 ∈ R such that |eit1 − eit2 |, a = |f (eit1 − f (eit2 )|. Hence, |t1 − t2 | ≤

π it1 π |e − eit2 | ≤ δ 2 2

and therefore a ∈ S0 (δπ/2). This means that S(δ) ⊆ S0 (δπ/2) and for this reason ω(f, δ) = sup S(δ) ≤ sup S0 (δπ/2) = ω0 (f, δπ/2).

36

2 Quasiconformal and Quasiregular Harmonic Mappings

If b ∈ S0 , then there are τ1 , τ2 ∈ R such that |τ1 − τ2 | ≤ δ and b = |f (eiτ1 ) − f (eiτ2 )|. Note that |eiτ1 − eiτ2 | ≤ |τ1 − τ2 | ≤ δ and therefore b ∈ S(δ). This means that S0 (δ) ⊆ S(δ) and ω0 (f, δ) = sup S0 (δ) ≤ sup S(δ) = ω(f, δ). Finally, ω0 (f, δ) ≤ ω(f, δ) ≤ ω0 (f, δπ/2) ≤ 2ω0 (δ).   From these lemmas it follows that ω(f, λδ) ≤ 2λω(f, δ),

λ ≥ 1, δ ≥ 0,

(2.21)

and ω(f, δ1 + δ2 ) ≤ 2ω(f, δ1 ) + 2ω(f, δ2 ),

δ1 , δ2 ≥ 0.

(2.22)

As a consequence of (2.21) we have, for 0 < p < 1, 

∞

x

ω(f, t)p ω(f, x)p dt ≤ C , x t2

x > 0,

(2.23)

where C depends only on p. For a continuous function f : D −→ C harmonic in D we already defined moduli of continuity ω(f, δ) and ω(f, ˜ δ). Clearly ω(f, δ) ≤ ω(f, ˜ δ), but the reverse inequality need not hold. To see this consider the function f (reiθ ) =

∞  (−1)n r n cos nθ n=1

n2

,

reiθ ∈ D.

Recall that the polar form of the Laplacian is Δf (r eiθ ) =

∂ 2f 1 ∂f 1 ∂ 2f + + 2 2. 2 r ∂r ∂r r ∂θ

(2.24)

2.4 Moduli of Continuity in the Plane

37

The series from (2.24) can be differentiated by variables r and θ item by item arbitrary many times inside disk r < 1. Any summand of this series is harmonic and therefore f is harmonic in the disk D. The same series is uniformly convergent in the disk r ≤ 1 and all summands are continuous in this disk. Therefore, f is continuous in D. The function v(θ ) = f (eiθ ), |θ | < π, is differentiable, and ∞

 (−1)n−1 sin nθ dv = dθ n n=1 θ |θ | < π. = , 2 The first equality follows from Abel’s test because 1/n (uniformly) and monotonously tends to zero and (−1)n−1 sin(nθ ) is uniformly bounded. The second equality is the expansion of the function θ/2 into Fourier series. Namely, using the fact that the function θ/2 is odd we can conclude that  π θ 1 a0 = dθ = 0, 2π −π 2 and for n ≥ 1 an = 1 bn = π



π −π

1 π



π −π

θ cos(nθ ) dθ = 0, 2

1 θ sin(nθ ) dθ = 2 π



π

θ sin(nθ ) dθ =

0

(−1)n−1 . n

The last equality is obtained by partial integration. Therefore, ∞

∞

n=1

n=1

  (−1)n−1 sin(nθ ) θ = a0 + , (an cos(nθ ) + bn sin(nθ )) = 2 n

|θ | < π.

From v  (θ ) = θ/2 for all θ ∈ (−π, π ) it follows that |v  (c)| < π/2 for all c ∈ (−π, π ). By Lagrange’s theorem we have |f (eiθ )−f (eit )| = |v(θ )−v(t)| = |v  (c)(θ −t)|≤(π/2)|θ −t|,

−π < θ, t < π.

If |eiθ − eit | ≤ δ, then we can choose x, y ∈ R such that eiθ = eix , eit = eiy and |x − y| ≤ π . For this reason and by Lemma 2.55, |f (eiθ )−f (eit )| = |f (eix )−f (eiy )| ≤ and therefore ω(f, δ) ≤

π2 4 δ,

δ > 0.

π 2 ix π π 2 iθ |x −y| ≤ |e −eiy | = |e −eit |, 2 4 4

38

2 Quasiconformal and Quasiregular Harmonic Mappings

On the other hand, the inequality ω(f, ˜ δ) ≤ CMδ, C = const, does not hold because it implies that |∂f/∂r| ≤ CM, which is not true (when r → 1−) because ∞

 r n−1 ln(1 − r) ∂ f (reiθ ) = =− , ∂r n r

for θ = π, 0 < r < 1.

n=1

The second equality is the direct consequence of the Taylor’s expansion − ln(1 − r) =

∞ n  r

n

n=1

,

−1 ≤ r < 1.

However, as was proved by Rubel, Shields, and Taylor [137] as well as Tamrazov [151], that if f is a holomorphic function, then ω(f, ˜ δ) ≤ Cω(f, δ), where C is independent of f and δ. The next theorem will be deduced from Theorem 2.53 by use of some simple properties of the modulus ω(f, δ). Let ω0 (f, δ) = sup{|f (eiθ ) − f (eit )| : |θ − t| ≤ δ, t, θ ∈ R}. To extend this result to HQR functions we need the following consequence of the harmonic Schwarz lemma (see [23, p. 125]). Lemma 2.59 If h : D −→ R is a function harmonic and bounded in the unit disk, with h(0) = 0, then |h(ξ )|  (4/π )||h||∞ , for ξ ∈ D. Theorem 2.60 ([88]) Let f : D −→ C be a k-quasiregular harmonic function which has a continuous extension on D, then there is a constant C depending only on k such that ω(f, ˜ δ) ≤ Cω(f, δ). Proof It is enough to prove that |f (z) − f (w)|  Cω(f, |z − w|) for all z, w ∈ D, where C depends only on k. Assume first that z = r ∈ (0, 1) and |w| = 1. Then, by Theorem 2.53, the function ϕ(ξ ) = |f (w) − f (ξ )|q , where q = 4k/(1 + k)2 < 1 is subharmonic in D and continuous on D, so that ϕ(ζ ) 

1 2π

 ∂D

(1 − r 2 )ϕ(ζ ) |dζ |. |ζ − r|2

For all a, b > 0 holds 

a a+b

q

 +

b a+b

q ≥

b a + = 1, a+b a+b

and therefore 

a a+b

q

 +

b a+b

q ≥1

2.4 Moduli of Continuity in the Plane

39

and (a + b)q ≤ a q + bq . The last inequality also holds if at least one of a, b is zero. Therefore, it holds for a, b ≥ 0. Since, by (2.22), ϕ(ζ )  (ω(f, |w − r| + |r − ζ |))q  2q ω(f, |w − r|)q + 2q ω(f, |r − ζ |)q , we have

 (1 − r 2 )ω(f, |r − ζ |)q 2q−1 |dζ | ϕ(z)  2 ω(f, |w − r|) + π ∂D |ζ − r|2 π q−1 2 it q (1 − r )ω(|r − e |) 2  2q ω(f, |w − r|)q + dt. π |eit − r|2 −π q

q

We have that + |r − eit | =

(1 − r)2 + 4r sin2 (t/2)  1 − r + |t| (0 < r < 1, |t| < π ).

From this, (*) from the Preliminaries and (2.23) it follows that 

π −π

 π (1 − r 2 )ω(f, |r − eit |)q (1 − r)ω(f, 1 − r + t)q dt  C dt 1 2 |eit − r|2 0  (1π−r + t) 2 1−r (1 − r )ω(f, |r − eit |)q = C1 + dt |eit − r|2 0 1−r ∞ ω(f, t)q  C2 (ω(1 − r))q + C2 (1 − r) dt t2 1−r  C3 (ω(f, 1 − r))q  C4 (ω(f, |w − z|))q .

Thus |f (w) − f (z)|  C5 ω(f, |w − z|) provided w ∈ ∂D and z ∈ (0, 1). Using a rotation and the continuity of f , we can extend this inequality to the case where w ∈ ∂D and z ∈ D. If 0 < |w| < 1, we look at the function h(ξ ) − f (ξ w/|w|) − f (ξ z/|w|) for |ξ |  1. Note that this function is harmonic in D, continuous on D, and h(0) = 0. Therefore, by the harmonic Schwarz lemma, the inequality (*) from the Preliminaries, and the previous case, we have |f (w) − f (z)| = |h(w)|  (4/π )|w| ||h||∞  C6 |w|ω(f, |w/|w| − z/|w||),  C7 ω(f, |w/|w| − z/|w||), = C7 ω(f, |w − z|).  

40

2 Quasiconformal and Quasiregular Harmonic Mappings

2.5 Moduli of Continuity in Higher Dimensions In [110], Olli Martio gave sufficient conditions for K-quasiconformality of a harmonic mapping from the unit disk onto itself. In the same paper he posed a question answered fully in the paper [14] where the following result is proved. Theorem 2.61 If u is harmonic in the unit disk D, and f is the restriction of u on S 1 = ∂D, find necessary and sufficient conditions that limz→ζ ur (z) and limz→ζ uθ (z) exist at each boundary point ζ ∈ S 1 . Here, ur and uθ are derivatives of u with respect to r and θ . Moreover, Martio’s paper [110] served as a motivation for further study of such phenomena. The following general, but natural question has been considered in several papers ([12–15]): Can one control the modulus of continuity ωu of a harmonic quasiregular mapping u in Bn , which is continuous on Bn , by the modulus of continuity ωf of its restriction to the boundary Sn−1 , i.e., is it true that ωu ≤ Cωf ? In fact this problem has been studied extensively for harmonic functions and mappings without assumption of quasiregularity. We recall some of the known results. For the unit ball the answer is positive in the case ω(δ) = δ α , 0 < α < 1 (Hölder continuity) and negative in the case ω(δ) = Lδ (Lipschitz continuity). In fact, for bounded plane domains the answer is always negative for Lipschitz continuity (see [7]). However, it is proved in [68] that for general plane domains one has “logarithmic loss of control": ωu (δ) ≤ Cωf (δ) log(1/δ) for δ > 0. For n ≥ 2,  P [φ](x) =

S n−1

P (x, ξ )φ(ξ )dσ (ξ ), x ∈ Bn

2

1−|x| n−1 → Rn where P (x, ξ ) = |x−ξ |n is the Poisson kernel for the unit ball, and φ : S is a continuous mapping. It was shown in [13] that Lipschitz continuity is preserved by harmonic extension, if the extension is quasiregular.

Theorem 2.62 ([13]) Assume φ : S n−1 → Rn satisfies a Lipschitz condition: |φ(ξ ) − φ(η)| ≤ L|ξ − η|, ξ, η ∈ S n−1 and assume u = P [φ] : Bn → Rn is K-quasiregular. Then |u(x) − u(y)| ≤ C  |x − y|, x, y ∈ Bn where C  depends on L, K, and n only.

2.5 Moduli of Continuity in Higher Dimensions

41

Proof We choose x0 = r ξ0 ∈ Bn , r = |x|, ξ0 ∈ S n−1 . Let T = Tx0 rS n−1 be the (n − 1)-dimensional tangent plane at x0 to the sphere rS n−1 . We want to prove that ||D(u|T )(x0 )|| ≤ C(n)L.

(2.25)

Without loss of generality we can assume ξ0 = en and x0 = r en by using a simple calculation −2xj xj − ξj ∂ P (x, ξ ) = − n(1 − |x|2 ) . ∂xj |x − ξ |n |x − ξ |n+2 Hence, for 1 ≤ j < n we have ξj ∂ P (x0 , ξ ) = n(1 − |x0 |2 ) . ∂xj |x0 − ξ |n+2 It is important to note that this kernel is odd in ξ (with respect to reflection (ξ1 , . . . , ξj , . . . , ξn ) → (ξ1 , . . . , −ξj , . . . , ξn )), a typical fact for kernels obtained by differentiation. This observation and differentiation under integral sign give, for any 1 ≤ j < n,  ξj ∂u 2 (x0 ) = n(1 − r ) φ(ξ ) dσ (ξ ) n−1 ∂xj |x − ξ |n+2 0 S ξj = n(1 − r 2 ) (φ(ξ ) − φ(ξ0 )) dσ (ξ ). n−1 |x − ξ |n+2 0 S Using the elementary inequality |ξj | ≤ |ξ − ξ0 |, (1 ≤ j < n, ξ ∈ S n−1 ) and Lipschitz continuity of φ we get $ $  $ $ ∂u |ξj | |ξ − ξ0 | $ ≤ Ln(1 − r 2 ) $ (x ) dσ (ξ ) 0 $ $ ∂x n−1 |x0 − ξ |n+2 j S |ξ − ξ0 |2 dσ (ξ ). ≤ Ln(1 − r 2 ) n+2 S n−1 |x0 − ξ | In order to estimate the last integral, we split S n−1 into two subsets E = {ξ ∈ : |ξ − ξ0 | ≤ 1 − r} and F = {ξ ∈ S n−1 : |ξ − ξ0 | > 1 − r}. Since |ξ − x0 | ≥ 1 − |x0 | for all ξ ∈ S n−1 we have S n−1

 E

 |ξ − ξ0 |2 2 −n−2 dσ (ξ ) ≤ (1 − r ) |ξ − ξ0 |2 dσ (ξ ) |x0 − ξ |n+2 E  1−r

≤ (1 − r 2 )−n−2

≤

0 −1

2 (1 − r) n+1

.

ρ 2 ρ n−1 dρ

42

2 Quasiconformal and Quasiregular Harmonic Mappings

On the other hand, |ξ − ξ0 | ≤ Cn |ξ − x0 | for every ξ ∈ F , so  F

|ξ − ξ0 |2 dσ (ξ ) ≤ Cnn+2 |x0 − ξ |n+2  ≤ Cn

 2

|ξ − ξ0 |−n dσ (ξ )

F

ρ −n ρ n−2 dρ

1−r

≤ Cn (1 − r)−1 . Combining these two estimates we get $ $ $ ∂u $ $ $ (x ) $ ∂x 0 $ ≤ LC(n) j

for 1 ≤ j < n. Due to rotational symmetry, the same estimate holds for every derivative in any tangential direction. This establishes estimate (2.25). Finally, Kquasiregularity gives ||Du(x)|| ≤ LKC(n). Now the mean value theorem gives Lipschitz continuity of u.

 

In [14], the authors gave the estimate Lu ≤ KLφ , where Lu , Lφ denote the Lipschitz constants of u, φ, respectively. Also in [14], Theorem 2.61 is extended to the n-dimensional case. In [15], the following results were proved. Theorem 2.63 ([15]) There is a constant q = q(K, n) ∈ (0, 1) such that |u|q is subharmonic in Ω ⊂ Rn whenever u : Ω → Rn is a K-quasiregular harmonic map. n

Theorem 2.64 ([15]) Let u : B → Rn is a continuous map which is Kquasiregular and harmonic in Bn . Then ωu (δ) ≤ Cωf (δ) for δ > 0, where f = u|Sn−1 and C is a constant depending only on K, ωf , and n. The proof of Theorem 2.63 is based on a linear algebra extremal problem. n It should be noted that every continuous map u : B → Ω, which is HQC in Bn and whose domain Ω is bounded and has C 2 boundary, is Lipschitz continuous, see [76]. Moreover, every holomorphic quasiregular mapping on a domain Ω ⊂ Cn (n > 1) with C 2 boundary is Lipschitz continuous, see [126]. This paper gives also an example of a holomorphic quasiregular map in a domain Ω ⊂ C2 (with non-smooth boundary) which is not Lipschitz. It is worth pointing out that a global holomorphic quasiregular map is affine by a result of Marden and Rickman [105].

2.6 An Example of Non-Lipschitz HQC Mapping on the Unit Ball

43

2.6 An Example of Non-Lipschitz HQC Mapping on the Unit Ball Given the facts presented above, one is tempted to make the following conjecture: every HQC map u : Bn → Ω is Lipschitz continuous. This is, however, false and an example for n = 3 is given in [15] on which we base this section. Our next lemma is true for vectors in every real inner product space, although for our purposes here M3 (R) ∼ = R9 is the only case of interest, where M3 (R) is a set of all square matrices of order 3. Recall that for T ∈ M3 (R), T = 0, its normalization T˜ is defined by T˜ = ||T ||−1 T . √ Lemma 2.65 Let A ∈ M3 (R), A = 0, and 0 <  < 2. Let B1 , . . . , Bk be ˜ < , where B = matrices satisfying A˜ − B˜ j  < , 1 ≤ j ≤ k. Then A˜ − B B1 + · · · + Bk . Proof The set ˜ < ε} S = {B ∈ M3 (R) : B = 0, ||A˜ − B|| is equal to the set {B ∈ M3 (R) : B = 0, (A, B) < arccos(1 − ε2 /2)}. This is a cone. It is convex and closed under multiplication by a positive scalar. So, if B1 , . . . , Bk ∈ S, then B1 + · · · + Bk = k(k −1 B1 + · · · + k −1 Bk ) ∈ S.   Example 2.66 Let us use the following notation: X = (x, y, z). We first find a mapping f : H3 → R3 such that 1. f is continuous on H3 . 2. f is not Lipschitz on L = {(0, 0, z) : 0 ≤ z ≤ 1}. 3. f is HQC on H3 . Note that the same is true for the restriction of f to the closed unit ball centered at (0, 0, 1). Let g(X) = X/|X|3 . This mapping is the gradient of a harmonic function 1/|X|, up to a constant, and therefore harmonic for X = 0. Note that Dg(X) =

1 (I − 3UXT · UX ), |X|3

44

2 Quasiconformal and Quasiregular Harmonic Mappings

where UX = X/|X|. Note also that |g(X)| ≤ 1/|X|2 and Dg(X) ≤ C0 /|X|3 . Now set f (X) = f0 (X) +

∞ 

fn (X),

n=1

where f0 (X) = (x, y, −2z), fn (X) = cn g(X − Xn ), and Xn = (0, 0, −rn ). We are going to show that the sequences rn and cn can be chosen so that the above three conditions are satisfied. We will require that they are strictly positive and that limn→∞ rn = 0 monotonically. First of all we impose the condition ∞  cn

r2 n=1 n

< +∞

(C)

that will be sufficient for continuity up to the boundary. To see this note that, for every X ∈ H3 , ∞ 

|fn (X)| =

n=1

∞  n=1

∞

 cn cn ≤ < +∞ 2 |X − Xn | rn2 n=1

 3 and therefore the series ∞ n=1 fn (X) converges absolutely and uniformly on H . Now we impose the condition ∞  cn

r3 n=1 n

= +∞

(N L)

that is sufficient for the property 2. More precisely, for X = (0, 0, z) ∈ H3 , we have ∂ 2cn fn (X) = − e3 , ∂z (z + rn )3 where e3 = (0, 0, 1) and   ∞  ∂ cn f (X) = − 2 + e3 . ∂z (z + rn )3 n=1

∂ f (0, 0, z)| → ∞ as z → 0 and this implies It follows that if (NL) holds, then | ∂z that f is not Lipschitz continuous on L. Let us now look for a sufficient condition for quasiconformality. Towards this, note that a matrix A ∈ M3 (R), A = 0 is K-quasiconformal iff its normalization

2.6 An Example of Non-Lipschitz HQC Mapping on the Unit Ball

45

A˜ = A/A is K-quasiconformal. Moreover, for each compact H ⊂ GL3 (R) there is a K ≥ 1 such that every A ∈ H is K-quasiconformal. √ Let A0 = diag(1, 1, −2) = Df0 . Choose 0 < δ < 2 such that H = {B : A˜ 0 − B ≤ δ} is a compact subset of GL3 (R). It follows that there is a K0 > 1 such that every B ∈ H is a K0 -quasiconformal matrix. Let AX = I − 3UXT · UX , X ∈ H3 , X = 0. Note that AX = AUX . It is clear that AX is continuous in X and A(0,0,1) = A0 and therefore there is an  > 0 such that A˜X − A˜0  < δ whenever |X − e3 | < . Putting it into an equivalent form, there ˜ 0  < δ or, in coordinates, is an η > 0 such that tan (e3 , X) < η implies A˜ X − A√ x 2 +y 2 < η. A˜ X − A˜ 0  < δ whenever X = (x, y, z) ∈ H3 satisfies z ˜ ˜ < δ. Now we choose β > 0 such that A0 − B ≤ β implies A0 − B Let us now show that the condition  √ cn 2 2C0 ≤ β for all ρ > 0 (QC1 ) 3 ρ (rn + ρ) rn ≤ η

is sufficient for quasiconformality of f . Note that for z > 0, Df , (0, 0, z) is a constant multiple of A0 . Consider now X = (x, y, z) ∈ H3 with ρ = x 2 + y 2 > 0. Then Df (X) = Df0 + = A0 +

∞ 

Dfn (X)

n=1 ∞ 

cn AX−Xn

n=1 

= A0 +

cn AX−Xn +

rn ≤ ρη

= A0 + R + T .



cn AX−Xn

rn > ρη

Note that the sum T is finite (possibly empty) and for each term in that sum we have ,

x2 + y2 ρ ≤ ≤ η. z + rn rn

It follows that A˜ X−Xn − A˜ 0  < δ for each term in that sum. Let us now estimate the norm of R:  cn AX−Xn  R ≤ rn ≤ ρη

≤ C0



rn ≤ ρη

cn |X − Xn |3

46

2 Quasiconformal and Quasiregular Harmonic Mappings



cn . 2 2 3/2 rn ≤ ρη ρ + (z + rn )  cn ≤ C0 2 + r 2 )3/2 (ρ n rn ≤ ρη  √ cn ≤ 2 2C0 (r + ρ)3 n ρ

= C0

≤ β.

rn ≤ η

It follows that (A0 + R) − A0  ≤ β and therefore A˜ 0 − A 0 + R < δ. Note that Df (X) can be represented as a sum of terms satisfying the assumptions of    Lemma 2.65, and so Df (X) − A˜ 0  < δ, and Df (X) ∈ H . It follows that Df (X) is a K0 -quasiconformal matrix and so is Df (X). It is easy to verify that f is oneto-one, hence it follows that f is a K0 -quasiconformal map. Note that the condition (QC1 ) is equivalent to the following: ∞ 

ck ≤ Mrn3

for all n ≥ 1

(QC),

k=n

where M is constant depending on η, C0 , and β. The exact value of M is not important for us here. This is so because once we have sequences rn and cn satisfying (C), (N L), and (QC) for some M, it suffices to multiply cn with a suitably small constant to get M as small as desired. This follows from that fact that the conditions (C) and (N L) are invariant under such change of cn . n n Note that that the sequences rn = 2−2 /3 and cn = 2−2 satisfy the conditions (C), (N L), and (QC). We can therefore conclude there is a non-Lipschitz qhc map on Bn continuous up to the boundary. In conclusion we note that an analogous construction can be carried in all dimensions k ≥ 2. Moreover, by multiplying  the z-component of our function by a factor −1/2 and taking a tail of the series ∞ n=1 fn (X), one can get the constant of quasiconformality as close to 1 as desired. We would like to emphasize that the harmonicity is important here because it is easy to construct non-Lipschitz quasiconformal mappings.

2.7 Hölder Continuity of HQC Mappings It is clear that for general quasiconformal mappings u : Ω1 → Ω2 one cannot expect that the modulus of continuity behaves as in the Theorem 2.64, even for Ω1 = Bn . However, for bounded Ω2 , Hölder continuity of u|∂Ω1 implies Hölder continuity of u, but with a possibly different Hölder exponent, see [120] and [111].

2.7 Hölder Continuity of HQC Mappings

47

The following theorem is the main result of the paper [111]. Theorem 2.67 Let D be a bounded domain in Rn and let f be a continuous mapping of D into Rn which is quasiconformal in D. Assume that for some M > 0 and 0 < α ≤ 1, |f (x) − f (y)| ≤ M|x − y|α

(2.26)

for x and y that lie on ∂D. Then |f (x) − f (y)| ≤ M  |x − y|β 1/(1−n)

for all x and y on D, where β = min(α, KI n, K(f ), and diam(D).

(2.27)

) and M  depends only on M, α,

It turns out that the exponent β is the best possible, since there is an example of a radial quasiconformal map f (x) = |x|α−1 x, 0 < α < 1, of Bn onto itself shows (see [155], p. 49). Moreover, the assumption of boundedness is essential. To see this, one can consider g(x) = |x|a x, |x| ≥ 1 where a > 0. Then g is quasiconformal in D = Rn \ Bn (see [155], p. 49), it is identity on ∂D and hence Lipschitz continuous on ∂D. However, |g(te1 ) − g(e1 )|  t a+1 , t → ∞, and therefore g is not globally Lipschitz continuous on D. Regarding this result, P. Koskela has suggested the following question: Question 2.68 Is it possible to replace β with α if we assume, in addition to quasiconformality, that f is harmonic? This question was answered in [16] on which we base the rest of this section and where Theorem 2.76 ( [16, Theorem 2.1]) below was proved showing that for a wide range of domains, including those with a uniformly perfect boundary, Hölder continuity on the boundary implies Hölder continuity with the same exponent inside the domain for the class of HQC mappings. In fact, we prove a more general result, including domains having a thin, (in the sense of capacity) portion of the boundary. However, this generality is in a sense illusory, because each HQC mapping extends harmonically and quasiconformally across such a portion of the boundary. We need a notion of capacity. Recall that a condenser is a pair (K, U ), where K is a nonempty compact subset of an open set U ⊂ Rn . The capacity of the condenser (K, U ) is defined as follows:  |∇u|n dV , cap(K, U ) = inf Rn

where infimum is taken over all continuous real-valued u ∈ ACLn (Rn ) such that u(x) = 1 for x ∈ K and u(x) = 0 for x ∈ Rn \ U . It turns out that the ACLn condition can be replaced with the Lipschitz continuity in this definition. It should be noted that for a compact K ⊂ Rn and open bounded sets U1 and U2 containing K, one has: cap(K, U1 ) = 0 iff cap(K, U2 ) = 0. It follows that the notion of a

48

2 Quasiconformal and Quasiregular Harmonic Mappings

compact set of zero capacity is well defined (see [158], Remarks 7.13), and that we can write cap(K) = 0 in this realm. It turns out that the notions of modulus of a curve family defined earlier and capacity are related since, by results of [67] and [164], we have cap(K, U ) = M(Δ(K, ∂U ; U )), where Δ(E, F ; G) denotes family of curves connecting E to F within G, see [155] or [158] for details. Beside this notion of capacity related to quasiconformal mappings, we need also the notion of Wiener capacity related to harmonic functions. For a compact K ⊂ Rn , n ≥ 3, the Weiner capacity is defined by  capW (K) = inf

Rn

|∇u|2 dV ,

where infimum is taken over all Lipschitz continuous compactly supported functions u on Rn such that u = 1 on K. It should be noted that every compact K ⊂ Rn which has capacity zero has Wiener capacity zero. To see this, choose an open ball BR = B(0, R) ⊃ K. Since n ≥ 2 we have, by the Hölder inequality, 

 Rn

|∇u|2 dV ≤ |BR |1−2/n

2/n Rn

|∇u|n dV

for each Lipschitz continuous u vanishing outside U , and our claim follows immediately from definitions. The condenser capacity is a conformal invariant and behaves nicely under K-quasiconformal and K-quasiregular mappings, almost like a moduli of curve families (see [113, Theorem 6.2, Theorem 7.1]). A compact set K ⊂ Rn , consisting of at least two points, is α-uniformly perfect (α > 0) if there is no ring R separating K (i.e., so that both components of Rn \ R intersect K) such that mod(R) > α. We say that a compact K ⊂ Rn is uniformly perfect if it is α-uniformly perfect for some α > 0. A characterization of uniform perfectness is given in the following theorem of P. Järvi and M. Vuorinen. Theorem 2.69 ([71, Theorem 4.1]) Let E ⊂ Rn be a closed set containing at least two points. Then the following properties are equivalent: 1. E is α-uniformly perfect for some α > 0; n 2. there exist positive constants β and C1 such that Λβ (B (x, r) ∩ E)  C1 r β for n x ∈ E ∩ R and r ∈ (0, d(E)); 3. there is a positive constant C2 such that cap(x, E, r)  C2 for x ∈ E ∩ Rn and r ∈ (0, d(E)). The constants α, β, C1 , and C2 depend only on n and each other.

2.7 Hölder Continuity of HQC Mappings

49

Uniformly perfect domains are important in Geometric Function Theory (see [80, Chapter 15], [47, pp. 343–345], [127]). We denote the α-dimensional Hausdorff measure of a set F ⊂ Rn by Λα (F ). If K is a compact set, then by [71, Corollary 4.2] Λβ (K) is positive for some positive β. Let D denote a bounded domain in Rn , n ≥ 3. Let n

Γ0 = {x ∈ ∂D : cap (B (x, ) ∩ ∂D) = 0 for some  > 0}, and Γ1 = ∂D \ Γ0 . Using this notation we can state the following facts which will be needed in the rest of this section. Lemma 2.70 ([158, Lemma 7.14, p. 86] and [132, p. 72]) Suppose that F is a compact set in Rn of capacity zero. Then for every α > 0, the α-dimensional Hausdorff measure Λα (F ) of F is zero. In particular, int F = ∅, and F is totally disconnected. Lemma 2.71 ([155, Theorem 35.1, p. 118]) Suppose that f : D −→ D  is a homeomorphism and that E ⊆ D is a set such that E is closed in D and such that E has a σ -finite (n − 1)-dimensional measure. Suppose also that every point in D \ E has a neighborhood U such that KI (f |U ) ≤ a and KO (f |U ) ≤ b. Then KI (f ) ≤ a and KO (f ) ≤ b. Lemma 2.72 ([158, Corollary 5.41, p. 63]) If x ∈ Rn , 0 < a < b < ∞, and F1 , F2 ⊆ B n (x, a), F3 ⊆ Rn \ B n (x, b), Γij = Δ(Fi , Fj ), then 1. M(Γ12 ) ≥ 3−n min{M(Γ13 ), M(Γ23 ), cn (log ab )}, 2. M(Γ12 ) ≥ d(n, b/a) min{M(Γ13 ) M(Γ23 )}. Lemma 2.73 ( [111, Lemma 8]) Let y be a point in a domain D, let x be a point in ∂D closest to y and let d = |x − y|. Suppose that n

cap(B (y, d/2), D) ≥ m > 0. If f is a continuous mapping of D into Rn which is quasiconformal in D and if the boundary mapping satisfies a Hölder condition at x with exponent α and constant M, then ˆ − y|α , |f (x) − f (y)| ≤ M|x where Mˆ > 0 depends only on m, n, K(f ), α, and M. Definition 2.74 We say that E is removable for the class H , where H is a class of functions harmonic in a bounded domain D, if it is possible to give metrical conditions on E which guarantee that every function in H can be extended to a harmonic function also in E.

50

2 Quasiconformal and Quasiregular Harmonic Mappings

Lemma 2.75 ([34, Theorem 2, p. 91]) A set E is removable for the class Hα of harmonic functions satisfying a Lipschitz condition of order α, 0 < α < 1 |u(x) − u(x  )| ≤ Const.|x − x  |α ,

x, x  ∈ D,

if and only if Λd−2+α (E) = 0. Theorem 2.76 ([16]) Let D be a bounded domain in Rn and assume f : D → Rn is continuous on D and harmonic and quasiconformal in D. Assume f is Hölder continuous with exponent α, 0 < α ≤ 1, on ∂D and Γ1 is uniformly perfect. Then f is Hölder continuous with exponent α on D. If Γ0 is empty, we obtain the following: Corollary 2.77 ([16]) Let D be a bounded domain in Rn . If f : D → Rn is continuous on D, Hölder continuous with exponent α, 0 < α ≤ 1 on ∂D, harmonic and quasiconformal in D, and if ∂D is uniformly perfect, then f is Hölder continuous with exponent α on D. The first step in the proof of Theorem 2.76 consists in reducing it to the case Γ0 = ∅. In fact, we show that the existence of a HQC extension of f across Γ0 follows from the well-known results. Let D  = D ∪ Γ0 . Then D  is an open set in Rn , Γ0 is a closed subset of D  , and ∂D  = Γ1 . Note that cap(K ∩ Γ0 ) = 0 for each compact K ⊂ D  , and therefore, by Lemma 2.70, Λα (K ∩ Γ0 ) = 0 for each α > 0. In particular, Γ0 has σ -finite (n − 1)-dimensional Hausdorff measure. Note that this set is closed in D  , so we can apply Lemma 2.71 to conclude that f has a quasiconformal extension F across Γ0 , which has the same quasiconformality constant as f . Our set Γ0 is a countable union of compact sets Kj of capacity zero and, therefore, of Wiener capacity zero, so we conclude that Γ0 has Wiener capacity zero. Therefore, by a Lemma 2.75 (see [34]), there is a (unique) extension G : D  → Rn of f which is harmonic in D  . Note that F = G is a harmonic quasiconformal extension of f to D  which has the same quasiconformality constant as f . Note that we have just reduced the proof of Theorem 2.76 to the proof of Corollary 2.77. So let us now start with the proof of Corollary 2.77. We need the following lemma. Lemma 2.78 Let D ⊂ Rn be a bounded domain with uniformly perfect boundary. There exists a constant m > 0 such that for every y ∈ D we have d n cap(B (y, ), D) ≥ m , 2

d = dist(y, ∂D).

(2.28)

Proof We start by fixing y ∈ D as above and z ∈ ∂D such that |y − z| = d ≡ r. Then diam(∂D) = diam(D) > 2r. Set F1 = B n (z, r) ∩ (∂D) and F2 = B n (z, r) ∩ B n (y, d2 ), F3 = S(z, 2r). Let Γi,j = Δ(Fi , Fj ; Rn ) for i, j = 1, 2, 3. By Theorem 2.69 there exists a constant a = a(E, n) > 0 such that M(Γ1,3 ) ≥ a,

2.7 Hölder Continuity of HQC Mappings

51

while by standard estimates for the spherical ring (see Example 2.9) there exists b = b(n) > 0 such that M(Γ2,3 ) ≥ b . Now, by Lemma 2.72 there exists m = m(E, n) > 0 such that M(Γ1,2 ) ≥ m . n

As the final step, with B = B (y, d/2), we have cap(B, D) = M(Δ(B, ∂D; Rn )) ≥ M(Γ1,2 ) ≥ m .   Using this lemma, our assumption |f (x1 ) − f (x2 )| ≤ C|x1 − x2 |α ,

x1 , x2 ∈ ∂D,

and Lemma 2.73, we conclude that there is a constant M, depending on m, n, K(f ), C, and α only such that |f (x) − f (y)| ≤ M|x − y|α ,

y ∈ D, x ∈ ∂D, dist(y, ∂D) = |x − y|.

It should be noted that an argument from [111] shows that the above estimate holds for y ∈ D, x ∈ ∂D without any further conditions, but with possibly different constant: |f (x) − f (y)| ≤ M  |x − y|α ,

y ∈ D, x ∈ ∂D.

(2.29)

The following lemma was proved in [32, Lemma A.1, p. 292] for real valued functions. The proof of this lemma relies on the maximum principle which holds also for vector valued harmonic functions, and so the lemma holds for harmonic mappings as well. Lemma 2.79 Assume h : D → Rn is continuous on D and harmonic in D, where D is bounded domain. Assume for each x0 ∈ ∂D we have sup

B n (x0 ,r)∩D 

|h(x) − h(x0 )| ≤ ω(r)

for 0 < r ≤ r0 .

Then |h(x) − h(y)| ≤ ω(|x − y|) whenever x, y ∈ D and |x − y| ≤ r0 . Now we combine (2.29) and the above lemma, with r0 = diam(D), to complete the proof of Corollary 2.77 and therefore of Theorem 2.76 as well.

52

2 Quasiconformal and Quasiregular Harmonic Mappings

Let ω is any nonnegative nondecreasing function satisfying ω(2t) ≤ 2ω(t) for t ≥ 0. Let D be a bounded domain in Rn , n ≥ 2 and f a continuous mapping from D into Rn which is quasiconformal in D and satisfies that |f (x) − f (y)| ≤ ω(|x −y|) for all x, y ∈ ∂D. In [17] it is shown that |f (x)−f (y)| ≤ C max{ω(|x − y|), |x − y|α } for all x, y ∈ D, where α = KI (f )1/(1−n) .

2.8 Subharmonicity of the Modulus of HQR Mappings In the paper [78] on which this section is based, Theorem 2.53 is extended to the n-dimensional setting and also Theorem 2.63 was improved by giving the optimal value of q. Moreover for the first time the case q < 0 was considered and the proofs in [78] are completely different from those presented above. Let 0 < λ21  λ22  · · ·  λ2n be the eigenvalues of the matrix Du(x)Du(x)t . Then Ju (x) =

n /

λk ,

k=1

|Du| = λn and l(Du) = λ1 . For the Hilbert–Schmidt norm of the matrix Du(x) defined by ||Du(x)|| =

,

Trace(Du(x)Du(x)t )

we have 0 0 1 n $ 1 n $ 1 $ ∂u $2 1 ∂u ∂u 2 2 $ $ · = ||Du(x)|| = $ ∂x $ ∂xk ∂xk k k=1

k=1

and 0 1 n 1 ||Du(x)|| = 2 λ2 . k

k=1

For a quasiregular mapping we have λn λk ,  K, λk λ1

k = 1, . . . , n.

2.8 Subharmonicity of the Modulus of HQR Mappings

53

Lemma 2.80 ([150, Chapter 7, 3.1.3, p. 218]) Let u : Ω −→ Rn be a harmonic mapping where Ω is a domain in Rn . Then for q ∈ R and all z ∈ Ω \ u−1 (0) $2 ⎤ $ $ $ n  $ ⎥ $ ⎢ Δ|u(z)|q = q ⎣|u(z)|q−2 Du(z)2 + (q − 2)|u(z)|q−4 $$ uj (z)∇uj (z)$$ ⎦ . $ $j =1 ⎡

(2.30) Proof Write v := |u|q = (u21 + · · · + u2n )p for p = q/2. A direct computation gives vxi = q(u21 + · · · + u2n )p−1 (u1 u1 xi + · · · + un un xi ) vxi xi = 2q(p − 1)(u21 + · · · + u2n )p−2 (u1 u1 xi + · · · + u2 u2 xi )2 + (u21 + · · · u2n )p−1 [u1 u1 xi xi + (u1 u1 xi )2 + · · · + un u1 xi xi + (un u1 xi )2 ]. Therefore  Δv = q{|u|

q−2

(u1 Δu1 + · · · + un Δun ) +

+ (q − 2)|u|q−4 = q{|u|q−2

n n  

 n 

k=1 j =1

j =1

= q|u|q−4 {|u|2

n  j =1

+ ··· +

n 

 u21 xk

k=1

uj u2j xk } 

k=1

= q|u|q−4 {|u|2

u21 xk

k=1

u21 xk + · · · + u2n xk

 n n  

 n 

 uj u2j xk

⎛ ⎞2 n n   ⎝ uj uj xk ⎠ } + (q − 2)|u|q−4 k=1

j =1

⎛ ⎞2 n n   ⎝ uj uj xk ⎠ } + (q − 2)

k=1

k=1

|∇uj |2 + (q − 2)

n 

j =1

⎛ ⎞2 n  ⎝ uj uj xk ⎠ }.

k=1

j =1

  1 − (n − 1)K 2

Lemma 2.81 ( [78]) For each < (linear) harmonic K-quasiconformal mapping u subharmonic.

q < 1 − n−1 K2 : Rn −→ Rn

and q = 0 there is a such that |u|q is not

Proof Assume first that q > 0. We will consider the linear mapping u : Rn −→ Rn defined by u(x1 , . . . , xn ) = (x1 , . . . , xn−1 , xn K)

(2.31)

54

2 Quasiconformal and Quasiregular Harmonic Mappings

where K  1. This is obviously harmonic and K-quasiconformal. Putting this into formula (2.30) we get $ $2 $ $ n−1 $ $ [(n − 1) + K 2 ]|u|2 + (q − 2) $$ xj ej + K 2 en xn $$ ≥ 0 $j =1 $ which is equivalent to $ $2 $ $ n−1 $ $ (n − 1 + K 2 ) ⎣ xj2 + K 2 xn2 ⎦ + (q − 2) $$ xj ej + K 2 en xn $$ ≥ 0. $j =1 $ n=1 ⎡

j −1 

⎤

By choosing x1 = · · · = xn−1 = 0 and xn = 1, we obtain (n − 1 + K 2 )K 2 ≥ (2 − q)K 4 which is equivalent to q ≥1−

n−1 . K2

For q < 0 we consider the linear mapping u : Rn −→ Rn defined by u(x1 , . . . , xn ) = (x1 , . . . , xn−1 , xn /K).

(2.32)  

Theorem 2.82 ([78]) Let Ω ⊂ Rn be a domain and let u : Ω → Rn be a K-quasiregular and harmonic mapping. Then the mapping g(x) = |u(x)|q is subharmonic in 1. Ω for q  max{1 − n−1 , 0}; K2 −1 2. Ω \ u (0), for q  1 − (n − 1)K 2 . Moreover, for 1 − (n − 1)K 2 < q < 1 − n−1 and q = 0 there exists a KK2 quasiconformal harmonic mapping such that |u|q is not subharmonic. Proof Let us fix such a map u : Ω → Rn and set Ω0 = Ω \ u−1 {0}. We have to find all positive real numbers q such that Δ|u|q ≥ 0 on Ω0 . Since u is quasiregular, the set Z = {x ∈ Ω0 : det Du(x) = 0} has measure zero (see [158]). The set Z is also closed since u is smooth. In particular, Ω1 = Ω0 \ Z is dense in Ω0 and thus it suffices to prove that Δ|u|q ≥ 0 on Ω1 .

2.8 Subharmonicity of the Modulus of HQR Mappings

55

By Lemma 2.80, we find all real q such that ⎛ $ $2 ⎞ $ n $  $ $ ⎟ ⎜ q−2 2 q−4 $ q ⎝|u| Du + (q − 2)|u| uj ∇uj $$ ⎠ ≥ 0. $ $j =1 $ If q ≥ 2, then Δ|u|q ≥ 0. Assume that q ≥ 0 and q < 2 such that $2 $ $ $ $ $ n 1 $ |u(x)|2 Du(x)2 , x ∈ Ω1 . uj (x)∇uj (x)$$ ≤ $ 2−q $ $j =1 After normalization, we see that it suffices to find a constant q = q(K, n) < 2 such that $2 $ $ $ $ $ n 1 $ Du(x)2 , x ∈ Ω1 . zj ∇uj (x)$$ ≤ (2.33) sup $ 2−q $ $ j =1

Let 0 < λ21 ≤ λ22 ≤ · · · ≤ λ2n be the eigenvalues of the matrix Du(x)Du(x)t . Then $2 $ $ $ $ $ n $ sup $ zj ∇uj (x)$$ = λ2n (2.34) $ z∈S n−1 $j =1 $2 $ $ $ $ $ n inf $$ zj ∇uj (x)$$ = λ21 z∈S n−1 $ $ j =1

(2.35)

and Du(x)2 =

n 

λ2k .

(2.36)

k=1

Because u is K-quasiregular, we have λn ≤ K, k = 1, . . . , n − 1. λk

(2.37)

Thus (2.33) can be written as λ2n ≤

n 1  2 λk . 2−q k=1

(2.38)

56

2 Quasiconformal and Quasiregular Harmonic Mappings

By (2.36) and (2.37) we get that the inequality (2.38) is satisfied whenever 1 1+

1 2−q

(2.39)

n−1 } ≤ q < 2. K2

(2.40)

n−1 K2

≤

i.e., max{0, 1 − If q < 0, then we should have $2 $ $ $ $ $ n 1 inf $$ Du(x)2 , x ∈ Ω1 , zj ∇uj (x)$$ ≥ 2 − q z∈S n−1 $ $ j =1

(2.41)

i.e., 2−q ≥

n  λ2 k

k=1

λ21

.

Because u is K-quasiregular λk ≤ K, k = 2, . . . , n. λ1

(2.42)

q ≤ 1 − (n − 1)K 2 ,

(2.43)

Thus if

then (2.41) holds. By Lemma 2.81, we only need to take u˜ = u|Ω , where u is defined in (2.31) respectively in (2.32).   Remark 2.83 If n = 2, 1 − Theorem 2.53.

n−1 K2

= 1 − K −2 . Thus, Theorem 2.82 is n-extension of

Remark 2.84 In the case 1  K  all q > 0.

√

n − 1, the function |u|q is subharmonic for

Chapter 3

Hyperbolic Type Metrics

The natural setup for our work here is a metric space (G, mG ) where G is a subdomain of Rn , n ≥ 2. For our studies, the distance mG (x, y), x, y ∈ G is required to take into account both how close the points x, y are to each other and the position of the points relative to the boundary ∂G. Metrics of this type are called hyperbolic type metrics and they are substitutes for the hyperbolic metric in dimensions n ≥ 3. The quasihyperbolic metric and the distance ratio metric are both examples of hyperbolic type metrics. A key problem is to study a quasiconformal mapping between metric spaces f : (G, mG ) → (f (G), mf (G) ) and to estimate its modulus of continuity. We expect Holder continuity, but a concrete form of these results may differ from metric to metric. Another question is the comparison of the metrics to each other. In this chapter we will introduce the following types of metrics: 1. 2. 3. 4.

Chordal metric q on Rn = Rn ∪ {∞}. Hyperbolic metric ρD on Bn or Hn . Quasihyperbolic metric kD of a domain D ⊂ Rn . A metric jD closely related to kD .

The last two are defined in all proper subdomains D ⊂ Rn . Both of them generalize hyperbolic metric (on Bn or Hn ) to arbitrary proper subdomain D ⊂ Rn . If D = Hn , then kD = ρ, but the quasihyperbolic and hyperbolic metrics are not equal on the unit ball. This chapter is based on [86] and [98].

© Springer Nature Switzerland AG 2019 V. Todorˇcevi´c, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, https://doi.org/10.1007/978-3-030-22591-9_3

57

58

3 Hyperbolic Type Metrics

3.1 Möbius Transformations Before discussing these metrics, we make a brief excursion into the theory of Möbius transformations [24]. We define the inversion with respect to the sphere S n−1 (a, r) by the formula  ϕ(x) = a +

r |x − a|

2 (x − a).

(3.1)

In the special case of the unit sphere, S n−1 , this becomes ϕ(x) = write x → x ∗ , where x ∗ = |x|x 2 . Then (3.1) becomes ϕ(x) = a + r 2 (x − a)∗ .

x . |x|2

We also

(3.2)

We extend ϕ by continuity to Rn : ϕ(a) = ∞, ϕ(∞) = a. Note that ϕ 2 (x) = (ϕ ◦ ϕ)(x) = x for all x ∈ Rn and ϕ(x) = x ⇔ x ∈ S n−1 (a, r). Lemma 3.1 If a ∈ Rn , r > 0 and ϕ is inversion with respect to the sphere S n−1 (a, r), then for all x, y ∈ Rn , we have |ϕ(x) − ϕ(y)| =

r 2 |x − y| . |x − a| |y − a|

(3.3)

Proof |ϕ(x) − ϕ(y)| = |(ϕ(x) − a) − (ϕ(y) − a)| = |r 2 (x − a)∗ − r 2 (y − a)∗ | $ $ $ y − a $$ 2$ x −a =r $ − |x − a|2 |y − a|2 $ 3 |x − a|2 |y − a|2 (x − a)(y − a) = r2 + −2 4 4 |x − a| |y − a| |x − a|2 |y − a|2 3 1 1 (x − a)(y − a) = r2 + −2 2 2 |x − a| |y − a| |x − a|2 |y − a|2 + r2 = |y − a|2 + |x − a|2 − 2(x − a)(y − a) |x − a| |y − a| =

r 2 |x − y| . |x − a| |y − a|  

We also consider hyperplanes P (a, t) = {x ∈ a ∈ Rn , a = 0, and t is a real number.

Rn | x, a

= t} ∪ {∞}, where

3.1 Möbius Transformations

59

A reflection in P (a, t) is a map ϕ defined as follows: ϕ(x) = x − 2[x, a − t] · a ∗ ,

x ∈ Rn .

(3.4)

We have that ϕ 2 (x) = x for x ∈ Rn and ϕ(x) = x for x ∈ P (a, t). Definition 3.2 A Möbius transformation in Rn is a finite composition of inversions (with respect to spheres or hyperplanes). Möbius transformations form a group, because ϕ 2 = id for each inversion ϕ. We denote the group of Möbius transformations by G M (Rn ). Definition 3.3 ( [158, p. 3]) A map f in G M (Rn ) with f (∞) = ∞ is called a similarity transformation if |f (x) − f (y)| = c|x − y| for all x, y ∈ Rn where c is a positive number. Theorem 3.4 ( [4, p. 19]) Each Möbius transformation is conformal. Proof It suffices, by the chain rule, to prove that each inversion is conformal. This is geometrically obvious for reflections with respect to hyperplanes. It remains to consider reflections in spheres. We shall consider the case of the unit sphere. The general case can be treated similarly. Set f (x) = x ∗ . Then ∂fi 1 (x) = ∂xj |x|2

  2xi xj δij − , |x|2

(3.5)

and therefore f  (x) =

1 (I − 2Q(x)), |x|2

where  Q(x) =

xi xj |x|2

n i,j =1

is a symmetric matrix (i.e., Q = QT ) and Q2 = Q. Then, we have the following: f  (x) · f  (x)T =

1 1 1 (I − 2Q(x))2 = (I − 4Q(x) + 4Q2 (x)) = I. |x|4 |x|4 |x|4

So, |x|2 f  (x) is an orthogonal matrix, hence f  (x) is a conformal matrix.

 

Note that since each inversion f changes orientation, it can be shown (see [129, pp. 137–145]) that we have the inequality det (f  (x)) < 0.

60

3 Hyperbolic Type Metrics

3.2 Chordal Metric The chordal metric q is defined by (Fig. 3.1) q(x, y) = |π(x) − π(y)|;

n

x, y ∈ R ,

(3.6)

where the stereographic projection π : Rn −→ S n (en+1 /2, 1/2) is defined by π(x) = en+1 +

x − en+1 , |x − en+1 |2

x ∈ Rn ,

π(∞) = en+1 .

(3.7)

From (3.7) and Lemma 3.1 follows ⎧ ⎪ ⎪ ⎨,

|x − y| , ,x= ∞ = y, 2 1 + |y|2 1 + |x| q(x, y) = 1 ⎪ ⎪ , y = ∞. ⎩, 1 + |x|2

(3.8)

Let (X1 , d1 ) and (X2 , d2 ) be metric spaces and let f : X1 → X2 be a homeomorphism. We call f an isometry if d2 (f (x), f (y)) = d1 (x, y) for all x, y ∈ n X1 . A map f in G M (R ) is called a spherical isometry if q(f (x), f (y)) = q(x, y) n for all x, y ∈ R . An orthogonal transformation preserves the chordal metric because |x|, |y| and |x −y| are invariant under orthogonal transformations. The same is true for inversion with respect to the unit sphere S n−1 .

Fig. 3.1 Formulas (3.6) and (3.7) visualized [158, Diagram 1.1, p. 5]

3.2 Chordal Metric

61

Lemma 3.5 (Symmetry Lemma) For all nonzero x and y in Rn , $ $ $ $ $ $ x $ $ y $=$ $. $ − |y|x − |x|y $ $ |x| $ $ |y| Lemma 3.6 The inversion with respect to the unit sphere S n−1 preserves the chordal metric. Proof Using the symmetry lemma, we have

$ $ $ x y $$ $ $ |x|2 − |y|2 $ |x ∗ − y ∗ | ∗ ∗ , , q(f (x), f (y)) = q(x , y ) = =3 3 1 + |x ∗ |2 1 + |y ∗ |2 1 1 1+ 2 1+ 2 |x| |y|

=

| |y|2 x − |x|2 y| 1 |x| |y| , ·3 =, 2 2 |x| |y| 1 + |x|2 1 + |y|2 (1 + |x|2 )(1 + |y|2 ) |x|2 |y|2 $ $ $ y $ x − |x| |y| $ |y| |x| $ |x − y| , , =, =, = q(x, y). 2 2 1 + |x| 1 + |y| 1 + |x|2 1 + |y|2

| |y|2 x

− |x|2 y|

  We now present some simple properties of the chordal metric q.

√ Lemma 3.7 ([158, Exercise 1.18 (1), p. 6]) For 0 < t < 1 let ω(t) = t/ 1 − t 2 , we have that q(0, ω(t) e1 ) = t and that ω(t) 2t t   s ω(s) s for 0 < s < t
0}. We start with Hn and define arc length element ds = |dx| xn . So the hyperbolic length of a curve γ : [a, b] → Hn is defined by  lh (γ ) =

 ds =

γ

b

a

|γ  (t)| dt . γn (t)

Hyperbolic distance ρ(a, b) between two points a, b ∈ Hn is defined as inf lh (γ ), where inf is taken over all rectifiable curves γ joining a and b. In fact this infimum is attained by a length minimizing curve called a geodesic. In the special case when one point is above the other (on the same vertical line; see [24, p.135]), the only geodesic arc connecting these points is line segment [a, b] = {(1 − t)a + tb : 0 ≤ t ≤ 1}. If a = r en , b = s en and r, s > 0, we see that $ s $ $ $ dt $$ $ r $$ (3.9) ρ(a, b) = ρ(ren , sen ) = $$ $ = $ln s $ (by a simple integration). r t

64

3 Hyperbolic Type Metrics

This can also be written as cosh ρ(a, b) = 1 +

|a − b|2 . 2rs

(3.10)

One can show that (3.10) is valid for all a, b ∈ Hn , using a suitable Möbius transformation ϕ : Hn → Hn , which is an isometry (preserves arc length element |dx|/xn ). In fact, we have the following theorem: Theorem 3.10 Every Möbius transformation ϕ which maps Hn onto Hn is an isometry with respect to the hyperbolic metric. We give an outline of the proof of this important fact. In view of formula (3.10) it suffices to show that the expression |x − y|2 , xn yn

x, y ∈ Hn

is invariant under the group G M (Hn ). For this, one has to consider inversions with respect to spheres S n−1 (a, r), a ∈ ∂Hn . The case of reflections in hyperplanes orthogonal to ∂Hn is trivial. (Note that inversion in S n−1 (a, r) preserves Hn iff an = 0.) So, let ϕ be such inversion. Then, for x, y ∈ Hn we have |ϕ(x) − ϕ(y)|2 = ϕ(x)n ϕ(y)n



r 2 |x − y| |x − a| |y − a|

2 ·

|x − a|2 |y − a|2 |x − y|2 · = , xn yn r 2 xn r 2 yn

where the first equality follows from (3.3) and the second equality from the fact that ϕ(x)n = 0 +

r 2 xn . |x − a|2

Since ϕ preserves angles and the ray {r en | r > 0} is orthogonal to ∂Hn , it follows that geodesics are semicircles orthogonal to ∂Hn (Fig. 3.3). Absolute (cross) ratio of an ordered quadruple a, b, c, d of distinct points in Rn is defined by |a, b, c, d| =

q(a, c) q(b, d) . q(a, b) q(c, d)

(3.11)

Using our formula for q(x, y) we see that for a, b, c, d ∈ Rn , we have |a, b, c, d| =

|a − c| · |b − d| . |a − b| · |c − d|

(3.12)

3.3 Hyperbolic Metric

65

Fig. 3.3 Some geodesics of Hn [158, Diagram 2.1, p. 20]

This notion is important in the study of Möbius transformation, and the main reason for this is the following theorem. Theorem 3.11 ([24, Theorem 3.2.7]) A mapping ϕ : Rn → Rn is a Möbius transformation iff it preserves absolute ratios. We need only the “only if” part of this theorem. It is clear from (3.12) that Euclidean isometries, or more generally similarity mappings, preserve absolute ratio. So, we have to show that inversion ϕ(x) = x ∗ preserves absolute ratios. However, |x ∗ − y ∗ | =

|x − y| |x| · |y|

and invariance again follows from (3.12). Note that the order of a, b, c, d is important. In fact, |0, e1 , x, ∞| = |x| =

1 , |0, x, e1 , ∞|

|0, e1 , ∞, x| = |x − e1 | =

1 , |0, ∞, e1 , x|

|0, ∞, x, e1 | =

1 |x| = . |x − e1 | |0, x, ∞, e1 |

Let z, w ∈ Hn and let L be an arc of a circle perpendicular to ∂Hn with z, w ∈ L and let {z∗ , w∗ } = L ∩ ∂Hn , the points being labeled so that z∗ , z, w, w∗ occur in this order on L. Now we prove that ρ(z, w) = log |z∗ , z, w, w∗ |.

(3.13)

66

3 Hyperbolic Type Metrics

We already know that both sides are invariant under ϕ ∈ G M (Hn ), so we can use an auxiliary Möbius map which sends z to s en and w to t en (s, t > 0). We can assume s < t. Then z∗ = 0, w∗ = ∞, and t |0, s en , t en , ∞| = , s and we know that ρ(z, w) = log st and that is sufficient (Fig. 3.4). Hyperbolic balls D(a, M) = {x ∈ Hn | ρ(a, x) < M} are also Euclidean balls. The balls of the same radii with center on a vertical line fill in a cone around that vertical line (Fig. 3.5).

Fig. 3.4 The quadruple z∗ , z, w, w∗ [158, Diagram 2.3, p. 22]

Fig. 3.5 The hyperbolic ball D(t en , M) as an Euclidean ball [158, Diagram 2.4, p. 22]

3.4 Distance Ratio Metric jD

67

3.4 Distance Ratio Metric jD For an open set D ⊂ Rn , D = Rn we define d(z) = d(z, ∂D) for z ∈ D and  jD (x, y) = log 1 +

|x − y| min{d(x), d(y)}

 (3.14)

for x, y ∈ D. It is immediate that jD has an invariant property in the sense that jD (x, y) = jf (D) (f (x), f (y)) if mapping f is a similarity. For a nonempty A ⊂ D, we define the jD -diameter of A by jD (A) = sup{jD (x, y) | x, y ∈ A}. Now, we show that jD is a metric on D. Obviously, (M1), (M2), and (M3) hold for jD . The proof of (M4) is divided into three cases: 1. In the case min{d(x), d(y), d(z)} = d(y), (M4) follows from    |x − y| |x − z| |y − z| |x − y| |y − z| 1+ + 1+ . 1+ 1+ d(y) d(y) d(y) d(y) d(y) 2. In the case that d(y) is between d(x) and d(z), we can assume that d(x)  d(z) because the other case is analogous. We have to prove that    |x − y| |y − z| |x − z| 1+ . 1+ 1+ d(x) d(y) d(x) This is equivalent to |y − z| |x − y| · |y − z| |x − z| − |x − y| +  . d(y) d(x) d(y) d(x) Because |z − y|  |x − z| − |x − y|, it is enough to prove that |z − y| |y − z| |x − y| · |y − z| +  . d(y) d(x) d(y) d(x) The last formula is equivalent to |y − z| (|x − y| − (d(y) − d(x)))  0, d(x) d(y) which is true, because |x − y|  d(y) − d(x) holds for all x, y ∈ Rn .

68

3 Hyperbolic Type Metrics

3. If d(y) = max{d(x), d(y), d(z)}, we may assume that d(x)  d(z), because the other case is analogous. We have to prove that    |x − y| |y − z| |x − z| 1+ . 1+ 1+ d(x) d(z) d(x) This is equivalent to |y − z| |x − y| · |y − z| |x − z| − |x − y| +  . d(z) d(x) d(z) d(x) Since |z − y|  |x − z| − |x − y|, it is enough to prove that |z − y| |y − z| |x − y| · |y − z| +  . d(z) d(x) d(z) d(x) The last formula follows from |y − z| |y − z| (|x − y| − (d(z) − d(x)))  (|x − y| − (d(y) − d(x)))  0. d(x) d(z) d(x) d(z) In the following lemma, we note some simple properties of jD . Lemma 3.12 ( [158, Lemma 2.36, p. 28]) For all x, y ∈ D the following inequalities hold: 1. jD (x, y)  | log d(x) d(y) |.

" 2. jD (x, y)  | log d(x) d(y) | + log 1 +

|x−y| d(x)

#

 2 jD (x, y).

Proof 1. Taking infimum over z ∈ ∂D in the inequality |y − z|  |x − z| + |x − y|, we conclude that d(y)  d(x) + |x − y|. 2. If d(x)  d(y), then the first inequality follows from the definition of jD , and the second one from 1) and definition. Now, let d(x)  d(y). Then, # " " d(x) d(x) jD (x, y) = log 1 + |x−y|  log d(y) + d(y) · $ $ " d(y) # $ $ |x−y| = $log d(x)  2 jD (x, y). d(y) $ + log 1 + d(y)

|x−y| d(y)

# (3.15)

The last inequality follows from (1).   For an open set D ⊂ 0 we define

Rn , D

=

Rn , and a nonempty A

rD (A) =

d(A) . d(A, ∂D)

⊂ D such that d(A, ∂D) >

3.4 Distance Ratio Metric jD

69

Now we prove the following inequalities:   1 1 log(1 + rD (A))  log 1 + rD (A)  jD (A)  log(1 + rD (A)). 2 2 The first one follows from " t #2 log(1 + t)  log 1 + . 2 For the second inequality, we have to show that     rD (A) |x − y|  sup log 1 + , log 1 + 2 min{d(x), d(y)} x,y∈A i.e., d(A) |x − y|  2 sup . d(A, ∂D) x,y∈A min{d(x), d(y)} Now, for a given ε > 0, we choose an x0 ∈ A such that d(x0 )  d(A, ∂D) + ε and x1 , y1 ∈ A such that |x1 − y1 |  d(A) − ε. Then max{|x0 − x1 |, |x0 − y1 |} 

d(A) − ε |x1 − y1 |  2 2

and further max{

d(A) − ε |x0 − x1 | |x0 − y1 | , } d(x0 ) d(x0 ) 2 (d(A, ∂D) + ε)

because ε > 0 is arbitrary, it follows that d(A) |x − y| |x − x0 | |x0 − x|  sup  sup  . 2 d(A, ∂D) x,y∈A min{d(x), d(y)} x∈A min{d(x0 ), d(x)} x∈A d(x0 ) sup

The remaining inequality reduces to |x − y| d(A)  , min{d(x), d(y)} d(A, ∂D)

for all x, y ∈ A,

which is obviously true because d(A)  |x − y| and d(A, ∂D)  d(x), d(y). For the case of the unit ball Bn one can develop the properties of the hyperbolic metric ρBn in the same way as for ρHn . For basic results we refer to [24] and [158, pp.19–32]. Lemma 3.13 ([10, Lemma 7.56] and [158, Lemma 2.41(2), p. 29]) 1. jBn (x, y)  ρBn (x, y)  2 jBn (x, y) for x, y ∈ Bn , 2. jHn (x, y)  ρHn (x, y)  2 jHn (x, y) for x, y ∈ Hn .

70

3 Hyperbolic Type Metrics

3.5 Quasihyperbolic Metric kD If ρ is a continuous function with ρ(x) > 0 for x ∈ D and if γ is a rectifiable curve in D, then we can define  lρ (γ ) = ρ ds. γ

The Euclidean length of a curve γ is denoted by l(γ ). Also, for x1 , x2 ∈ D we define dρ (x, y) = inf lρ (γ ),

(3.16)

where the infimum is taken over all rectifiable curves from x1 to x2 . It is easy to show that dρ is a metric in D. A special case of this metric is the hyperbolic metric in Hn (ρ(x) = 1/xn ) and in Bn (ρ(x) = 2/(1 − |x|2 )). Now we take each proper domain D ⊂ Rn and set ρ(x) = 1/(d(x, ∂D)). The corresponding metric, denoted by kD , is called quasihyperbolic metric in D. Since, ρ(ϕ(x)) =

1 1 = = ρ(x), d(ϕ(x), ∂ (ϕD)) d(x, ∂D)

for Euclidean isometry ϕ, we have that kD  (x  , y  ) = kD (x, y),

where D  = ϕ(D), x  = ϕ(x), y  = ϕ(y).

Moreover, we have that both metrics jD and kD are invariant under similarity transformations. Using a compactness argument (the Helly selection principle), one proves the existence of geodesics: Lemma 3.14 ([53, Lemma 1]) For each pair of points x1 , x2 ∈ D there exists a quasihyperbolic geodesic γ with x1 , x2 as its end points. For all x3 ∈ γ , we have kD (x1 , x2 ) = kD (x1 , x3 ) + kD (x3 , x2 ). We give a simple proof of the inequality kD (x, y)  jD (x, y) for x, y ∈ D.

3.5 Quasihyperbolic Metric kD

71

Lemma 3.15 ([54, Lemma 2.1]) For x, y ∈ D  kD (x, y) ≥ log 1 +

|x − y| min{d(x), d(y)}

 ≥ jD (x, y).

Proof We can assume 0 < d(x)  d(y). Choose a rectifiable arc γ : [0, s] → D from x to y, parametrized by arc length: γ (0) = x,

γ (s) = y.

Obviously, s  |x − y|. For each 0  t  s, we have the key observation, d(γ (t))  d(x) + t, so,  lρ (γ )  0

s

d(x) + s d(x) + |x − y| dt = log  log = jD (x, y). d(x) + t d(x) d(x)  

Lemma 3.16 ([158, Lemma 3.7, p. 34]) 1. If x ∈ D, y ∈ Bx = B n (x, d(x)), then  |x − y| kD (x, y)  log 1 + . d(x) − |x − y| 

2. If s ∈ (0, 1) and |x − y|  s d(x), then kD (x, y) 

1 jD (x, y). 1−s

Proof 1. Select z ∈ ∂Bx such that y ∈ [x, z], see Fig. 3.6. Because [x, y] ∈ Γxy , we obtain  d(x) |dw| dt = kD (x, y)  kBx (x, y)  |w − z| [x,y] [x,y] d(x)−|x−y| t   d(x) |x − y| = log = log 1 + d(x) − |x − y| d(x) − |x − y|

 = jRn \{z} (x, y) . 

|dw|  d(w)



72

3 Hyperbolic Type Metrics

Fig. 3.6 Lemma 3.16

2. For the proof of this part we apply the previous part, the Bernoulli inequality, and the definition of jD , to obtain  kD (x, y)  log 1 + 

|x − y| (1 − s) d(x)



  |x − y| 1 1 log 1 + jD (x, y),  1−s d(x) 1−s

as desired.   We note that, in general, there is no constant C, such that for all x, y ∈ D kD (x, y)  C · jD (x, y). A domain D ⊂ Rn , for which the above inequality holds for some C ≥ 1, is called a uniform domain. Furthermore, the best possible number CG := inf{C ≥ 1 : kG ≤ C jG } is called the uniformity constant of G. An example of domain which is not uniform is {x ∈ R2 : |x| < 1} \ {t e1 : t ≥ 0}. This wide and useful class of uniform domains was introduced by O. Martio and J. Sarvas (see [112]). There are other equivalent definitions for uniform domains; see, for example, the book [52, Definition 3.5.1, p. 40] of F. W. Gehring and K. Hag. Uniformity of different domains and respective uniformity constants has been studied by H. Linden (see [97]).

3.6 Other Hyperbolic Type Metrics

73

3.6 Other Hyperbolic Type Metrics One can define hyperbolic type metrics in several ways. For example, these metrics can be defined as weighted metrics such as the quasihyperbolic metric or by a formula involving Euclidean distances. Let us introduce first the Seittenranta metric δG [141]. For an open set G ⊂ Rn with card ∂G  2 we set mG (x, y) = sup |a, x, b, y| a,b∈∂G

and δG (x, y) = log(1 + mG (x, y)) for all x, y ∈ G. The case of an unbounded domain G ⊂ Rn , ∞ ∈ ∂G , should now be considered. Note that if a or b in the supremum is equal to the infinity, then we get exactly the jG metric. This implies that we always have jG  δG . Let us also introduce the Apollonian metric considered by Beardon [25], (see also [10, 7.28 (2)]) defined in open proper subsets G ⊂ Rn as follows: αG (x, y) = sup log |a, x, y, b|

for all x, y ∈ G.

a,b∈∂G

This formula defines a metric if and only if Rn \ G is not a proper subset of an (n − 1)-dimensional sphere in Rn . For a domain G ⊂ Rn and x, y ∈ G, the triangular ratio metric sG is defined as follows: |x − y| ∈ [0, 1]. |x − z| + |z − y| z∈∂G

sG (x, y) = sup

This metric was introduced by P. Hästö [59] and studied in [58]. Another group of hyperbolic type metrics may again be classified by the number of boundary points used in their definition. So for instance, the j metric is a onepoint metric, while the Apollonian metric is a two-point metric. Also, it is natural to introduce conformal invariants λG (x, y) and μG (x, y) defined for a domain G ⊂ Rn and x, y ∈ G. A basic fact is that λG (x, y)1/(1−n) and μG (x, y) are metrics [10, Remark 16.18 (2), p. 320]. If G is a proper subdomain of Rn , then for x, y ∈ G with x = y we define λG (x, y) = inf M(Δ(Cx , Cy ; G)) Cx ,Cy

(3.17)

74

3 Hyperbolic Type Metrics

where Cz = γz [0, 1) and γz : [0, 1) −→ G is a curve such that γz (0) = z and γz (t) → ∂G when t → 1, z = x, y. This conformal invariant was introduced by J. Ferrand (see [159]). If G is a proper subdomain of Rn , then for x, y ∈ G, we define μG (x, y) = inf M(Δ(Cxy , ∂G; G)),

(3.18)

Cxy

where the infimum is taken over all continua Cxy such that Cxy = γ [0, 1] and γ is a curve with γ (0) = x and γ (1) = y. In the case G = Bn the function μBn (x, y) is the extremal quantity of H. Grötzsch (see [159]). For further relations between other hyperbolic type metrics the reader is referred to [25, 61, 101, 128] and [141].

3.7 Quasiconformal Mappings and kD and jD Metrics Gehring and Osgood [53] proved a quasi-invariance property of the quasihyperbolic metric. To prove this property, we need the following lemma, obtained by combining (2.16) with the distortion theorem 2.48. Lemma 3.17 ( [53, Lemma 2]) For n ≥ 2 there exists a constant, a, depending only on n with the following property. If f is a K-quasiconformal mapping of D onto D  , then |f (x1 ) − f (x2 )| a d(f (x1 ), ∂D  )



|x1 − x2 | d(x1 , ∂D)

α ,

α = K 1/(1−n)

(3.19)

for all x1 , x2 ∈ D with |x1 − x2 |  a −1/α . d(x1 , ∂D) Theorem 3.18 ([53, Theorem 3]) For n  2, K  1, there exists a constant c depending only on n and K with the following property. If f is a K-quasiconformal mapping of D onto D  , then kD  (f (x1 ), f (x2 ))  c · max{kD (x1 , x2 ), kD (x1 , x2 )α },

α = K 1/(1−n) , (3.20)

for all x1 , x2 ∈ D. Proof We split the proof into two cases. Case A Suppose that |x1 − x2 |  (2 a)−1/α < 1. d(x1 , ∂D)

(3.21)

3.7 Quasiconformal Mappings and kD and jD Metrics

75

By the previous lemma |f (x1 ) − f (x2 )| a d(f (x1 ), ∂D  )



|x1 − x2 | d(x1 , ∂D)

α 

1 2

(3.22)

and d(y, ∂D  ) 

1 d(f (x1 ), ∂D  ) 2

(3.23)

for all y on the segment joining f (x1 ) and f (x2 ). Hence, 2|f (x1 ) − f (x2 )|  1. d(f (x1 ), ∂D  )

(3.24)

 1 |x1 − x2 | |x1 − x2 | +1  . d(x1 , ∂D) 2 d(x1 , ∂D)

(3.25)

kD  (f (x1 ), f (x2 ))  Next,

 kD (x1 , x2 )  log

The first inequality follows from Lemma 3.15. The second inequality follows from the following simple consequence of the concavity of the logarithm function, log(1 + t)  t log 2  applied with t =

|x1 −x2 | d(x1 ,∂D) .

t , 2

for 0  t  1,

From (3.22) and (3.25) we obtain

kD  (f (x1 , f (x2 ))  4 a kD (x1 , x2 )α .

(3.26)

Case B Now suppose that (3.21) is not true. Then join x1 , x2 by a quasihyperbolic geodesic curve γ and pick y1 = x1 , . . . , ym+1 = x2 and so that |yj − yj +1 | = (2 a)1−α , d(yj , ∂D)

|ym − ym+1 |  (2 a)−1/α d(ym , ∂D)

for j = 1, . . . , m − 1. Then k (f (x1 ), f (x2 ))  D

m 

kD  (f (yj ), f (yj +1 ))  m,

j =1

because each term is  1 by (3.24). Since the points yj lie on geodesics, we have by additivity on geodesics (Lemma 3.14) kD (x1 , x2 ) =

m  j =1

kD (yj , yj +1 ) 

m−1 (2 a)−1/α 2

76

3 Hyperbolic Type Metrics

by (3.25). Thus kD  (f (x1 ), f (x2 ))  4 (2 a)1/α kD (x1 , x2 )

(3.27)

since m  2. Inequality (3.20) then follows from (3.26) and (3.27) with c = 4 (2 a)1/α .   It was proved in [83, Theorem 1.4] that the quasihyperbolic metric is not invariant under Möbius transformations of the unit ball onto itself, and hence the constant c in Theorem 3.18 cannot be asymptotically sharp when K → 1. In other words, c  1 when K → 1. This is studied also in [9] where it is shown that c → ∞ when K → ∞ and moreover that c can be chosen independently on n. This means that there is a dimension-free variant of Theorem 3.18 [158, Cor. 12.19, 12.20]. In the Theorem 3.19 below, for the special domain G = Rn \ {0}, a quasiinvariance property of the quasihyperbolic metric with the asymptotically sharp constants is established. We remark that for this domain Martin and Osgood [109] proved that 3 $2   $ 1 $$ x y $$ 2 |x| + 2 arcsin − kD (x, y) = log , (3.28) |y| 2 $ |x| |y| $ for all x, y ∈ D. Theorem 3.19 ([84]) For given K ∈ (1, 2] and n ≥ 2 there exists a constant ω(K, n) such that if G = Rn \ {0} and f : Rn → Rn is a K-quasiconformal mapping with f (0) = 0 , then for all x, y ∈ G kG (f (x), f (y)) ≤ ω(K, n) max{kG (x, y)α , kG (x, y)} where α = K 1/(1−n) and ω(K, n) → 1 when K → 1. We note that Agard and Gehring have studied the angle distortion under quasiconformal mappings of the plane [2]. Motivated by their work we state the following corollary of Theorem 3.19. Corollary 3.20 ([84]) Suppose that under the hypotheses of Theorem 3.19, x, y ∈ S n−1 and f (x), f (y) ∈ S n−1 . Let φ and ψ be the angles between the segments [0, x], [0, y] and [0, f (x)], [0, f (y)] respectively. Then ψ ≤ ω(K, n) max{φ α , φ} . Proof The proof follows easily from the Martin–Osgood formula (3.28) . Since x, y, f (x), f (y) ∈ S n−1 holds |x| = |y| = |f (x)| = |f (y)| = 1 and log(|x|/|y|) = 0. Then kG (x, y) = 2 arcsin(|x − y|/2) = φ

3.8 Quasiconformal Mappings with Identity Boundary Values

77

and kG (f (x), f (y)) = 2 arcsin(|f (x) − f (y)|/2) = ψ.   The proof of the following Theorem 3.21 [84] relies on two ingredients: a sharp version of the Schwarz lemma for quasiconformal mappings from [9], and a sharp bound for the linear dilatation from [159]. We note that Theorem 3.21 represents an analogue of Theorem 3.19 for the distance ratio metric. Theorem 3.21 ([84]) For given K ∈ (1, 2] and n ≥ 2 there exist a constant c(K) such that if G = Rn \ {0} and f : Rn −→ Rn is a K-quasiconformal mapping with f (0) = 0, then for all x, y ∈ G jG (f (x), f (y)) ≤ c(K) max{jG (x, y)α , jG (x, y)}, where α = K 1/(1−n) , and c(K) → 1 as K → 1.

3.8 Quasiconformal Mappings with Identity Boundary Values The paper [84] considers the standard normalization which requires that the mapping keeps two points fixed and proves a stability result for dimensions n ≥ 2 , which is a counterpart of O. Teichmüller’s result in the case n = 2. We recall that the well-known Verschiebungssatz of Teichmüller [152] was a surprisingly new phenomenon of quasiconformality which later became a useful tool in many applications. It concerns the minimal dilatation of quasiconformal automorphisms of the disk, which have identical boundary values and move one prescribed inner point onto another. The extremal dilatation is given in terms of hyperbolic distance between these points (with the explicit description of the extremal map) and so the result is extended to each hyperbolic domain on the Riemann sphere. Namely, for a domain G ⊂ Rn , n  2, let I d(∂G) = {f : Rn → Rn homeomorphism : f (x) = x,

∀x ∈ Rn \ G}.

Recall that here Rn stands for the Möbius space Rn ∪ {∞}. We shall always assume that card{Rn \ G} ≥ 3. If K  1, then the class of K-quasiconformal maps in I d(∂G) is denoted by I dK (∂G). O. Teichmüller proved the following theorem with a sharp bound for K(f ). Theorem 3.22 Let G = R2 \ {0, 1}, a, b ∈ G. Then there exists f ∈ I dK (∂G) with f (a) = b iff log(K(f ))  sG (a, b), where sG (a, b) is the hyperbolic metric of G.

78

3 Hyperbolic Type Metrics

The search for a multidimensional analog of this result and questions related to this have been investigated for quite some time, but so far only special results have been obtained. For example, in [104] the author and M. Vuorinen have studied the problem of multidimensional version of Teichmüller’s theorem. The main result of [104] is an upper bound for the hyperbolic distance ρ(f (x), x) for admissible Kquasiconformal automorphisms of the unit ball Bn in Rn . This bound holds for all points of Bn . , Lemma 3.23 ([104]) For x, y ∈ Bn let t = (1 − |x|2 )(1 − |y|2 ). Then for x, y ∈ Bn th2

|x − y|  2 th

|x − y|2 ρBn (x, y) = , 2 |x − y|2 + t 2

2|x − y| ρBn (x, y) =, , 4 |x − y|2 + t 2 + t

(3.29)

(3.30)

where equality holds for x = −y. Theorem 3.24 ([104]) If f ∈ I dK (∂Bn), then for all x ∈ Bn ρBn (f (x), x)  log

1−a , a

√ a = ϕ1/K,n (1/ 2)2 ,

(3.31)

where ϕK,n is as in (2.7). Proof Fix x ∈ Bn and let Tx denote a Möbius transformation of Rn with Tx (Bn ) = Bn and Tx (x) = 0. Define g : Rn −→ Rn by setting g(z) = Tx ◦ f ◦ Tx−1 (z) for z ∈ Bn and g(z) = z for z ∈ Rn \ Bn . Then g ∈ I dK (∂Bn )with g(0) = Tx (f (x)) (Fig. 3.7). By the invariance of ρBn under the group G M (Bn ) of Möbius selfautomorphisms of Bn , we see that for x ∈ Bn , ρBn (f (x), x) = ρBn (Tx (f (x)), Tx (x)) = ρBn (g(0), 0).

(3.32)

Choose z ∈ ∂Bn such that g(0) ∈ [0, z] = {tz : 0  t  1}. Let E  = {−sz : s  1}, Γ  = Δ([g(0), z], E  ; Rn ) and Γ = Δ(g −1 [g(0), z], g −1 E  ; Rn ). The spherical symmetrization with center at 0 yields by [10, Theorem 8.44] M(Γ )  τn (1)

√ (= 21−n γn ( 2))

because g(x) = x for x ∈ Rn \ Bn . Next, we see by the choice of Γ  that 

M(Γ ) = τn



 1 + |g(0)| . 1 − |g(0)|

3.8 Quasiconformal Mappings with Identity Boundary Values

79

Fig. 3.7 The mapping g with identity boundary values

By K-quasiconformality we have M(Γ )  K M(Γ  ) implying exp(ρBn (0, g(0))) =

1 + |g(0)| 1−a  τn−1 (τn (1)/K) = . 1 − |g(0)| a

(3.33)

The last equality follows from (2.11). Finally, (3.32) and (3.33) complete the proof.   √ 2 Lemma 3.25 ([104]) If a = ϕ1/K,n (1/ 2) is as in Theorem 3.24, then for M > 1 and β ∈ [1, M], 

1−a log a

 ≤ log(λn2(β−1) 2β − 1) ≤ V (n)(β − 1),

(3.34)

where V (n) = (2 log(2λ2n ))(2λ2n )M−1 . Furthermore, for K ∈ [1, 17],  log

1−a a

  (K − 1)(4 + 6 log 2) < 9(K − 1),

(3.35)

80

3 Hyperbolic Type Metrics

with equality holding only for K = 1. For n = 2, 

1−a log a



 √ ϕK,2 (1/ 2)2  b(K − 1), = log √ ϕ1/K,2 (1/ 2)2 

(3.36)

√ where b = (4/π )K (1/ 2)2 ≤ 4.38. Here K (r) is Legendre’s complete elliptic integral of the first kind (see [10, Chapter 3]). Theorem 3.26 ([104]) If f ∈ I dK (∂Bn ), then for all x ∈ Bn , n ≥ 2, and K ∈ [1, 17] 9 |f (x) − x| ≤ (K − 1) . (3.37) 2 For n = 2 we have |f (x) − x| 

b (K − 1), 2

b  4.38.

Proof We have  |f (x) − x|  2 th

ρBn (f (x), x) 4



⎛  2 th ⎝

(3.38)

" log

1−a a

4

#⎞ ⎠

 (K − 1)(4 + 6 log 2)  2 th 4 9  (K − 1)(2 + 3 log 2)  (K − 1). 2 The first inequality follows from (3.30), the second one from Theorem 3.24, the third one from Lemma 3.25, and the last one from the inequality th(t)  t for t  0. For n = 2 we use the same first two steps and the planar case of Lemma 3.25 to derive the inequality 

|f (x) − x| 

b (K − 1). 2

 

We next consider the class I dK (∂Z) for the case when the domain Z is an infinite cylinder. Theorem 3.27 ([98]) Let Z = {(x, t) ∈ Rn : |x| < 1, t ∈ R}, f ∈ I dK (∂Z). Then kZ (0, f (0))  c(K) where c(K) → 0 when K → 1. Proof Let f (0) = (y, t), E  = [w, f (0)], F  = {w + s(y, 0) : s  0}, where w = (y/|y|, t), w = (−y/|y|, t) (Fig. 3.8). Then E  and F  are the complementary components of a Teichmüller ring and writing Γ  = Δ(E  , F  ; Rn ), we have

3.8 Quasiconformal Mappings with Identity Boundary Values

81

Fig. 3.8 Maps of cylinder



M(Γ )  τn



 1 + |y| . 1 − |y|

The modulus of the family Γ = Δ(E, F ; Rn ), E = f −1 E  , F = f −1 F  can be estimated by use of spherical symmetrization with the center at 0. Note that E = E  because E  ⊂ Rn \ Z and f ∈ I dK (∂Z). By [159, 7.34], we have M(Γ )  τn (1). By K-quasiconformality M(Γ )  K M(Γ  ) implying 1 + |y|  τn−1 exp(ρB n−1 (0, y)) = 1 − |y|



 τn (1) . K

Next we shall estimate t. First fix z in {w ∈ ∂Z : wn = 0} such that |f (0) − z| is maximal. Then choose a point w on the line through f (0) and z such that |z−w| = 1 and [z, w] ⊂ Rn \ Z. Let E  = [z, w] and F  = {f (0) + t (f (0) − z) : t  0}. Then E  and F  are the complementary components of a Teichmüller ring and writing Δ = Δ(E  , F  ; Rn ) (Fig. 3.9), we have M(Δ ) = τn (|f (0) − z|).

82

3 Hyperbolic Type Metrics

Fig. 3.9 Illustration for the proof of Theorem 3.27

Observing that E  = f −1 E  (because f ∈ I dK (∂Z)) and carrying out a spherical symmetrization with center at z we see that if E = f −1 E  , F = f −1 F  then M(Δ)  τn (1),

Δ = Δ(E, F ; Rn ).

By K-quasiconformality we have 1 + t 2  |f (0) − z|2  τn−1



τn (1) K

2 .

The triangle inequality for kZ yields kZ (0, f (0))  kZ (0, (0, t)) + kZ ((0, t), (y, t)) = t4+ kB n−1 (0, y)  |t| + 2 ρB n−1 (0, y) " #2 " ## "  τn−1 τnK(1) − 1 + 2 log τn−1 τnK(1) ,  e18(K−1) − 1 + 18(K − 1). The last inequality follows from (2.11) and Lemma 3.25.

 

3.8 Quasiconformal Mappings with Identity Boundary Values

83

We finish this chapter by remarking that recently several other authors have studied extensions, ramifications, and generalizations of Teichmüller’s problem. For example, the case of this problem for the unit ball in Rn is considered in [160, 163] and the case of Riemann surfaces in [27].

Chapter 4

Distance Ratio Metric

The basic distortion results about quasiconformal mappings such as the Schwarz lemma and the Gehring–Osgood theorem say that these mappings are Hölder continuous with respect to the hyperbolic and the quasihyperbolic metric respectively. In this chapter we analyze the modulus of continuity in the case of the distance ratio metric. The natural question is to find Lipschitz constants for this metric under Möbius transformations or arbitrary holomorphic mappings. The domains we work with here are the unit ball, the punctured ball, and the upper half space. Recall that a ∗ = |a|a 2 for a ∈ Rn \ {0} and that 0∗ = ∞, ∞∗ = 0. For fixed a ∈ Bn \ {0}, let σa (x) = a ∗ + s 2 (x − a ∗ )∗ , s 2 = |a|−2 − 1

(4.1)

be an inversion in the sphere S n−1 (a ∗ , s) orthogonal to S n−1 see (3.1) and (3.2). Then σa (a) = 0, σa (0) = a, σa (a ∗ ) = ∞ and |σa (x) − σa (y)| =

s 2 |x − y| |x − a ∗ ||y − a ∗ |

(4.2)

(see Lemma 3.1). Lemma 4.1 ([24, Theorem 3.5.1, p. 40]) Let f be a Möbius transformation and f (Bn ) = Bn . Then f (x) = (σ x)A, where σ is an inversion in some sphere orthogonal to S n−1 and A is an orthogonal matrix. Since the j -metric is invariant under orthogonal transformations, by Lemma 4.1, for x, y, a ∈ Bn , we have © Springer Nature Switzerland AG 2019 V. Todorˇcevi´c, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, https://doi.org/10.1007/978-3-030-22591-9_4

85

86

4 Distance Ratio Metric

jBn (f (x), f (y)) = jBn (σa (x), σa (y)), where σa (x) is defined as above and f is a Möbius transformation. For given domains D, D  ⊂ Rn and an open continuous mapping f : D → D  with f D ⊂ D  , let us consider the following condition: There exists a constant C ≥ 1 such that for all x, y ∈ D we have jD  (f (x), f (y)) ≤ CjD (x, y) ,

(4.3)

or, equivalently, that the mapping f : (D, jD ) → (D  , jD  ) between metric spaces is Lipschitz continuous with the Lipschitz constant C. The process of determining the best possible j -Lip constants is characterized by a series of subtle inequalities that are of interest to study. It is well known that the hyperbolic metric in the unit ball or half space is Möbius invariant. This is not true for the distance ratio metric jG , and so it is natural to ask what is the Lipschitz constant for this metric under conformal mappings or Möbius transformations in higher dimension. F. W. Gehring and B. G. Osgood proved that this metric is not changed by more than a factor of 2 under Möbius transformations (see [53, proof of Theorem 4]). n

n

Theorem 4.2 If D and D  are proper subdomains of Rn and if f : R → R is a Möbius transformation with f (D) = D  , then for all x, y ∈ D 1 jD (x, y) ≤ jD  (f (x), f (y)) ≤ 2jD (x, y). 2 The purpose of this chapter examines the challenging task of improving this estimation for some particular domains of Rn . Our presentation here is based on the papers [146, 147] and [143].

4.1 Refinements of the Gehring–Osgood Result One of the refinements of the Gehring–Osgood result is given by S. Simi´c and M. Vuorinen [146] and is of a two-fold nature. First, the constant 2 can be essentially improved in the cases of the unit ball and the punctured unit ball in Rn with the best possible constants provided. On the other hand, it happens that Theorem 4.2 is valid on the half-plane or the unit disk not only for Möbius transformation but also for arbitrary holomorphic mappings. Proofs of those facts are based on the famous Schwarz–Pick Lemma. The following theorem conjectured in [83] and proved by S. Simi´c, M. Vuorinen, and G. Wang [147] yields a sharp form of Theorem 4.2 for Möbius automorphisms of the unit ball.

4.1 Refinements of the Gehring–Osgood Result

87

Theorem 4.3 ([147]) A Möbius transformation f : Bn → Bn = f (Bn ), f (0) = a ∈ Bn , satisfies 1 jBn (x, y) ≤ jBn (f (x), f (y)) ≤ (1 + |a|)jBn (x, y) 1 + |a| for all x, y ∈ Bn . The constants are best possible. In order to prove this result, we introduce a series of lemmas. Lemma 4.4 ( [158, Exercise 2.52 (2), p. 32]) For the hyperbolic metric there holds |x| − |y| 1 |x| + |y| ≤ th ρ(x, y) ≤ . 1 − |x| |y| 2 1 + |x| |y|

(4.4)

Proof Starting with the identity 1 |x − y| th ρ(x, y) = , 2 2 |x − y| + (1 − |x|2 )(1 − |y|2 )

(4.5)

it suffices to prove |x − y| |x| + |y| |x| − |y| ≤, . ≤ 2 2 2 1 − |x| |y| 1 + |x| |y| |x − y| + (1 − |x| )(1 − |y| )

(4.6)

If |x| < |y|, then the first inequality holds because the left-hand side is a negative real. Otherwise, the double inequality (4.6) is about nonnegative reals. For this reason, by squaring this double inequality we obtain the following equivalent inequality: |x|2 + |y|2 − 2|x| |y| |x|2 + |y|2 − 2x, y |x|2 + |y|2 + 2|x| |y| ≤ ≤ . 2 2 2 2 1 + |x| |y| − 2|x| |y| 1 + |x| |y| − 2x, y 1 + |x|2 |y|2 + 2|x| |y| (4.7) Recall the Cauchy inequality |x, y | ≤ |x| |y|. Applying it, we get that − 2|x| |y| ≤ −2x, y ≤ 2|x| |y|. Because |x|, |y| < 1, we have 1 + |x|2 |y|2 = (1 − |x| |y|)2 + 2|x| |y| > 2|x| |y|. It follows that the function ϕ(t) =

|x|2 + |y|2 + t 1 + |x|2 |y|2 + t

(4.8)

88

4 Distance Ratio Metric

is defined for t ≥ −2|x| |y|, and it is increasing because ϕ(t) = 1 −

(1 − |x|2 )(1 − |y|2 ) . 1 + |x|2 |y|2 + t

From this and the inequality (4.8), we get (4.7).

 

We shall need the following simple fact about the unit ball. Lemma 4.5 Let a, b ∈ Bn . Then 1. |a|2 |b − a ∗ |2 − |b − a|2 = (1 − |a|2 )(1 − |b|2 ). 2. |b − a| |b| + |a| ||b| − |a|| ≤ ≤ . ∗ 1 − |a||b| |a||b − a | 1 + |a||b| Proof 1. Using a simple calculation, we get |a|2 |b − a ∗ |2 − |b − a|2 = |a|2 [|b|2 +

1 2(b · a) − ] − [|b|2 + |a|2 − 2(b · a)] 2 |a| |a|2

= 1 + |a|2 |b|2 − |a|2 − |b|2 = (1 − |a|2 )(1 − |b|2 ). 2. This can be obtained by Lemma 4.4 directly.   We shall need the following monotone form of l’Hôpital’s rule. Lemma 4.6 ([10, Theorem 1.25]) For −∞ < a < b < ∞, let f, g : [a, b] → R be continuous on [a, b] and differentiable on (a, b), and suppose g  (x) = 0 on (a, b). If f  (x)/g  (x) is increasing/deceasing on (a, b), then so are f (x) − f (a) g(x) − g(a)

and

f (x) − f (b) . g(x) − g(b)

If f  (x)/g  (x) is strictly monotone, then the monotonicity in the conclusion is also strict. It should be mentioned that Lemma 4.6 has found numerous applications. See, for example, [43] and the bibliography of [11] for a long list of applications towards various kinds of inequalities.

4.1 Refinements of the Gehring–Osgood Result

89

Lemma 4.7 Let c, d ∈ (0, 1), θ ∈ (0, 1]. Then " # 1. f (θ ) ≡

2cdθ log 1+ 1−cd

"

# is increasing. In particular,

2dθ log 1+ 1−d

# # " " 2cdθ 2cd log 1 + 1−cd log 1 + 1−cd # ≤ #. " " 2dθ 2d log 1 + 1−d log 1 + 1−d 2. g(θ ) ≡

arth(cθ) arthθ

is a decreasing function. In particular, arth(cθ ) ≤ c. arthθ

3. "

1+

c(1 − d) # c(1 − d) + 2cdθ 2cdθ #" 1+ ≤1+ . 1 − cd 1 + cd 1 − cd

Proof

# # " " 2cdθ 2dθ and f2 (θ ) = log 1+ 1−d . Then we have f1 (0+ ) = 1. Let f1 (θ ) = log 1+ 1−cd f2 (0+ ) = 0 and

f1 (θ ) 1−c =1− , f2 (θ ) 1 − cd + 2cdθ which is increasing in θ . Hence, the monotonicity of f follows from Lemma 4.6 above. The inequality follows by the monotonicity of f . 2. Let g1 (θ ) = arth(cθ ) and g2 (θ ) = arthθ . Then we have g1 (0+ ) = g2 (0+ ) = 0 and g1 (θ ) 1 1 − c2 = (1 − ) g2 (θ ) c 1 − c2 θ 2 which is clearly decreasing in θ . Hence, the monotonicity of g follows from Lemma 4.6. The inequality follows by the monotonicity of g and l’Hôpital’s rule. 3. The proof of this is straightforward and left to the reader.   We are now in a position to give a short proof of Theorem 4.3. Proof The conclusion is trivial for a = 0, so we only need to consider a = 0. Since the j -metric is invariant under orthogonal transformations, using Lemma 4.1 for x, y, a ∈ Bn , we have

90

4 Distance Ratio Metric

jBn (f (x), f (y)) = jBn (σa (x), σa (y)), , where σa (x) is an inversion in the sphere S n−1 (a ∗ , |a|−2 − 1) orthogonal to S n−1 . Hence, it suffices to estimate the expression " # |σa (x)−σa (y)| log 1 + min{1−|σ (x)|,1−|σ (y)|} jBn (σa (x), σa (y)) a a # . " J (x, y; a) ≡ = |x−y| jBn (x, y) log 1 + min{1−|x|,1−|y|} Let r = max{|x|, |y|} and suppose |σa (x)| ≥ |σa (y)|. Then by (4.2), we have min{1 − |σa (x)|, 1 − |σa (y)|} = 1 − |σa (x)| =

|a||x − a ∗ | − |x − a| . |a||x − a ∗ |

Let us first prove the right-hand side of the inequality. By Lemma 4.5, we get " jBn (σa (x), σa (y)) = log 1 + " = log 1 +

# (1 − |a|2 )|x − y| |a||y − a ∗ |(|a||x − a ∗ | − |x − a|) |x − y|(|a||x − a ∗ | + |x − a|) # |a||y − a ∗ |(1 − |x|2 )

" |x − a| # |x − y||x − a ∗ |) ) (1 + = log 1 + |a||x − a ∗ | (1 − |x|2 )|y − a ∗ | " |x| + |a| |x − y| |x − y| )(1 + ) (1 + ≤ log 1 + |y − a ∗ | 1 + |a||x| 1 − r2 " |x − y| |a||x − y| |a|(1 − r) # ≤ log 1 + (1 + )(1 + ) . 1−r 1 − |a|r 1 + |a|r Then J (x, y; a) ≤

=

" log 1 + " log 1 +

|x−y| |a||x−y| |a|(1−r) 1−r (1 + 1−|a|r )(1 + 1+|a|r )

" log 1 +

|x−y| 1−r

#

2|a|rθ |a|(1−r) 2rθ 1−r (1 + 1−|a|r )(1 + 1+|a|r )

" log 1 +

2rθ 1−r

#

where θ = |x−y| 2r . By Lemma 4.7 it follows that J (x, y; a) ≤

log(1 +

|a|(1−r) 2rθ 1−r (1 + 1−|a|r

" log 1 +

2rθ 1−r

#

+

2|a|rθ 1−|a|r ))

# ,

#

4.1 Refinements of the Gehring–Osgood Result

91

2|a|rθ 2|a|r ) ) log(1 + 1−|a|r log(1 + 1−|a|r # ≤1+ # " " = 1+ 2rθ 2r log 1 + 1−r log 1 + 1−r

= 1+

arth(|a|r) ≤ 1 + |a|. arthr

Therefore, we get jBn (f (x), f (y)) ≤ (1 + |a|)jBn (x, y). That the upper bound 1 + |a| is sharp is already proved in [83]. Looking at the left-hand side of inequality, note that f −1 (x) = A−1 σa−1 (x) = A−1 σa (x), where σa (x) and A are as above. Note that since A is an orthogonal matrix, so is A−1 . Applying the above proof, for x, y ∈ Bn , we get jBn (f −1 (x), f −1 (y)) jBn (σa (x), σa (y)) = ≤ 1 + |a|. jBn (x, y) jBn (x, y) Therefore, we have jBn (f (x), f (y)) ≥

1 jBn (x, y). 1 + |a|

(4.9)  

Lemma 4.8 For a ≥ 0, q ∈ [0, 1], we have log

" q + ea # 1 − q a. ≤ 1 + qea 1+q

Proof Denote f (a, q) := log

" q + ea # 1 − q a. − 1 + qea 1+q

By differentiation, we have fa (a, q) = −

(ea − 1)2 q(1 − q) . 1 + q (1 + qea )(q + ea )

Therefore we conclude that f (a, q) ≤ f (0, q) = 0.  

92

4 Distance Ratio Metric

Looking at the definition of the distance ratio metric, it is natural to expect that some properties of the logarithm will be essential for establishing properties of this metric. In fact, in [147], the classical Bernoulli inequality [158, (3.6)] was used for this purpose. However, it seems that now some stronger inequalities are needed and in particular we need to use the following result, which gives good estimates whenever the Lipschitz constant C satisfies 1 ≤ C ≤ 2, and it also allows us to eliminate logarithms in further considerations. Theorem 4.9 Let D and D  be proper subdomains of Rn . For an open continuous mapping f : D → D  denote X = X(z, w) :=

Y = Y (z, w) :=

|z − w| ; min{dD (z), dD (w)}

|z − w| min{dD  (f (z)), dD  (f (w))} . |f (z) − f (w)| min{dD (z), dD (w)}

If there exists a number q, 0 ≤ q ≤ 1 , such that q≤Y+

Y −1 , X+1

(4.10)

then the inequality

jD  (f (z), f (w)) ≤

2 jD (z, w), 1+q

holds for all z, w ∈ D. Proof Observe that X=

|z − w| = exp(jD (z, w)) − 1, min{dD (z), dD (w)}

and Y =

min{dD  (f (z)), dD  (f (w))} exp(jD (z, w)) − 1 |z − w| = . |f (z) − f (w)| min{dD (z), dD (w)} exp(jD  (f (z), f (w))) − 1

Hence the condition (4.10) is equivalent to exp(jD  (f (z), f (w))) ≤ exp(jD (z, w)) Therefore, by Lemma 4.8, we get

" q + ejD (z,w) # . 1 + qejD (z,w)

4.1 Refinements of the Gehring–Osgood Result

jD  (f (z), f (w)) ≤ jD (z, w) + log ≤ jD (z, w) +

93

" q + ejD (z,w) # 1 + qejD (z,w)

1−q 2 jD (z, w) = jD (z, w). 1+q 1+q  

Lemma 4.10 ([146, Lemma 2.4]) For positive numbers A, B, D and 0 < C < 1, θ ≥ 0, we have the following: 1. The inequality 1+

      B D B B B θ 1+ θ ≤ 1+ θ 1+ θ , 1+ D 1+A 1−C D 1−C

holds if and only if Bθ ≤ A + C; 2. The function log(1 +

B 1−C θ ) B log(1 + D θ)

is monotone increasing (decreasing) in θ if C + D < 1 (C + D > 1). Proof The proof of the first part uses a simple direct argument and is therefore left to the reader. To prove the second part, let  f1 (θ ) = log 1 +

   B B θ , f1 (0) = 0; f2 (θ ) = log 1 + θ , f2 (0) = 0. 1−C D

Since f1 (θ ) D + Bθ C+D−1 = =1+ , f2 (θ ) 1 − C + Bθ 1 − C + Bθ the proof follows from Lemma 4.6.

 

The following theorem was first conjectured in [147] in the special case of the punctured disk. Theorem 4.11 ([146]) Let a ∈ Bn and h : Bn → Bn = h(Bn ) be a Möbius transformation with h(0) = a. Then h(Bn \ {0}) = Bn \ {a} and for x, y ∈ Bn \ {0} jBn \{a} (h(x), h(y)) ≤ C(a)jBn \{0} (x, y), where the constant C(a) = 1 + (log 2+|a| 2−|a| )/ log 3 is best possible.

94

4 Distance Ratio Metric

Note that C(a) < 1 + |a| < 2 for all a ∈ Bn , and hence the constant C(a) is smaller than the constant 1+|f (0)| in Theorem 4.3 and far smaller than the constant 2 in Theorem 4.2. Proof Without loosing generality, we may assume that h(z) = σa (z) and suppose in the rest of the proof that |z| ≥ |w|. Let G = Bn \ {0} and G = Bn \ {a}. Then # " |z − w| jG (z, w) = log 1 + min{|z|, |w|, 1 − # |z|, 1 − |w|} " |z − w| , = log 1 + min{|w|, 1 − |z|} and " |σa (z) − σa (w)| # jG (σa (z), σa (w)) = log 1 + , T where T = Ta (z, w) := min{|σa (z) − a|, |σa (w) − a|, 1 − |σa (z)|, 1 − |σa (w)|}. Depending on the values of the number T , the proof is divided into four cases. We consider each case separately, applying the Bernoulli inequality in the first case, the assertion from Theorem 4.9 in the second case, and a direct approach in the last two cases. 1. T = |σa (z) − a|. Since |σa (z) − a| = |σa (z) − σa (0)| = s 2 |z−w| |z−a ∗ ||w−a ∗ | ,

s 2 |z| |a ∗ ||z−a ∗ |

and |σa (z) − σa (w)| =

we have " jG (σa (z), σa (w)) = log 1 +

|z − w| # . |a||z||w − a ∗ |

Suppose first that |w| ≤ 1 − |z|. Since also |w| ≤ 1 − |z| ≤ 1 − |w|, we get that 0 ≤ |w| ≤ 1/2. Therefore, by the Bernoulli inequality (see, e.g., [158, (3.6)]), we get " jG (σa (z), σa (w)) ≤ log 1 +

≤

1 1−

|a| 2

# " |z − w| # |z − w| ≤ log 1 + |z|(1 − |a||w|) |w|(1 − |a| 2 )

" 1 |z − w| # = log 1 + jG (z, w). |w| 1 − |a| 2

Assume now 1 − |z| ≤ |w|(≤ |z|). Then 1/2 ≤ |z| < 1.

4.1 Refinements of the Gehring–Osgood Result

95

Note that in this case (|z| − 12 )(2 − |a|(1 + |z|)) ≥ 0, so we obtain that 1 ≤ |z|(1 − |a||z|) (1 −

1 |a| 2 )(1 − |z|)

.

Hence, " jG (σa (z), σa (w)) ≤ log 1 + " ≤ log 1+

#

|z − w| (1 −

|a| 2 )(1 − |z|)

# " |z − w| # |z − w| ≤ log 1 + |z|(1 − |a||w|) |z|(1 − |a||z|)

≤

1 1−

|a| 2

" |z − w| # 1 = log 1+ jG (z, w). 1 − |z| 1 − |a| 2

2. T = |σa (w) − a|. This case can be considered using Theorem 4.9 with the same resulting 2 constant C1 (a) = 2−|a| . Namely, in terms of Theorem 4.9, we consider first the case |w| ≤ 1 − |z|. We get X=

|z| − |w| |z| |z − w| ≥ = − 1 = X∗ , |w| |w| |w|

and Y =

|z − a ∗ | ≥ 1 − |a||z| = Y ∗ . |a ∗ |

Therefore, Y+

Y −1 1 − Y∗ ≥ Y∗ − = 1 − |a|(|w| + |z|) ≥ 1 − |a| = q . X+1 1 + X∗

In the second case, i.e., when 1 − |z| ≤ |w|, we want to show that Y+

Y −1 ≥ 1 − |a| X+1

which is equivalent to (Y − (1 − |a|))(1 + X) + Y ≥ 1. Since in this case X=

|z| − |w| |z − w| ≥ := X∗ 1 − |z| 1 − |z|

and |w| |w||z − a ∗ | ≥ (1 − |a||z|) ∗ |a |(1 − |z|) 1 − |z| 1 − |a||z| = 1 − |a||z| + (|w| + |z| − 1) := Y ∗ , 1 − |z|

Y =

96

4 Distance Ratio Metric

we get (Y − (1 − |a|))(1 + X) + Y − 1 ≥ (Y ∗ − (1 − |a|))(1 + X∗ ) + Y ∗ − 1 5 1 − |a||z| 1 − |a||z| 6 1 − |w| −|a||z|+(|w|+|z|−1) = |a|(1−|z|)+(|w|+|z|−1) 1 − |z| 1 − |z| 1 − |z| ≥ |a|(1 − |w| − |z|) + (|w| + |z| − 1)

1 − |a| 1 − |a||z| = (|w| + |z| − 1) ≥ 0. 1 − |z| 1 − |z|

Hence, by Theorem 4.9, in both cases we get jG (σa (z), σa (w)) ≤

2 2 jG (z, w) = jG (z, w) = C1 (a)jG (z, w). 1+q 2 − |a|

3. T = 1 − |σa (z)|. In this case, combining a well-known fact which can be found, for example, in [147, Lemma 3.2], Lemma 4.4 above, the facts |a|2 |z − a ∗ |2 − |z − a|2 = (1 − |a|2 )(1 − |z|2 ) and |σa (z)| ≤

|a| + |z| , 1 + |a||z|

and the inequalities |a||w − a ∗ | ≥ 1 − |a||w| ≥ 1 − |a||z|, we get " |σa (z) − σa (w)| # jG (σa (z), σa (w)) = log 1 + 1 − |σa (z)| # " |a|s 2 |z − w| = log 1 + |w − a ∗ |(|a||z − a ∗ | − |z − a|) " |z − w|(|a||z − a ∗ | + |z − a|) # = log 1 + |a||w$ − a ∗ |(1$  − |z|2 ) # " |z − a| |z − w| $$ z − a ∗ $$ 1+ = log 1 + |a||z − a ∗ | 1 − |z|2 $ w − a ∗ $    " |z − w| |a| + |z|) # |z − w| 1 + 1 + ≤ log 1 + |w − a ∗ | 1 + |a||z| 1 − |z|2    " |z − w| |a||z − w| |a|(1 − |z|) # ≤ log 1 + . 1+ 1+ 1 − |z| 1 − |a||z| 1 + |a||z| We use here Lemma 4.10 part 1, together with the equalities A = |a||z|, B = |a|, C = |a||w|, D = |a|(1 − |z|), θ = |z − w|,

4.1 Refinements of the Gehring–Osgood Result

97

we obtain jG (σa (z), σa (w)) ≤ log

5" |a||z − w| #6 |z − w| #" 1+ . 1+ 1 − |z| 1 − |a||w|

(4.11)

Assume that 1 − |z| ≤ |w| (≤ |z|). Using Lemma 4.10 part 2 and the equalities B = |a|, C = |a||z|, D = |a|(1 − |z|), θ = |z − w|, we get J (z, w; a) :=

log(1 + |a||z−w| jG (σa (z), σa (w)) 1−|a||z| ) # " ≤1+ jG (z, w) log 1 + |z−w| 1−|z| ≤1+

2|a||z| ) log(1 + 1−|a||z| #, " 2|z| log 1 + 1−|z|

since in this case, we have C + D = |a| < 1 and |z − w| ≤ 2|z|. Since the last function is monotone decreasing in |z| and |z| ≥ 1/2, we get 1+ 1 |a|

log( 21 ) 2 + |a| 1− 2 |a| )/ log 3 := C2 (a). J (z, w; a) ≤ 1 + " 1 # = 1 + (log 1+ 2 2 − |a| log 1 1− 2

Now suppose |w| ≤ 1 − |z|(≤ 1 − |w|). The estimate (4.11) and Lemma 4.10 part 2, with B = |a|, C = D = |a||w|, θ = |z − w|, yield 5" J (z, w; a) ≤ 5" log

1+

log

|z−w| |w|

1+

|z−w| 1−|z|

#" 1+

" log 1 +

#" 1+

|z−w| |w| #6 |a||z−w| 1−|a||w|

|a||z−w| 1−|a||w|

#6

#

# " log 1 + |z−w| |w|# " # " |a||z−w| |a| log 1 + 1−|a||w| log 1 + 1−|a||w| # ≤1+ # , " " =1+ 1 log 1 + |z−w| log 1 + |w| |w| ≤

98

4 Distance Ratio Metric

since C + D = 2|a||w| ≤ |a| < 1 and 0 ≤ |z − w| ≤ |z| + |w| ≤ 1. Let us denote the last function as g(|w|) and let |w| = r, 0 < r ≤ 1/2. Since g  (r) =

|a|2 (1 − r|a|)(1 + (1 − r)|a|) log(1 + 1/r) " # |a| log 1 + 1−|a|r > 0, + r(1 + r) log2 (1 + 1/r)

it follows that g(r) is a monotone increasing function and we finally obtain

J (z, w; a) ≤ 1 +

" log 1 +

|a| 1−|a|/2

# = C2 (a).

log(1 + 2)

4. T = 1 − |σa (w)|. This case is treated similar to the previous case, " jG (σa (z), σa (w)) = log 1 +

|z

# |a|s 2 |z − w| ∗ − a | − |w − a|)

− a ∗ |(|a||w

" |z − w|(|a||w − a ∗ | + |w − a|) # = log 1 + |a||z − a ∗ |(1 − |w|2 )    " |a||z − w| |a|(1 − |w|) # |z − w| . 1+ 1+ ≤ log 1 + 1 − |w| 1 − |a||z| 1 + |a||w| Applying Lemma 4.10 part 1 with A = |a||w|, B = |a|, C = |a||z|, D = |a|(1 − |w|), θ = |z − w|, we obtain

jG (σa (z), σa (w)) ≤ log

5" |a||z − w| #6 |z − w| #" 1+ . 1+ 1 − |w| 1 − |a||z|

(4.12)

Assume that 1 − |z| ≤ |w|(≤ |z|). We get log jG (σa (z), σa (w)) J (z, w; a) := ≤ jG (z, w)

5" 1+

|z−w| 1−|z|

#" 1+

" log 1 +

|z−w| 1−|z|

|a||z−w| 1−|a||z|

#

#6

4.1 Refinements of the Gehring–Osgood Result

99

log(1 + |a||z−w| 1−|a||z| ) #, " =1+ log 1 + |z−w| 1−|z| and we have already considered this inequality above. In the case |w| ≤ 1−|z| ≤ 1 − |w|, we have J (z, w; a) ≤ 5" ≤

log

log

1+

5" 1+

|z−w| 1−|w|

#"

" log 1 +

|z−w| |w|

#"

1+

" log 1 +

1+

|z−w| |w|

|a||z−w| 1−|a||z|

#

|a||z−w| 1−|a|(1−|w|)

|z−w| |w|

#6

#6

#

# # " " |a||z−w| |a| log 1 + 1−|a|(1−|w|) log 1 + 1−|a|(1−|w|) # # " " =1+ ≤1+ , 1 log 1 + |z−w| log 1 + |w| |w| where the last inequality follows from Lemma 4.10 part 2 for B = |a|, C = |a|(1 − |w|), D = |a||w|, θ = |z − w|, since C + D = |a| < 1 and |z − w| ≤ |z| + |w| ≤ 1. Put now |w| = r and let k(r) = k1 (r)/k2 (r) where " k1 (r) = log 1 +

# " |a| 1# ; k2 (r) = log 1 + . 1 − |a|(1 − r) r

Let us show now that the function k(r) is monotone increasing on the positive part of the real axis. Namely, since k1 (∞) = k2 (∞) = 0 and k1 (r)/k2 (r) =

|a|r |a|(1 + r) |a|2 r(1 + r) = (1 + |a|r)(1 − |a| + |a|r) 1 + |a|r 1 − |a| + |a|r

   1 − |a| 1 − |a| 1− , = 1− 1 + |a|r 1 − |a| + |a|r with both functions in parenthesis increasing on R+ , the conclusion follows from Lemma 4.6. This is so because in this case 0 < r ≤ 1/2, we also obtain that " # |a| log 1 + 1−|a|/2 " # J (z, w; a) ≤ 1 + = C2 (a). log 1 + 2

100

4 Distance Ratio Metric

In order to verify that the constant C2 (a) is sharp, we first calculate a −a Ta ( 2|a| , 2|a| ) = 1−|a| 2+|a| . Using this, it follows that  J

 a −a , ; a = C2 (a). 2|a| 2|a|

Since C2 (a) = 1 + (log 2+|a| 2−|a| )/ log 3 >

1 1− |a| 2

= C1 (a), we conclude that the

best possible upper bound C is C = C2 (a).

  Bn

Hn ,

Since proper domains for Möbius transformation are and it follows that for dimension n = 2, much more can be said. For example, as a special case K = 1 of Theorem 2.45, we obtain Proposition 4.12 below. This is so because in a half plane H2 , by [24, Theorem 7.2.1, p. 130], we have " ρ (z, w) # |z − w| H = . th 2 |z − w| As another example, in dimension n = 2, one can prove an analogue of Theorem 4.2 valid for arbitrary holomorphic mappings of a half-plane into itself. Hence in this case we get a substantial generalization of Theorem 4.2. The main tool in the proof of such generalization will be the following Julia’s variant of the famous Schwarz–Pick Lemma for the half-plane [26]. Proposition 4.12 For all holomorphic mappings f, f : H2 → H2 , and all z, w ∈ H2 , we have |

f (z) − f (w) f (z) − f (w)

|≤|

z−w |. z−w

Proof As indicated above, this follows from Theorem 2.45 when K = 1.

 

Because the j -metric is invariant under similarities, we can restrict the proof to the case of the upper half-plane H2 := {z : Im z > 0}. A generalization of Theorem 4.2 for this domain is given by the following theorem. Theorem 4.13 ([144]) If f : H2 → H2 is a holomorphic mappings and z, w ∈ H2 , then jH2 (f (z), f (w)) ≤ 2jH2 (z, w), where the constant 2 is best possible. Proof Denote s = min{Im z, Im w} and suppose that Im f (z) ≤ Im f (w). Then

4.1 Refinements of the Gehring–Osgood Result

101

" " |z − w| # |f (z) − f (w)| # ; jH2 (f (z), f (w)) = log 1 + . jH2 (z, w) = log 1 + s Im f (z) Applying Proposition 4.12, we get |

f (z) − f (w) 2 z−w 2 | −1≥| | − 1, f (z) − f (w) z−w

|

f (z) − f (w) 2 Im f (z)Im f (w) | ≤ , z−w Im zIm w

that is,

since for all x, y ∈ C, the identity |x −y|2 −|x −y|2 = 4Im xIm y holds. Therefore, |f (z) − f (w)|2 |z − w|2 |z − w|2 Im f (w) Im (f (w) − f (z)) = ), ≤ (1 + 2 2 2 Im f (z) Im f (z) Im f (z) s s and so 3 |f (z) − f (w)| |z − w| |f (z) − f (w)| ≤ , 1+ Im f (z) s Im f (z)

(*)

since Im x ≤ |x|, x ∈ C. (z)−f (w)| Denote |z−w| := 2X and 1 + |f Im := Y . From the relation (*) we get that s f (z) √ √ √ √ 2 Y − 2X Y ≤ 1, and so ( Y − X) ≤ 1 + X2 and Y ≤ X + 1 + X2 , which is the same as 1+

, |f (z) − f (w)| ≤ (X + 1 + X2 )2 . Im f (z)

Hence " |f (z) − f (w)| # jH2 (f (z), f (w)) = log 1 + Im f (z) , ≤ 2 log(X + 1 + X2 ) ≤ 2 log(1 + 2X) = 2jH2 (z, w), and the proof is complete. In order to establish that the constant 2 is best possible, choose f0 (z) = a −

1 , b+z

102

4 Distance Ratio Metric

Im z , it follows that where a and b are arbitrary real numbers. Since Im f0 (z) = |b+z| 2 f0 : H2 → H2 . A calculation of j values along the line ζ ⊂ H2 given by ζ := {z = i − b + t, t ∈ R} with w = i − b yields √t jH2 (z, w) = log(1+t); jH2 (f0 (z), f0 (w)) = log(1+

1+t 2 1 1+t 2

, ) = log(1+t 1 + t 2 ).

Hence, √ log(1 + t 1 + t 2 ) jH2 (f0 (z), f0 (w)) = , jH2 (z, w) log(1 + t) and this expression tends to 2 as t → ∞. Therefore the constant 2 is best possible.

 

A similar result for the unit disk D is given in [145]. Theorem 4.14 For every holomorphic mapping f : D → D and all z, w ∈ D, we have jD (f (z), f (w)) ≤ 2jD (z, w). Remark 4.15 It cannot be claimed in this case that the best possible Lipschitz constant is C ∗ = 2, since an example of an analytic function f0 : D → D which satisfies jD (f0 (z), f0 (w)) =2 jD (z, w) z,w∈D sup

is not known. This question was treated in [142] where the following estimation for C ∗ = is proved:

C ∗ (a)

+ 1 + |a| ≤ C ∗ (a) ≤ min{2(1 + |a|), 5 + 2|a| + |a|2 }; a := f (0). By Theorem 4.14, this reduces to 1 + |a| ≤ C ∗ (a) ≤ 2, but the problem of determining the best possible Lipschitz constant in this case remains open.

4.2 Lipschitz Continuity and Analytic Functions

103

4.2 Lipschitz Continuity and Analytic Functions We begin this section by defining the sector as follows: Sϕ = {reiθ ∈ C : 0 < θ < ϕ , r > 0}. π ) and r ∈ (0, 1). Then Lemma 4.16 ([147]) Let k ∈ (1, ∞), θ ∈ (0, 2k

1. f1 (θ ) ≡

k sin θ 1−sin θ

−

2. f2 (r) ≡

1−(1−sin θ)r 1−(1−sin kθ)r k

sin θ is decreasing from (0, 1) to ( sin kθ , 1).

3. f3 (r) ≡

log(1+

1−r k ) r k sin kθ log(1+ r1−r sin θ )

is decreasing from (0, 1) to ( ksinsinkθθ , k).

sin kθ 1−sin kθ

π is decreasing from (0, 2k ) to (−∞, 0).

Proof 1. Differentiating, we get f1 (θ ) = k[g(1) − g(k)], where g(k) =

cos kθ . (1−sin kθ)2

Since

g  (k) =

θ (1 − sin kθ + cos2 kθ ) > 0, (1 − sin kθ )3

π ). The limiting values are we have f1 (θ ) < 0 and hence f1 is decreasing in (0, 2k clear. 2. Differentiating again, we get

[1 − (1 − sin kθ )r k ]2 f2 (r) = −(k − 1)(1 − sin θ )(1 − sin kθ )r k + k(1 − sin kθ )r k−1 − (1 − sin θ ) ≡ h(r). Since h (r) = k(k − 1)(1 − sin kθ )r k−2 [1 − r(1 − sin θ )] > 0, we have that h is increasing in (0, 1) and hence h(r) ≤ h(1). By (1), h(1) = (1 − sin θ )(1 − sin kθ )f1 (θ ) < 0. Therefore f2 (r) < 0, and hence f2 is decreasing in (0, 1). The limiting values are also clear.

104

4 Distance Ratio Metric k

1−r 3. Let f3 (r) ≡ gg12 (r) (r) , where g1 (r) = log(1 + r k sin kθ ) and g2 (r) = log(1 + − − Then g1 (1 ) = g2 (1 ) = 0. Differentiating, we have

1−r r sin θ ).

g1 (r) = kf2 (r). g2 (r) Therefore, f3 is increasing in (0, 1) by Lemma 4.6. By l’Hôpital’s rule and (2) we get the limiting values easily.   Lemma 4.17 ([147]) Let f : Sπ/k → H2 with f (z) = zk (k ≥ 1). Let x, y ∈ Sπ/k with arg(x) = arg(y) = θ . Then k sin θ jS (x, y) ≤ jH2 (f (x), f (y)) ≤ kjSπ/k (x, y). sin kθ π/k Proof Since the j -metric is invariant under similarity, it can be assumed that x = π reiθ and y = eiθ , 0 < r < 1. By symmetry, we also assume 0 < θ ≤ 2k . Then k

log(1 + r k1−r ) jH2 (f (x), f (y)) sin kθ = . 1−r jSπ/k (x, y) log(1 + r sin θ )  

By Lemma 4.16(3), the result follows. Lemma 4.18 ([147]) Let n ∈ N, 0 < θ ≤ 1+

π 2n .

Then for x, y ∈

Rn

\ {0},

|x − y| n |x n − y n | ≤ (1 + ) . n |x| sin nθ |x| sin θ

Proof It is clear that (4.18) holds if n = 1. Next, suppose that (4.18) holds when n = k. Namely, 1+ where 0 < θ ≤ (1 +

|x k − y k | |x − y| k ≤ (1 + ) , |x|k sin kθ |x| sin θ

π 2k . Then, if n

= k + 1, we have 0 < θ ≤

π 2(k+1)

(4.13)
0 (note that g  (z) = 0). Furthermore,   " # |h (z)|2  2 2 = |g 1 − |ω(z)| , Jf (z) = |g  (z)|2 1 −  (z)| |g (z)|2 where ω(z) =

h (z) g  (z)

is analytic and |ω| < 1. Now we have

log

1 = −2 log |g  (z)| − log(1 − |ω(z)|2 ). Jf (z)

5.1 Bi-Lipschitz Property of HQC Mappings in Plane

113

Note that the first term is a harmonic function. It is well known that the logarithm of the modulus of an analytic function is harmonic everywhere except where that analytic function vanishes, but g  (z) = 0 everywhere (see [124, p. 141]). The second term can be expanded into the series ∞  |ω(z)|2k k=1

k

,

and each term is subharmonic (note that ω is analytic). So, − log(1 − |ω(z)|2 ) is a continuous function represented as a locally uniform sum of subharmonic functions. Thus it is also subharmonic. It follows that log

1 is a subharmonic function. Jf (z)

(5.3)

It should be noted that the representation f (z) = g(z) + h(z) is local, but this suffices for our conclusion (5.3). From (5.3), we have  1 1 1 dm(w) ≥ log . log m(Bz ) Bz Jf (w) Jf (z) Applying this with (5.2), we have   1 1 1 1 ≥ exp log =, , αf (z) 2 Jf (z) Jf (z) and therefore +

Jf (z)  αf (z).

From the first inequality in Theorem 5.1 we have +

Jf (z) ≥

1 d(f (z), ∂D  ) . c d(z, ∂D)

(5.4)

Note that Jf (z) = |g  (z)|2 − |h (z)|2 ≤ |g  (z)|2 and by K-quasiconformality of f , |h | ≤ k|g  |, 0 ≤ k < 1, where K = (see (2.17)). This gives Jf ≥ (1 − k 2 )|g  |2 . Hence

1+k 1−k

114

5 Bi-Lipschitz Property of HQC Mappings

,

Jf  |g  |  |g  | + |h | = L(f, z),

where L(f, z) = max |f  (z)h|. |h|=1

We finish by observing that (5.4) and the above asymptotic relation give us L(f, z) ≥

1 d(f (z), ∂D  ) , c d(z, ∂D)

c = c(k).

To establish the reversed inequality, we again use Jf (z) ≥ (1 − k 2 )|g  (z)|2 , i.e., + , Jf (z) ≥ 1 − k 2 |g  (z)|.

(5.5)

Furthermore, we know that for n = 2    + 1 αf (z) = exp log Jf (x) dm(w) . m(Bz ) Bz Using (5.5)   + , 1 1 log Jf (x) dm(w) ≥ log 1 − k 2 + log |g  (w)| dm(w) m(Bz ) Bz m(Bz ) Bz  , 1 = log 1 − k 2 + log |g  (w)| dm(w) m(B ) z B z √ = log 1 − k 2 + log |g  (z)|. Now we have the following by the harmonicity of log |g  | 

αf (z) = ≥ = ≥ =

  + 1 exp log Jf (x) dm(w) m(B , z ) Bz exp(log 1 − k 2 + log |g  (z)|) , 1 − k 2 |g  (z)| 1, 1 − k 2 (|g  (z)| + |h (z)|) 2√ 1 − k2 L(f, z). 2

Applying the second inequality of Theorem 5.1, we get + d(f (z), ∂D  ) , L(f, z)  c Jf (z)  c αf (z)  c d(z, ∂D)

c = c(k).

5.2 When Part of Boundary Is Flat

115

Summarizing, L(f, z) 

d(f (z), ∂D  ) , d(z, ∂D)

however, quasiconformality gives us L(f, z)  l(f, z), where l(f, z) = min |f  (z)h|. |h|=1

Therefore, we have l(f, z) 

d(f (z), ∂D  ) . d(z, ∂D)

This pointwise result, via integration along curves, easily gives kD  (f (z1 ), f (z2 ))  kD (z1 , z2 ). This completes the proof.

 

5.2 When Part of Boundary Is Flat In this section we give a method of achieving local bi-Lipschitz behavior when part of the boundary is flat. This is a local generalization of the work of Kalaj and Pavlovi´c [79]. Our approach is to use the boundary Harnack inequality for this problem. The following theorem will play an important role in our proofs. Theorem 5.3 ([99]) Let f : Ω −→ C be a harmonic map whose Jacobian determinant J = |fz |2 − |fz¯ |2 is positive everywhere in Ω. Then log J is a superharmonic function. We note that this theorem has also been used in establishing the minimum principle for the Jacobian determinant, which is a novelty in the new analytic proof of the celebrated Radó–Kneser–Choquet theorem given by T. Iwaniec and J. Onninen [70]. We will also need the following boundary Harnack inequality ([47], exercise 6, p. 28).

116

5 Bi-Lipschitz Property of HQC Mappings

Theorem 5.4 Let u and v be positive harmonic functions on unit disk D in R2 with u(0) = v(0) and let I ⊂ ∂D be an open arc and assume lim u(z) = lim v(z) = 0

z→ζ

z→ζ

for all ζ ∈ I . Then for every compact A ⊂ D ∪ I there is a constant C(A) independent of u and v such that on A ∩ D 1 u(z) ≤ ≤ C(A). C(A) v(z) Proof We will consider the case I = ∂D ∩ H− where H− = {z : I m(z) < 0}.

By our assumption u is positive and harmonic, so we

have u(z) 1= P (t)dμ(t), where μ is a positive measure with u(0) = 1 z S S 1 dμ = μ(S ). Applying the similar argument to v, we get that it is defined via a corresponding positive measure ν. $ $ $ $ Assume v, u ≥ 0 are harmonic in D and u$ = v $ = 0, u(0) = v(0) = 1, I

I

i.e., μ(S 1 ) = ν(S 1 ) = 1. Since u is harmonic and μ is supported on {z : Im(z) ≥ 1 , we have 0, |z| = 1} = S+  u(z) =

S1

1 − |z|2 dμ(ξ ) = |ξ − z|2

 1 S+

1 − |z|2 dμ(ξ ). |ξ − z|2

1 ) ≥ δ > 0 and For δ0 = dist (A, supp(μ)) and z ∈ A we have dist (z, S+ 0

 u(z) ≤ (1 − |z|2 )

1 S+

1 2(1 − |z|) dμ(ξ ) ≤ . |ξ − z|2 δ02

Since |ξ − z| ≤ 2 we have  v(z) ≥ (1 − |z|)

1 S+

1 1 − |z| dν(ξ ) ≥ 4 |ξ − z|2

and we conclude that for z ∈ A we have u(z)/v(z) ≤ v(z)/u(z) ≤

8 , δ02

8 δ02

and analogously

and hence δ02 u(z) 8 ≤ ≤ 2. 8 v(z) δ0  

Using Theorem 5.4 we obtain the following special case of this result.

5.2 When Part of Boundary Is Flat

117

Theorem 5.5 ([103]) Suppose that D is the unit disk and H2 is the upper-half plane in R2 . If f : D −→ H2 is HQC homeomorphism with f (∂(D− )) ⊆ R, then f |D− is bi-Lipschitz with respect to the Euclidean metric, where D− = {z : z ∈ D, I m(z) < 0}. Proof It is clear that we loss no generality by assuming that f (0) = i. We take the Möbius transformation M(z) =

1 − iz z−i

such that M(±1) = ±1, M(0) = i, M(−i) = 0 and choose u = I m(f ),

v = I m(M(z)) =

1 − |z|2 . |z − i|2

Note that u(0) = v(0) = 1 and that for each ξ, such that I m(ξ ) < 0, |ξ | = 1 lim u(z) = lim v(z) = 0.

z→ξ

z→ξ

Note that in our setting I m(f (z)) ≡ d(f (z), ∂H2 ), so applying Theorem 5.4 we now have d(f (z), ∂H2 ) 1 ≤ ≤ C(A) 1−|z|2 C(A) |z−i|2

on A ∩ D for some constant C(A) and for every compact A ⊂ D ∪ I, where I = ∂D ∩ H− . Since |z − i|2 ≤ 4, d(z, ∂D) = 1 − |z|, it follows that 1 d(f (z), ∂H2 ) ≤ ≤ 4C(A). 2C(A) d(z, ∂D) Using Theorem 5.1 we conclude that 1 ≤ αf (x) ≤ c, c where c is a constant which depends only on A. We finish by noting that from the proof of the main theorem in [99] it follows that αf (x)  f  (x), and since f is quasiconformal, it follows that it is bi-Lipschitz.

 

118

5 Bi-Lipschitz Property of HQC Mappings

We give a local version of Theorem 5.5. Lemma 5.6 ([103]) Let D+ = D ∩ H2 and let g : D → D+ be a harmonic Kquasiconformal mapping with g(±1) = ±1,

g(−i) = 0.

If A ⊂ D is a compact subset with δ0 := dist(A, S 1 ∩ H2 ) > 0, then dist(g(z), ∂D+ ) 1 ≤ ≤ c(K, δ0 ), c(K, δ0 ) 1 − |z|

z ∈ A.

The constant c(K, δ0 ) < ∞ depends only on K and δ0 . Proof First note that the map g : D → D+ is η-quasisymmetric, where η depends only on K. To see this note that every K-quasiconformal mapping of the unit disk D fixing ±1 and −i is η-quasisymmetric, and so the case of our mapping is reduced to this fact using, for example, an appropriate bi-Lipschitz mapping from D+ to D. Note that if u(z) = I m(g(z)), z ∈ A, then we have c(K) ≤ u(0) ≤ 1, and that, u(z) 1 ≤ ≤ c(K, δ0 ) c(K, δ0 ) dist(g(z), ∂D+ ) for some constant c(K) < ∞ depending only on K and δ0 . Hence, we can argue similarly as in Theorem 5.5 to finish the proof.   By further developing the above ideas, we can consider local questions of biLipschitz property phenomena when only part of the boundary is flat. In this case we need to use some quasiconformal geometry. Definition 5.7 We say that ∂Ω is flat at some x0 ∈ ∂Ω if, up to a rotation around the point x0 , we have ∂Ω ∩ B 2 (x0 , ρ) = [x0 − ρ, x0 + ρ] for some ρ > 0. Theorem 5.8 ([103]) Suppose that D is the unit disk, Ω is simply connected, and f : D −→ Ω is a harmonic and quasiconformal mapping such that f (D) = Ω. Suppose also that ∂Ω is flat at x0 , and that f is normalized so that f (±1) = x0 ± ρ with f (−i) = x0 . If Ω1 = f −1 (B 2 (x0 , ρ/2) ∩ Ω), then f : Ω1 −→ B 2 (x0 , ρ/2) ∩ Ω is biLipschitz. Indeed, 1 |f (x) − f (y)| ≤ L0 ≤ L0 ρ|x − y| for some L0 depending only on K(f ).

5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions

119

Proof By Lemma 5.6 for the conformal mapping φ : D → Ω˜ = f −1 [B 2 (x0 , ρ) ∩ Ω] with φ(±1) = ±1 and φ(−i) = −i, the statement follows as it is enough to notice that φ is bi-Lipschitz on A = f −1 [B 2 (x0 , ρ/2) ∩ Ω].  

5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions In this section we study counterparts in higher dimensions of the following result of M. Pavlovi´c [123]. Theorem 5.9 ([123]) Let u be a harmonic homeomorphism of D that extends continuously to the boundary, and f : S1 → S1 be its boundary function. Then the following conditions are equivalent: 1. u is quasiconformal; 2. u is bi-Lipschitz in the Euclidean metric; 3. f is bi-Lipschitz and the Hilbert transformation of its derivative is in L∞ . In [121], Partyka and Sakan gave explicit estimations of the bi-Lipschitz constants for u expressed by means of the maximal dilatation K of u and |u−1 (0)|. Additionally, if u(0) = 0, they have asymptotically sharp estimates as K → 1. In order to prove Theorem 5.9 Pavlovi´c had to make a deep and detailed analysis of the boundary values of f . By analyzing these boundary values, he achieved the Lipschitz-property for every harmonic quasiconformal mapping of the disk. In higher dimensions, Pavlovi´c’s approach seems difficult to generalize. Instead it would seem possible that the Lipschitz-property follows by the regularity theory of elliptic PDEs. Indeed, such an approach was taken by Kalaj [76] but the proof in [76] is quite long and technical. A simple and self-contained argument showing the Lipschitz property in all dimensions is given in [21]. Recall that Lp (Ω, Rn ) for p ∈ [1, ∞) is the Banach space of all measurable functions f : Ω −→ Rn such that  |f |p < ∞ Ω

with the norm defined as 1

 ||f ||p =

|f |

p

p

.

Ω

Recall that L∞ is the Banach space of all measurable functions f : Ω −→ Rn such that there is a constant c ∈ [0, ∞) such that |f | ≤ c,

a.e. in Ω

120

5 Bi-Lipschitz Property of HQC Mappings

with the norm defined as ||f ||∞ = inf{c ≥ 0 : |f | ≤ c a.e. in Ω}. p

Recall also that Lloc (Ω, Rn ) for p ∈ [1, ∞] is the vector space of all measurable functions f : Ω −→ Rn such that for every compact set K ⊆ Ω holds f |K ∈ Lp (K). Instead of Lp (Ω, R) we write Lp (Ω). Definition 5.10 ([65, Definition A.13, p. 148]) Let Ω ⊆ Rn be open and u ∈ L1loc (Ω). A function v ∈ L1loc is called a weak derivative of u if 

 ϕ(x)v(x) dx = −

Ω

u(x)∇ϕ(x) dx Ω

for every ϕ ∈ CC∞ (Ω). We denote such a function v by Du. For 1 ≤ p ≤ ∞ we define the Sobolev space W 1,p (Ω) = {u ∈ Lp (Ω) : Du ∈ Lp (Ω, Rn )} and we define the norm  ||u||W 1,p (Ω) =

1

 |u| +

|Du|

p

Ω

p

p

.

Ω

For harmonic K-quasiconformal mappings f = (f 1 , . . . , f n ) : Bn → Bn by harmonicity we have Δ(f j )2 (x) = 2|∇f j (x)|2 ,

j = 1, . . . , n.

We say that a mapping f defined in a domain Ω ⊂ Rn has the co-Lipschitz property with constant L ≥ 1, if |f (x) − f (y)| ≥

1 |x − y| L

∀ x, y ∈ Ω.

(5.6)

Naturally, for mappings in (5.6) the Jacobians are nonvanishing everywhere. Lemma 5.11 (Wood [42, p. 26]) In dimensions n ≥ 3, the Jacobian of a harmonic homeomorphism may vanish. Example 5.12 Let f : R3 −→ R3 be as follows: f (x, y, z) = (u, v, w), where u = x 3 − 3xz2 + yz,

v = y − 3xz,

w = z.

5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions

121

It is clear that f is harmonic. To see that f is univalent suppose that f (x1 , y1 , z1 ) = f (x2 , y2 , z2 ) = (u, v, w). Then w = z1 = z2 and v = y1 − 3x1 w = y2 − 3x2 w, which implies u = x13 + w(y1 − 3x1 w) = x23 + w(y2 − 3x2 w). It follows that x13 + vw = x23 + vw, so x1 = x2 and y1 = y2 . The calculations show that the mapping (u, v, w) → (x, y, z) is defined by x = (u − vw)1/3 ,

y = v + 3w(u − vw)1/3 ,

z = w,

is an inverse for f . Thus, f is a (univalent) harmonic mapping of R3 onto R3 . A straightforward calculation reveals that f has the Jacobian Jf (x, y, z) = 3x 2 , which vanishes on the plane x = 0. Note that the main difficulty here is in finding lower bounds for |f (x) − f (y)| in terms of the distance between x and y. However, in general dimensions it is not known if harmonic quasiconformal mappings of the ball have nonvanishing Jacobian. Note now that, by the Lewy theorem [95], if the gradient f = ∇u of a (real valued) harmonic function defines a homeomorphism f : Ω → Ω  where Ω, Ω  ⊂ R3 , then the Jacobian Jf does not vanish. Moreover, Jf = Hu , where Hu denotes the Hessian of u, and here we have the theorem of Gleason and Wolff [56, Theorem A] which shows that in dimension three, the function log det (Hu ) is superharmonic outside the zeroes of the Hessian. The following result collects these facts. Theorem 5.13 (Lewy–Gleason–Wolff) Suppose u : Ω → R is a harmonic function, such that f (x) := ∇u(x) defines a homeomorphism between the domains Ω and Ω  ⊂ R3 . Then log Jf (z) = log det (Hu ) is superharmonic in Ω.

(5.7)

122

5 Bi-Lipschitz Property of HQC Mappings

The usual Sobolev embedding theorem in whole space is given in [93, Theorem 1, p. 263]. In the paper [21], the local form of Sobolev embedding for the unit ball is proved using Green’s function and the estimates on Riesz potential. Lemma 5.14 ([21]) Suppose that w ∈ C 2 (Bn ) ∩ C(Bn ), that h ∈ Lp (Bn ) for some 1 < p < ∞ and $ Δw = h in Bn , with w $S n−1 = 0,

(a) If 1 < p < n, then 1 1 1 = − . q p n

∇wLq (Bn ) ≤ c(p, n)hLp (Bn ) , (b) If n < p < ∞, then

∇wL∞ (Bn ) ≤ c(p, n)hLp (Bn ) . Proof The function w can be represented in terms of the Green function GBn (x, y) of the unit ball:  GBn (x, y)h(y)dm(y), x ∈ Bn . w(x) = Bn

The Green function and its gradient 

2 x−y n $ y − |y| x$ ∇x GBn (x, y) = c1 (n) + |y| $y − |y|2 x $n |x − y|n



can be explicitly calculated. Details of calculation can be found in [93, p. 40]. Since |y||x − y| ≤ |y − |y|2 x| for all x, y ∈ Bn , the gradient is bounded by |∇x GBn (x, y)| ≤ 2c1 (n)|y − x|1−n

for

x, y ∈ Bn .

Therefore ∇wLq (Bn ) ≤ cI1 hLq (Rn ) , where Is h denotes the Riesz potential of order s. It follows that Lemma 5.14(a) reduces to the well-known boundedness properties of the Riesz potentials, Is hLq (Rn ) ≤ c(s, p, q)hLp (Rn ) ,

1 s 1 = − , q p n

given, e.g., in [150, p. 119]. We note that the bound in b) is easier and it follows n from Hölder’s inequality, since x → |x − y|1−n ∈ Lq (Bn ) for every 1 ≤ q < n−1 .  

5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions

123

Lemma 5.14 leads us to a quick proof for the following result. 2,1 n Corollary 5.15 ([21, Lemma 2.1]) Suppose w ∈ Wloc (B ) ∩ C(Bn ), n ≥ 2, is such that  $ w $S n−1 = 0, with |∇w|p0 dm < ∞ for some n < p0 < ∞. (5.8) Bn

If w satisfies the following uniform differential inequality, |Δw(x)| ≤ a|∇w(x)|2 + b,

x ∈ Bn ,

(5.9)

for some constants a, b < ∞, we then have ∇wL∞ (Bn ) ≤ M < ∞,

(5.10)

where M = M(a, b, p0 , n, ∇wp0 ). In particular, w is Lipschitz continuous. Proof Applying (5.9), we have Δw(x) = h(x), where

for x ∈ Bn ,

# " h(x) = c(x) |∇w(x)|2 + 1

(5.11)

(5.12)

and c∞ ≤ max{a, b}; one can simply define #−1 " c(x) := Δw(x) |∇w(x)|2 + 1 for almost every x ∈ Bn . Note that here, by our assumptions, ∇w ∈ Lp0 (Bn ), but with Sobolev embedding one can improve this integrability up to ∇w ∈ Ls (Bn )

where s > 2n. (5.13)

 Namely, if p0 /2 < n < p0 , then h(x) = c(x) |∇w(x)|2 + 1 ∈ Lp0 /2 (Bn ) and (5.11) with Lemma 5.14(a) give us ∇w ∈ Lp1 (Bn ),

p1 =

p0 n > p0 , 2n − p0

which is a strict improvement in the integrability. Note that if initially p0 = n(1 + ε), ε > 0, then p1 =

p0 n 1+ε > n(1 + 2ε). =n 2n − p0 1−ε

(5.14)

124

5 Bi-Lipschitz Property of HQC Mappings

So one can iterate this feedback argument, getting ∇w ∈ Lp (Bn ),  = 0, 1, 2, . . . with p > n(1 + 2 ε), until the condition (5.13) is reached. (If it turns out that for some exponent p = 2n, we can choose p0 a little smaller so that this degeneracy does not happen.) Once (5.13) is achieved, (5.11)–(5.12) with Sobolev embedding and Lemma 5.14(b) give ∇w∞ < ∞. Note that the proof also gives a bound for ∇w∞ that depends only on the constants a and b, the exponent p0 , and the initial norm hp0 /2 ≤ max{a, b}(∇w2p0 + 1).   It should be noted that the above iteration argument fails if in (5.8) one assumes integrability only for some 1 ≤ p0 < n. Thus, the following Gehring’s theorem [50] in case of quasiconformal mappings becomes particularly useful here. Lemma 5.16 ([50]) For every quasiconformal mapping f : Rn → Rn we have the following higher integrability property  |Df (x)|p dm ≤ C < ∞, p = p(n, K) > n, (5.15) Bn

where for mappings of the whole space Rn , the constant C depends only on n and distortion K(f ). Theorem 5.17 ([21]) If n ≥ 2 and f : Bn → Bn is a harmonic and Kquasiconformal mapping, then |f (x) − f (y)| ≤ L|x − y|,

x, y ∈ Bn ,

where L depends only on the distortion K, dimension n, and dist(f (0), S n−1 ). Proof In case f : Bn → Bn is K-quasiconformal, we can compose it with a Möbius transform ψ preserving the ball, such that f ◦ ψ(0) = 0. Using Schwarz reflection one can then extend f ◦ ψ to Rn and apply (5.15) to this mapping. Unfolding the Möbius transformation after a change of variables, we see that each K-quasiconformal mapping f : Bn → Bn satisfies (5.15) with C = C n, K, dist(f −1 (0), S n−1 ) Moreover, if f is harmonic, consider the function w(x) = 1 − |f (x)|2 ,

x ∈ Bn .

Recall that quasiconformal mappings of Bn extend continuously to the boundary (see [155, Theorem 17.20, p. 61]), so we have that w(x) satisfies the assumptions of Corollary 5.15. To check the condition (5.9) note that w = u ◦ f where u(x) = 1 − |x|2

with ∇u(x) = −2x,

x ∈ Bn .

Hence ∇w(x) = Df t (x)∇u(f (x)) satisfies 2 |f (x)| |Df (x)| ≤ |∇w(x)| ≤ 2|f (x)| |Df (x)| K

(5.16)

5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions

125

with |Δw(x)| = 2||Df (x)||2 ≤ 2n2 |Df (x)|2 ,

x ∈ Bn ,

where ||Df (x)||2 denotes the Hilbert–Schmidt norm of the differential matrix. We note that the above argument already establishes (5.9) but to see the explicit dependence of a and b on properties of the mapping  f we first need to note that there is a constant δ = δ n, K, a, dist(f (0), S n−1 ) such that 1 − |x| + |f (x)| ≥ δ > 0,

for all x ∈ Bn .

(5.17)

To see this note that quasiconformal mappings of Bn are quasi-isometries in the

hyperbolic metric [158]. Hence either hBn f (x), f (0) ≥ 1 + hBn (f (0), 0) (which implies hBn (f (x), 0) ≥ 1) and |f (x)|  e−1 e+1 hold, or else we have hBn (x, 0) ≤ c(K)(1 + hBn (f (x), f (0)))  c(K)(2 + hBn (f (0), 0)). In the latter case 1 − |x|  e−M , where M = c(K)(2 + hBn (f (0), 0)). Hence (5.17) holds, and we have 4n2 [(1 − |x|)2 + |f (x)|2 ]|Df (x)|2 δ2 2Kn2 4n2 ≤ |∇w|2 + 2 (1 − |x|)2 |Df (x)|2 . 2 δ δ |Δw| ≤

The last term is controlled by basic ellipticity bounds [55, p.38], or more precisely, the Bloch norm bounds (1 − |x|)|Df (x)| ≤ c(n)f ∞

(5.18)

which is valid for every harmonic function. Hence, (5.9) holds with a = K 2 n2 δ −2 , b = 4n2 c(n)2 δ −2 , so that ∇w ∈ L∞ (Bn ) by Corollary 5.15. In a similar manner (5.16)–(5.18) give us |Df (x)| 

c(n) K||∇w||∞ 1 (1 − |x| + |f (x)|)|Df (x)|  + . δ δ 2δ

Thus f is a Lipschitz mapping, with an explicit bound L = L(n, K, dist (f (0), S n−1 )) for its Lipschitz constant.

 

Definition 5.18 (Muckenhoupt Classes) Let w be a locally integrable nonnegative function in Rn such that 0 < w < ∞ almost everywhere. Then w belongs to the

126

5 Bi-Lipschitz Property of HQC Mappings

Muckenhoupt class Ap for 1 < p < ∞, or that is an Ap -weight whenever there is a constant cp,w such that 1 |B|

 w dx ≤ cp,w ( B

1 |B|



1

w 1−p dx)1−p B

for all balls B in Rn . We let w belong to A1 , or that w is an A1 -weight if there is a constant c1,w ≥ 1 such that 1 |B|

 w dx ≤ c1,w ess infB w B

for all balls B in Rn . The union of all Muckenhoupt classes Ap is denoted by A∞ , or more precisely, A∞ =



Ap .

p>1

We shall say that w is an A∞ -weight if it belongs to A∞ , i.e., if w is an Ap -weight for some p > 1. Recall that for a quasiconformal mapping, the Jacobian Jf is an A∞ -weight (see # "

p 1/p 1 [64, Theorem 15.32]), αf (z) is comparable to m(B J for every 0 < p ≤ z ) Bz f 1, and hence we could have used such averages as well. From the other side, in the case n = 2 and f conformal, we have αf (z) = |f  (z)|, and therefore the choice (5.1) above appears to be a natural one. We now need a version of the classical Hopf lemma. Lemma 5.19 (Hopf Lemma [23, Exercise I-25]) Suppose that u is real valued, n n nonconstant, and harmonic on B . Show that if u attains its maximum value on B at ζ ∈ S, then there is a positive constant c such that u(ζ ) − u(rζ ) ≥ c(1 − r) for all r ∈ (0, 1). Conclude that (Dn u)(ζ ) > 0. It should be noted that this lemma is frequently applied in various contexts for estimating superharmonic functions in terms of boundary distance. For example, in [116] M. Mateljevi´c used this well-known approach for harmonic quasiconformal mappings. Theorem 5.20 ([21]) Suppose f : Bn −→ Ω is a harmonic quasiconformal mapping, with Ω ⊂ Rn a convex subdomain. Then

5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions

αf (x) ≥ c0 d(f (0), ∂Ω) > 0,

x ∈ Bn ,

127

(5.19)

where the constant c0 = c0 (n, K) depends only on the dimension n and distortion K = K(f ). Proof For z ∈ Bn , we have d(f (z), ∂Ω) = inf d(f (z), p), p

where the infimum is taken over all lines p outside the domain. Note that the function hp (z) = d(f (z), p) is positive and harmonic in Bn . Applying the usual Harnack inequality in Bn to each hp , we get, hp (z) ≥

1 − |z| hp (0). (1 + |z|)n−1

Since d(f (0), p) ≥ d(f (0), ∂Ω), we have hp (z) ≥

1 − |z| d(f (0), ∂Ω). (1 + |z|)n−1

Taking the infimum of the hp (z) over all p gives us d(f (z), ∂Ω) ≥

1 − |z| d(f (0), ∂Ω). (1 + |z|)n−1

Note that since d(z, ∂Bn ) = 1 − |z|, the last inequality can be rewritten as follows: d(f (z), ∂Ω) d(f (0), ∂Ω) ≥ . d(z, ∂Bn ) (1 + |z|)n−1 Applying Theorem 5.1 and quasiconformality of f, we conclude that αf (z) ≥ c(n, K)d(f (0), ∂Ω).  

128

5 Bi-Lipschitz Property of HQC Mappings

It follows that the co-Lipschitz property can be established if the usual derivative can be estimated from below by the average derivative. We note that in dimension n = 2 this can be done using the main result of the paper [99]. Theorem 5.21 Suppose Ω, Ω  ⊂ R2 are planar domains and f : Ω → Ω  a harmonic quasiconformal mapping. Then log Jf is superharmonic in Ω. Using the superharmonicity of log Jf for the harmonic quasiconformal mapping f defined in the unit disk B2 , we get log |Df (x)|2 ≥ log Jf (z) ≥

1 m(Bz )

 log Jf dm = log αf (z)2 ,

z ∈ B2 .

Bz

(5.20) Combining this estimate with Theorem 5.20 proves that for every harmonic quasiconformal mapping from the disk onto a convex domain the lower bound inf |Df (x)h| ≥ |Df (x)|/K ≥ αf (x)/K ≥ cd(f (0), ∂Ω)

|h|=1

(5.21)

holds for some constant c > 0. It follows from this that f is co-Lipschitz. This is a new proof of theorem [73, Th 3.5]. In fact we have here the following fact. Corollary 5.22 ([21]) Suppose Ω, Ω  ⊂ R2 are simply connected domains and f : Ω → Ω  is a harmonic quasiconformal mapping. If Ω  is convex and the Riemann map of Ω has derivative bounded from above, then f has the co-Lipschitz property (5.6). The proof relies on applying (5.21) to f ◦ g, where g : D → Ω is the Riemann map. Hence, in particular, in Corollary 5.22 the boundary of Ω need not be C 1 or even Lipschitz. For example, g(z) = 2z − z2 is a conformal map from D onto a cardioid, with cusp at 1 = g(1). In a similar manner, combining (5.20) with Theorems 5.17 and 5.20, we have a new proof for Pavlovi´c’s theorem for B2 (see Theorem 5.9). Finally, to prove the higher dimensional version of the Pavlovi´c theorem, we will use an argument analogous to (5.20) and Theorem 5.21. Theorem 5.23 ([21]) Suppose f : B3 → B3 is a harmonic quasiconformal mapping, which is also a gradient mapping, that is f = ∇u for some function u harmonic in the unit ball B3 , then f is a bi-Lipschitz mapping. Proof Note that if f = ∇u is a quasiconformal harmonic gradient mapping in B3 , then as in (5.20), using Theorem 5.13 we conclude that αf (x)3 ≤ Jf (x) ≤ K(f )2 inf |Df (x)h|3 . |h|=1

Hence, when f (B3 ) = B3 , or more generally when the target domain is convex, Theorem 5.20 gives us that f is co-Lipschitz. The Lipschitz-property follow from Theorem 5.17. This finishes the proof.  

5.3 Bi-Lipschitz Property of HQC Mappings in Higher Dimensions

129

The above approach has a few consequences also in more general subdomains of Rn . We have a generalization of a theorem 5.2. Theorem 5.24 ([21]) Consider domains Ω, Ω  ⊂ R3 , and let f : Ω → Ω  be a harmonic quasiconformal homeomorphism which is also a gradient mapping, say f = ∇u for some function u harmonic in Ω. Then f is bi-Lipschitz with respect to the corresponding quasihyperbolic metrics, 1 kΩ (x, y) ≤ kΩ  (f (x), f (y)) ≤ MkΩ (x, y), M

x, y ∈ Ω,

where the constant M depends only on the distortion K(f ). Proof Using (5.7) and Theorem 5.13, we get that αf (x)3 ≤ Jf (x). Looking from the other side, Df (x) is a vector valued harmonic function, whose norm is subharmonic and therefore   1 K Df (x)3 ≤ Df 3 dm ≤ Jf dm m(Bx ) Bx m(Bx ) Bx ≤ C(K, n) exp[

1 m(Bx )

 log Jf dm] = C(K, n)αf (x)3 , Bx

where the last inequality follows from the fact that Jf is an A∞ -Muckenhoupt weight. Thus αf (x) # inf|h|=1 |Df (x)h| # sup|h|=1 |Df (x)h|, and the conclusion follows as in [99].   The proof of Theorem 5.23 gives immediately the following consequence. Corollary 5.25 ([21]) Suppose f : B3 → Ω is quasiconformal. If Ω is convex and f = ∇u is the gradient of a harmonic function, then f has the co-Lipschitz property (5.6). We note that the method of the proof of Theorem 5.17 works also for more general domains. We also note that we have the following result of Kalaj [76] as consequence. Corollary 5.26 ([21]) If f : B3 → Ω is a harmonic quasiconformal mapping, where Ω ⊂ Rn is a domain with C 2 -boundary, then f is a Lipschitz mapping. Proof Consider this time w(x) = dist(f (x), ∂Ω) near ∂Ω, and choose some smooth extension to Ω. Then w satisfies the inequality (5.9) (see [76]), so ∇w∞ < ∞ by Corollary 5.15, and we obtain the Lipschitz bounds for f as in the proof of Theorem 5.17.   Finally we mention that combining the above results, we have the following theorem as consequence. Theorem 5.27 ([21]) Suppose Ω is a convex subdomain of R3 with C 2 -boundary, and let f : B3 → Ω be a harmonic quasiconformal homeomorphism. If f = ∇u is a harmonic gradient mapping, then f is bi-Lipschitz with respect to the Euclidean metric.

Chapter 6

Quasi-Nearly Subharmonic Functions and QC Mappings

Let G be a domain in Rn , f : G → Rn a harmonic map, and A a class of self-homeomorphisms of G. We study in this chapter what can be said about the functions of the form f ◦ h, h ∈ A . For example, we show that if n = 2 and A is the class of conformal maps, then the functions in this class are also harmonic. However, if A is the class of harmonic maps, or quasiconformal harmonic maps, this statement is no longer true. Our presentation here is based on [38, 87, 90] and the survey paper [153] which has a slightly different focus.

6.1 Quasi-Nearly Subharmonic Functions and Conformal Mappings Let B 2 (z, r) denote the Euclidean disk with center z and radius r, and let m be the Lebesgue measure in C normalized so that the measure of the unit disk equals 1. Then if u : Ω −→ R is a subharmonic nonnegative function on a domain Ω ⊂ C and p  1, then u(z)p 

1 r2

 up dm.

(6.1)

B 2 (z,r)

If 0 < p < 1, then (6.1) need not hold, but there is a constant C = C(p)  1 such that  C p u(z)  2 up dm. (6.2) r B 2 (z,r) This fact, essentially due to Hardy and Littlewood [57], was first proved by Fefferman and Stein [44, Lemma 2]. The proof of Fefferman and Stein is reproduced by Garnett [46, Lemma 3.7]. It should be noted that Fefferman and Stein originally © Springer Nature Switzerland AG 2019 V. Todorˇcevi´c, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, https://doi.org/10.1007/978-3-030-22591-9_6

131

132

6 Quasi-Nearly Subharmonic Functions and QC Mappings

considered only the case when u = |v| and v is harmonic, but their proof applies also in the case of nonnegative subharmonic functions. For proofs of (6.2) see [134], [122, Theorem 1], [135, Lemma 2.1], [38, 125, 136]. That the validity of (6.2) for some p implies its validity for all p was observed in [3], [122, Theorem 1], and [135, Lemma 2.1]. Following [125], we call a Borel measurable function u : Ω −→ [0, ∞] on a subdomain Ω of the complex plane C quasi-nearly subharmonic if u ∈ L1loc (Ω) and if there is a constant K = K(u, Ω)  1 such that  K u(z)  2 u(w) dm(w) (6.3) r B 2 (z,r) for each disk B 2 (z, r) ⊂ Ω. In [135], the term pseudoharmonic functions is used, while in [122], the condition (6.3) is called shK -condition. If K = 1 and if u takes its values in [−∞, ∞], then u is called nearly subharmonic (see [66]). Let QNSK (Ω) denote the class of all functions satisfying (6.3) (for a fixed K), and by QNS(Ω) the class of all quasi-nearly subharmonic functions defined in Ω; so  QNSK (Ω). QN S(Ω) = K1

The following fact generalizes the above-mentioned result of Fefferman and Stein [44] and gives one of the most important properties of QNS. Theorem 6.1 If u ∈ QNSK (Ω) and p > 0, then up ∈ QNSC (Ω), where C is a constant depending only on p and K. In particular, if up is quasi-nearly subharmonic for some p > 0, then it is so for every p > 0. The list of relevant papers related to Theorem 6.1 includes the following [3, 38, 122, 125, 134–136]. Theorem 6.2 ([41, Theorem 2.3] Koebe One-Quarter Theorem) Let ϕ be a conformal mapping from the disk B 2 (z0 , R) into C. Then the image ϕ(B 2 (z0 , R)) contains the disk B 2 (ϕ(z0 ), ρ), where ρ = R|ϕ  (z0 )|/4. Theorem 6.3 ([41, Theorem 2.5] Koebe Distortion Theorem) Let ϕ be a conformal mapping from the disk B 2 (z0 , R) into C. Then there holds inequalities R 2 (R + |z − z0 |) R 2 (R − |z − z0 |) |ϕ  (z)|    , 3 |ϕ (z0 )| (R + |z − z0 |) (R − |z − z0 |)3 Consequently, if |z − z0 | < R/2, then 4 |ϕ  (z)|  .  |ϕ (z0 )| 27

z ∈ B 2 (z0 , R).

6.2 Regularly Oscillating Functions and Conformal Mappings

133

Theorem 6.4 ([87, Theorem 1]) Let u ∈ QNSK (Ω). If ϕ is a conformal mapping from a domain G onto Ω, then the composition u ◦ ϕ belongs to the QNSC (G), where C depends only on K. Proof Let u ∈ QNSK (Ω) and let ϕ be a conformal mapping from G onto Ω. We will find a constant C such that  u(ϕ(z))dm(z)  u(ϕ(z0 ))r 2 /C, (6.4) B 2 (z0 ,r)

for r < dist (z, ∂G). Let w0 = ϕ(z0 ) and ρ = r|ϕ  (z0 )|/4, and let ψ : Ω −→ G denote the inverse of ϕ. Then   u(ϕ(z))dm(z) = u(w)|ψ  (w)|2 dm(w) B 2 (z0 ,r)

ϕ(B 2 (z0 ,r))

 

u(w)|ψ  (w)|2 dm(w),

B 2 (w0 ,ρ/2)

where Theorem 6.2 above has been applied. Apply now Theorem 6.3 to the function ψ and get |ψ  (w)|  (4/27)|ψ  (w0 )|, for |w − w0 | < ρ/2. Hence  B 2 (z0 ,r)

u(ϕ(z))dm(z)  (4/27)2 |ψ  (w0 )|2

 u(w) dm(w) B 2 (w0 ,ρ/2)

 (4/27)2 |ψ  (w0 )|2 (ρ/2)2 u(w0 )/K = (4/27)2 |ψ  (w0 )|2 |ϕ  (z0 )|2 u(w0 )r 2 /(16K). We now use the identity ψ  (w0 )ϕ  (z0 ) = 1 and get (6.4) with C = 272 K. This concludes the proof.  

6.2 Regularly Oscillating Functions and Conformal Mappings A function f : Ω −→ R defined in Ω ⊆ C is called regularly oscillating (see [125]) if f is of class C 1 (Ω) and if for some K, |∇f (z)|  Kr −1 sup |f − f (z)|,

B 2 (z, r) ⊆ Ω.

(6.5)

B 2 (x,r) 1 (Ω) (O = oscillation). The In [122] the class of such functions is denoted by OCK class of all regularly oscillating functions will be denoted by RO(Ω).

134

6 Quasi-Nearly Subharmonic Functions and QC Mappings

Theorem 6.5 ([122, Theorem 3]) If f is regularly oscillating, then |f | and |∇f | 1 (Ω), then |f | and |∇f | are are quasi-nearly subharmonic. Moreover, if f ∈ OCK in QNSC (Ω), where C depends only on K. Example 6.6 Harmonic functions are regularly oscillating. Example 6.7 ([122]) Convex functions are regularly oscillating. It follows that the modulus of the gradient of a convex function is quasi-nearly subharmonic. We are ready now for the following theorem. Theorem 6.8 ([87, Theorem 2]) If f ∈ RO(Ω), and ϕ is a conformal mapping 1 (Ω), then f ◦ ϕ is in from G onto Ω, then f ◦ ϕ ∈ RO(G). Moreover, if f ∈ OCK 1 OCK1 (Ω), where K1 depends only on K. 1 (Ω) and let ϕ be a conformal mapping from G onto Ω. We Proof Let u ∈ OCK will find a constant K1 such that for B 2 (z0 , ε) ⊂ G, we have

|∇u(ϕ(z0 ))| · |ϕ  (z0 )| 

K1 ε

sup

|u(ϕ(z)) − u(ϕ(z0 ))|.

(6.6)

z∈B 2 (z0 ,ε)

Let ψ be the inverse of ϕ and let w0 = ϕ(z0 ) and ρ = ε|ϕ  (z0 )|/4. Then recalling 1 and applying Theorem 6.2, we get the definition of OCK sup{|u(ϕ(z)) − u(ϕ(z0 ))| : z ∈ B 2 (z0 , ε)} = sup{|u(w) − u(w0 )| : w ∈ ϕ(B 2 (z0 , ε))}  sup{|u(w) − u(w0 )| : w ∈ B 2 (w0 , ρ)}  |∇u(w0 )|ρ/K = |∇u(w0 )| · |ϕ  (z0 )|ε/(4K). This gives us (6.6) with K1 = 4K. The proof is completed.

 

6.3 Quasi-Nearly Subharmonic Functions and QC Mappings If h is a function harmonic in a domain Ω in the Euclidean space Rn , then the function |h|p , which need not be subharmonic in Ω for 0 < p < 1, behaves like a subharmonic function in the sense that the inequality |h(a)|p ≤

C rn

 B n (a,r)

|h|p dm,

B n (a, r) ⊂ Ω, 0 < p < ∞,

(6.7)

holds for some constant C ≥ 1, where B n (a, r) = {x : |x − a| < r} and where m is the Lebesgue measure normalized so that |Bn | := m(Bn ) = 1. The constant C in (6.7) depends only on n and p when p < 1, and C = 1 when p ≥ 1.

6.3 Quasi-Nearly Subharmonic Functions and QC Mappings

135

From the Fefferman–Stein proof it follows that (6.7) remains true if |h| is replaced by a nonnegative subharmonic function. So we have the following result. Theorem 6.9 If u ≥ 0 is a function subharmonic in a domain Ω ⊂ Rn , then u(a)p ≤

C rn

 up dm, B n (a,r)

B n (a, r) ⊂ Ω, 0 < p < ∞,

(6.8)

where C depends only on p and n, when p < 1, and C = 1 when p ≥ 1. Remark 6.10 The list of references related to Theorem 6.9 includes the following: [3, 38, 122, 125, 134–136]. Definition 6.11 Let u ≥ 0 be a locally bounded, measurable function on Ω. We say (see [122, 125]) that u is C-quasi-nearly subharmonic (abbreviated C-QNS) if the following condition is satisfied: u(a) ≤

C rn

 whenever B n (a, r) ⊂ Ω.

u dm, B n (a,r)

(6.9)

One can view (6.9) as a weak sub-mean value property. Besides nonnegative subharmonic functions, it also holds for nonnegative subsolutions to a large family of second order elliptic equations, see [64]. In fact, (6.9) is typically proven as a step towards Harnack inequalities for second order elliptic equations, using the Moser iteration scheme. Notice that u is quasi-nearly subharmonic if and only if u is everywhere dominated by its centered minimal function [36]. The paper [37] gives a partial generalization of the invariance under conformal mappings for both function classes QNS and RO, a result originally proven in [87]. More precisely, O. Dovgoshey and J. Riihentaus have shown in [37] that in Rn , n  2, both classes QNS and RO are invariant under bi-Lipschitz mappings. Since we know that if f is L-bi-Lipschitz, then it is L2n−2 -quasiconformal, it is natural to try to generalize this result to the whole class of quasiconformal mappings, and even more generally, to the class of all quasiregular mappings of bounded multiplicity. To prove our next theorem we need the following lemmas. Lemma 6.12 ([133, Proposition 4.14, pp. 20–21]) Let f : G −→ Rn be quasiregular. Then the transformation formula 

 (h ◦ f )Jf dm =

E

Rn

h(y)N(y, f, E) dy

holds for every measurable h : Rn −→ [0, ∞] and all measurable E ⊆ G. Lemma 6.13 Let f : Ω → Ω  be K-quasiregular and of bounded multiplicity N. Let x ∈ Ω and 0 < r ≤ 12 d(x, ∂Ω)). Then

136

6 Quasi-Nearly Subharmonic Functions and QC Mappings

 d f (x), ∂f (B n (x, r)) ≥ δ

sup n

|f (y) − f (x)|,

y∈B (x,r)

where δ = δ(n, K, N ). Proof Let x ∈ Ω and let 0 < r ≤ 12 d(x, ∂Ω). Note that f (x) is an interior point of f (B n (x, r)), because f is open. Moreover |f (y) − f (x)| = |f (z) − f (x)|

sup n

y∈B (x,r)

for some z ∈ ∂B n (x, r), and 0 < d(f (x), ∂f (B n (x, r)) = |f (ω) − f (x)| for some ω ∈ ∂B n (x, r). Let E = [f (x), f (ω)] be the segment between f (w) and f (ω), and F be a segment that joins f (z) to ∂Ω  (or to infinity) outside the ball B n (f (x), |f (z) − f (ω)|). We may assume that |f (z) − f (x)| ≥ 2|f (ω) − f (x)|. Let

u(y) =

⎧ ⎪ ⎨ ⎪ ⎩ log |f (z)−f (x)| |y−f (x)|

1, 0, 7

n

|f (z)−f (x)| log |f (ω)−f (x)|

y ∈ B (f (x), |f (ω) − f (x)|), y ∈ (B n (f (x), |f (z) − f (x)|))c , elsewhere.

Then, by Lemma 6.12, the following holds: 

|∇(u ◦ f )|n dm ≤ K Ω  |(∇u)(f (y))|n Jf (y) dm(y) Ω

≤ KN Ω  |∇u|n dm KN Cn ≤ . n−1 |f (z)−f (x)| log |f (ω)−f (x)| Note that since f is open, the set f −1 ([f (x), f (ω)]) contains a continuum joining x to ∂B n (x, r), and the set f −1 (F ) a contains a continuum joining ∂B n (x, r) to ∂B n (x, 32 r). By the capacity estimates (see, e.g., [89, p. 59]) we have  |∇(u ◦ f )|n dm ≥ δ0 (n, K) > 0. Ω

6.3 Quasi-Nearly Subharmonic Functions and QC Mappings

137

 

Now the lemma follows. The following lemma is an immediate consequence of Lemma 6.13.

Lemma 6.14 Let f : B n (0, 2) → Ω  be a K-quasiregular mapping with bounded multiplicity N such that f (0) = 0. Then there exist ρ ∈ (0, 1) and R > 0 such that B n (0, R) ⊃ f (Bn ) ⊃ B n (0, ρ), where R/ρ ≤ 1/δ, and δ depends only on K, n, and N. We shall also need the following fundamental fact. Lemma 6.15 ([63, p. 258]) Under the hypotheses of Lemma 6.14, there exists p > 1 such that  " #1/p J (y, f )p dm ≤C J (y, f ) dm, Bn

Bn

where p depends only on K, N, and n. We now need to introduce the following norm. Definition 6.16 (QNS-Norm) Let  ! C uQNS = inf C ≥ 0 : u(a) ≤ n u dm for all a ∈ Ω  , 0 < r ≤ d(a, ∂Ω  ) . r B n (a,r) Theorem 6.17 ([90, Theorem 1.2]) If u ≥ 0 is a C-quasi-nearly subharmonic function defined on a domain Ω  ⊂ Rn , n ≥ 2, and f is a K-quasiregular mapping with bounded multiplicity N from a domain Ω onto Ω  , then the function u ◦ f is C1 -quasi-nearly subharmonic in Ω, where C1 only depends on K, C, N , and n. Proof The proof reduces to the case Ω = B n (0, 2), f (0) = 0. Let v = u ◦ f. Using rotations and translations, it can be seen that it is enough to prove that  v(0) ≤ C

Bn

v(y) dm(y),

for C = C(K, n, uQNS ). By Theorem 6.1, it suffices to find q = q(K, N, n) ≥ 1 such that " #1/q v(0) ≤ C v(y)q dm(y) . (6.10) Bn

Towards the proof of this, we start from the Hölder inequality:  Bn

v(y)J (y, f ) dm(y) ≤

" Bn

#1/q "  v(y)q dm(y)

Bn

#1/p J (y, f )p dm(y) ,

138

6 Quasi-Nearly Subharmonic Functions and QC Mappings

where p = q/(q − 1). Using Lemma 6.12, we have 

 Bn

v(y)J (y, f ) dm(y) =  ≥

f (Bn )

u(y) N(y, f, Bn ) dm(y) u(y) dm(y)

f (Bn )

 ≥

u(y) dm(y) B n (0,ρ) n

≥ cρ u(0) = cρ n v(0). Note that we have used here Lemma 6.14 and the hypothesis that u is QNS. On the other side, by Lemmas 6.15 and 6.14, we have "

 #1/p J (y, f ) dm(y) dy ≤ C p

B

Bn

J (y, f ) dm(y)

 =C

f (Bn )

N (y, f, Bn ) dm(y)

≤ CN |f (Bn )| ≤ CN |B n (0, R)| = CN kn R n . Combining these inequalities, we obtain cρ v(0) ≤ CN kn R n

n

" Bn

#1/q v(y)q dm(y) .

Hence CN kn R n " v(0) ≤ cρ n

 Bn

#1/q v(y)q dm(y) .

The desired result now follows from the inequality R/ρ ≤ 1/δ, where δ depends only on K, n, and N.   It should be noted that the hypothesis of bounded multiplicity of f in Theorem 6.17 is necessary as the following example shows. Let f (z) = ez , Ω = C, Ω  = C \ {0}, E=



[exp(2j ), exp(2j + 1)],

j ≥2

and u(w) = χE (|w|). Then it can be checked that u is quasi-nearly subharmonic in Ω  but, on the other hand, u ◦ f is not quasi-nearly subharmonic in Ω. We shall now consider the above morphism property in more detail.

6.3 Quasi-Nearly Subharmonic Functions and QC Mappings

139

Definition 6.18 Let Ω and Ω  be subdomains of Rn . A mapping f : Ω → Ω  is a quasi-nearly subharmonic morphism (QNS-morphism) if there is a constant C < ∞ such that for every quasi-nearly subharmonic u defined in Ω  we have u ◦ f QNS ≤ CuQNS , where uQNS is the quasi-norm defined above in Definition 6.16. If the inequality holds with a constant C, we call f a C-quasi-nearly subharmonic-morphism. Finally, f is a strong QNS-morphism if there is a constant C so that f restricted to each domain G ⊂ Ω, f : G → G , is a C-QNS-morphism. Theorem 6.19 ([90, Theorem 1.3]) Let Ω, Ω  ⊂ Rn , n ≥ 2, be domains. Then a homeomorphism f : Ω → Ω  is a strong QNS-morphism if and only if f is quasiconformal. Proof It is an immediate consequence of Theorem 6.17 that the quasiconformality of the homeomorphism f is a sufficient condition for f to be a strong QNSmorphism. For the other direction, it suffices to prove that f −1 : Ω  → Ω is quasiconformal. Thus it suffices to verify the existence of H < ∞ such that n

lim sup r→0

diam (f −1 (B (y, r)))n n

|f −1 (B (y, r))|

≤H

(6.11)

for all y ∈ Ω  (see page 64 of [62]). To simplify our notation, we write x  = f (x) for x ∈ Ω in what follows. Fix y  ∈ Ω  and let r > 0. Towards proving (6.11), we may assume that r is sufficiently small such that B n (y, 2 diam (f −1 (B (y  , 2r)))) ⊂ Ω. n

Fix y0 ∈ ∂B n (y  , 2r) and pick y1 ∈ ∂B n (y  , r) so that |y0 − y1 | =

max

ω ∈∂B n (y  ,r)

|ω − y0 |.

Set G = Ω  \ {y0 } and G = Ω \ {y0 }. Now B n (y1 , |y1 − y0 |/2) ⊂ G and diam (f −1 (B (y  , r)) ≤ 2|y1 − y0 |. n

(6.12)

Define u(ω ) = χB n (y  ,r) (ω ) for ω ∈ G . Then u is QNS in G with uQNS ≤ 3n . Since f is C-QNS-morphism in G, we conclude that  1 u ◦ f dm |B n (y1 , |y1 − y0 |/2)| B n (y1 ,|y1 −y0 |/2)| . n −1 (B (y  , r)) ∩ B n (y , |y − y |/2)| 1 1 0 n |f = C3 |B n (y1 , |y1 − y0 |/2)| u ◦ f (y1 ) ≤ C3n

140

6 Quasi-Nearly Subharmonic Functions and QC Mappings

Recalling (6.12) and that u ◦ f (y1 ) = u(y1 ) = 1, we arrive at diam (f −1 (B (y  , r))n ≤ CCn |f −1 (B (y  , r))| n

n

 

as desired.

1,n It should be noted that if we assume the Wloc regularity for f, a version of Theorem 6.19 holds also for QNS-morphism. The reader may wish to compare this with related quasiconformal invariance properties for other function classes: BMO-functions, weight functions of the class A∞ , and doubling measures; see [18, 131, 148, 149, 154]. To introduce A∞ -measures, we need the following theorem of Coifman and Fefferman:

Theorem 6.20 ([35]) When Q is a cube with sides parallel to coordinates axis and μ is a measure defined on the Borel sets of Rn , the following conditions are equivalent: 1. There exist δ1 > 0 and C1 > 0 such that for all measurable E ⊆ Q the following holds: μ(E)/μ(Q)  C1 (|E|/|Q|)δ1 . 2. There exist δ2 > 0 and C2 > 0 such that for all measurable E ⊆ Q the following holds: |E|/|Q|  C2 (μ(E)/μ(Q))δ2 . 3. dμ = w(x) dx and there exist C > 0 and a > 0 such that for all Q |Q|−1



w(x) dx  C |Q|−1 Q



w(x)−a dx

−1/a

.

Q

Definition 6.21 The class of all measures μ that satisfy the conditions of the previous theorem is denoted by A∞ . In his paper [154] Uchiyama gave a characterization of quasiconformal mappings using A∞ measures by proving that if a homeomorphism of Rn ϕ is ACL and differentiable a.e., then ϕ is quasiconformal if and only if for all μ ∈ A∞ , the measures μ ◦ ϕ and μ ◦ ϕ −1 are also in A∞ . Now we are ready to introduce the BMO class of functions. Definition 6.22 A locally integrable real valued function u is said to be of bounded mean oscillation (BMO) in Rn , if $  $  $ $ 1 $u(x) − 1 u(x) dx $$ dx  K $ |Q| Q |Q| Q

6.3 Quasi-Nearly Subharmonic Functions and QC Mappings

141

for every cube Q ⊆ Rn and some constant K. On the space of BMO-functions, the BMO-norm can be defined by ||u||BMO = sup

Q⊆Rn

1 |Q|

 |u(x) − uQ | dx, Q

where 1 uQ = |Q|

 u(x) dx. Q

It should be noted that Uchiyama’s characterization is based on the result of Reimann [131], where under the similar conditions on ϕ (with the additional assumption that |ϕ(·)| and |ϕ −1 (·)| are absolutely continuous set functions) it follows that ϕ is quasiconformal iff there exists C > 0 such that ||f ◦ ϕ −1 ||BMO  C||f ||BMO

(6.13)

for each BMO function f . Later Astala proved that the local variant of (6.13) holds without analytic conditions on homeomorphism ϕ. More precisely, he proved the following theorem: Theorem 6.23 ([18]) Let ϕ : G → G be an orientation preserving homeomorphism. If there exists a constant C such that 1 ||u||BMO,D   ||u ◦ ϕ||BMO,D  C||u||BMO,D  C

(6.14)

holds for all subdomains D ⊆ G and for all u ∈ BMO(D  ), D  = ϕ(D), then ϕ is quasiconformal. Definition 6.24 Let D be a domain in Rn and let μ be a Borel measure defined on D. Let 2Q denote the cube concentric with Q and of the side length twice that of Q. We say that μ is a doubling on D and write μ ∈ D(D) if there exists a constant c > 0 such that μ(2Q)  c μ(Q) for all cubes Q with 2Q ⊆ D. In his paper [149], Staples gave a characterization of quasiconformal mappings f : G → G in terms of doubling measures using, additionally to ACL and differentiability a.e. of f, that f and f −1 satisfy the condition (N) (that the image of a null set is a null set), and the assumption that ν = μ ◦ f is a doubling measure on D ⊆ G for every doubling measure μ on D  = f (D). Similarly as in Theorem 6.23 this is a local type condition that is assumed on every subdomain D of G. Theorem 6.25 ([90, Theorem 1.4]) Let n ≥ 2 and let f : Ω → Ω  be a quasi1,n nearly subharmonic-morphism that belongs to Wloc . If, additionally, J (x, f ) ≥ 0 almost everywhere, then f is quasiregular.

142

6 Quasi-Nearly Subharmonic Functions and QC Mappings

Proof Our definition of quasiregular mappings requires them to be continuous; however, this condition is superfluous here and it suffices to show that there exists K ≥ 1 so that |Df (x)|n ≤ KJ (x, f ) holds almost everywhere; see e.g., [133, p. 177]. Furthermore, every mapping f 1,1 is approximatively differentiable almost everywhere. with Sobolev regularity Wloc More precisely, for almost every x0 and every  > 0, the set A = x :

! |f (x) − f (x0 ) − Df (x0 )(x − x0 )| 0 is such that B n (x0 , r) ⊂ Ω, we conclude from the morphism property of f that

6.3 Quasi-Nearly Subharmonic Functions and QC Mappings

143

 C u ◦ f dm r 2 B n (0,r) C1 a = 2 r 2 arctan + o(r) b r 2 Ca , → 2b

1≤

when r → 0, where C > 0 comes from the morphism property. Hence b/a ≤ C/2. Case (b) Suppose now that f is differentiable at x0 with Jf (x0 ) = 0. We want to prove that Df (x0 ) = 0. We argue by contradiction: suppose Df (x0 ) = 0. We may assume   00 Df (x0 ) = . 01 Define u(ω) = χ{ω=x  +iy  :0≤|y  |≤x  } (ω − f (x0 )). Then uQNS = 1, and f −1 (t + it + f (x0 )) = s1 (t) + is2 (t) + x0 , where lim

s2 (t)

t→0 s1 (t)

= 0. But then there is no C > 0 such that (u ◦ f )(x0 ) ≤

C r2

 B n (x0 ,r)

u ◦ f dm

for all small r > 0, which contradicts the morphism property of f.   Each quasiregular mapping f is either constant or both open and discrete; in the latter case the multiplicity of f is locally finite. The condition J (x, f ) ≥ 0 in Theorem 6.25 cannot be dropped, as the mapping f (x, y) = (x, |y|) is a planar (strong) quasi-nearly subharmonic-morphism. It turns out that the Sobolev regularity assumption can be slightly relaxed: if f above is a C-quasi-nearly subharmonic-morphism, then local p-integrability of the distributional derivatives suffices for p = p(n, C) < n. This can be deduced using [69] and the proof of 1,1 Theorem 6.25. In the injective planar case even the regularity Wloc suffices.

144

6 Quasi-Nearly Subharmonic Functions and QC Mappings

6.4 Regularly Oscillating Functions and QC Mappings In this section we emphasize the invariance of a related function class, introduced in [122]. Definition 6.26 A function u : Ω  → Rk is said to be regularly oscillating if Lip u(x) ≤ Cr −1

sup

y∈B n (x,r)⊂Ω 

|u(y)−u(x)|,

x ∈ Ω  , B n (x, r) ⊂ Ω  ,

(6.15)

where C ≥ 0 is a constant independent of x and r. Here Lip u(x) = lim sup y→x

|u(y) − u(x)| . |y − x|

(We borrow this notation from [89].) Note that Lip u(x) = |grad u(x)| if u is differentiable at x. The smallest C satisfying (6.15) will be denoted by uRO . We will now show the following invariance property of regularly oscillating functions. Theorem 6.27 ([90, Theorem 1.5]) Let f : Ω → Ω  be quasiregular, regularly oscillating, and of bounded multiplicity in Ω. If u is regularly oscillating in Ω  , then u ◦ f is regularly oscillating in Ω with u ◦ f RO ≤ C  uRO , where C  depends only on the multiplicity of f , K, n, and ||f ||RO . Proof Let x ∈ Ω and 0 < r < 12 d(x, ∂Ω). Since the mapping f is regularly oscillating and quasiregular we have, by Lemma 6.13, Lip f (x) ≤ Cr −1 supy∈B n (x,r) |f (y) − f (x)| C ≤ r −1 d(f (x), ∂f (B n (x, r))). δ We now recall that nonconstant quasiregular mappings are open. Since u is regularly oscillating and d(f (x), ∂f (B(x, r))) > 0, we have that ˆ Lip u(f (x)) ≤ Cd(f (x), ∂f (B n (x, r)))−1 supz∈B n (f (x),d(f (x),∂f (B n (x,r))) |u(f (x)) − z| ˆ ≤ Cd(f (x), ∂f (B n (x, r)))−1 supy∈B n (x,r) |u ◦ f (y) − u ◦ f (x)|.

It follows that Lip (u ◦ f )(x) ≤ Lip (u(f (x)) Lip f (x) ˆ ≤ Cd(f (x), ∂f (B n (x, r)))−1 supy∈B n (x,r) |u ◦ f (y) − u ◦ f (x)| C −1 × r d(f (x), ∂f (B n (x, r))) δ = C  r −1 supy∈B n (x,r) |u ◦ f (y) − u ◦ f (x)|. This completes the proof of the theorem.

 

6.5 Some Generalizations and Examples

145

It should be noted that the assumption that f is regularly oscillating is necessary in this theorem. This can be seen by observing that the coordinate projections are regularly oscillating. Recall that not all quasiregular mappings are regularly oscillating. However, in the case of analytic functions, this assumption can be dropped. Moreover, similarly as in Theorem 6.25, quasiregularity is also necessary if we assume that J (x, f ) ≥ 0 almost everywhere. However, no Sobolev regularity is needed because regularly oscillating functions and mappings are locally Lipschitz continuous. In case n = 2 and when f is conformal the invariance property of Theorem 6.27 was established in [87]. It should be also remarked that the assumption of bounded multiplicity of f in Theorem 6.27 is necessary as in the case of Theorem 6.17. To see this simply let f (z) = ez , Ω = C, Ω  = C \ {0}, E=



[exp(2j ), exp(2j + 1)],

j ≥2

and v(w) =

|w| 0

χE (t) dt. Then v is regularly oscillating but v ◦ f is not.

6.5 Some Generalizations and Examples It is natural to pose the following question: Question 6.28 Can these results be generalized for metric spaces? Recently, in the paper [38], Dovgoshey and Riihentaus gave some generalizations of Definition 6.11 as well as some examples related to these generalizations. Let Ω be a domain in Rn , n ≥ 2. Definition 6.29 ([38], see also the reference therein) Let u : Ω → [−∞, ∞) be a Lebesgue measurable function on Ω. We say that u is C-quasi-nearly subharmonic 1 (Ω) and the following condition in narrow sense for a constant C  1 if u+ ∈ Lloc is satisfied:  C u(a) ≤ n u dm, whenever B n (a, r) ⊆ Ω. (6.16) |B (a, r)| B n (a,r) A function u is quasi-nearly subharmonic in narrow sense if it is C-quasi-nearly subharmonic in narrow sense for at least one C  1. Definition 6.30 ([38]) Let u : Ω → [−∞, ∞) be a Lebesgue measurable function 1 (Ω) and for all on Ω. We say that u is C-quasi-nearly subharmonic if u+ ∈ Lloc M  0 the following condition is satisfied:

146

6 Quasi-Nearly Subharmonic Functions and QC Mappings

uM (a) ≤

C |B n (a, r)|

 B n (a,r)

whenever B n (a, r) ⊂ Ω,

uM dm,

(6.17)

where uM (x) = max{u(x), −M} + M. A function u is quasi-nearly subharmonic if it is C-quasi-nearly subharmonic for at least one C  1. The class of functions from Definition 6.11 is a proper subclass of the class of functions from Definition 6.29. The class of functions from Definition 6.29 is a proper subclass of the class of functions from Definition 6.30. However, all three definitions match in the class of nonnegative functions. Example 6.31 ([38]) The function u : R2 → R  u(x, y) =

−1, y < 0, 1, y  0

is 2-quasi-nearly subharmonic, but not quasi-nearly subharmonic in a narrow sense. Example 6.32 ([38]) The function u : R2 → R  u(x, y) =

3, x = 0, 1, x = 0

is 3-quasi-nearly subharmonic. The constant function v : R2 → R, v(x, y) = −2 is harmonic. Then  1, x = 0, (u + v)(x, y) = −1, x = 0 and (u + v)M = max{u + v, −M} + M = (u + v + M)+ for every M  0. In particular for M = 1 we obtain  (u + v)1 (x, y) =

2, x = 0, 0, x = 0.



Since (u+v)1 (0, 0) > 1 and the double integral B 2 (a,r) (u+v)1 (x, y) dx dy is zero for every a ∈ R2 and r > 0, the function (u + v)1 is not quasi-nearly subharmonic. Hence (u + v) is also not quasi-nearly subharmonic.

Chapter 7

Possible Research Directions

7.1 Characterizations of Boundary Values We have already mentioned that Pavlovi´c [123] made a deep and detailed analysis of the boundary values of harmonic quasiconformal mappings of the unit disk D by proving the following theorem. Theorem 7.1 Let u be a harmonic homeomorphism of D that extends continuously to the boundary, and f : S 1 → S 1 be its boundary function. Then the following conditions are equivalent: 1. u is quasiconformal; 2. u is bi-Lipschitz in the Euclidean metric; 3. f is bi-Lipschitz and the Hilbert transformation of its derivative is in L∞ . This result has initiated an extensive line of research between the theories of bi-Lipschitz conditions and HQC mappings (see [15, 79, 118] and [76]). In [99], the author has shown that HQC mappings between every two proper domains in the plane are bi-Lipschitz with respect to the corresponding quasihyperbolic metrics. More precisely, in [99] the following result was obtained. Theorem 7.2 Suppose D and D  are proper domains in R2 . If f : D → D  is K-quasiconformal and harmonic, then it is bi-Lipschitz with respect to the quasihyperbolic metrics on D and D  . The proof of this result was based on a counterpart of Koebe theorem, established by Astala and Gehring, using estimates of a geometric notion of average derivative on a ball 1 αf (z) = exp (log Jf )Bz . n

© Springer Nature Switzerland AG 2019 V. Todorˇcevi´c, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, https://doi.org/10.1007/978-3-030-22591-9_7

(7.1)

147

148

7 Possible Research Directions

This leads us to the following natural problem about the co-Lipschitz condition in higher dimensions. Problem 7.3 Does a HQC map have a nonvanishing Jacobian in dimensions greater than two? It should be noted that this problem is closely related to properties of the function (7.1), so this will be the first thing to be examined here. Another line of research initiated by Theorem 7.1 deals with characterizations of boundary values of HQC mappings. In [79], the analogous result was proved for the half-plane. A natural question arises: what happens in higher dimensions, both in the case of the unit ball and in the case of the upper half space? Since the answer to this question in dimension n = 2 involves Hilbert’s transform, we expect that in higher dimensions it will involve singular integral operators, like Riesz transforms. A variant of this problem is to replace the condition of harmonic quasiconformal (HQC) with the condition of harmonic quasiregular (HQR), as well as with the condition of p-harmonic quasiconformal mappings, which involve nonlinear potential theory. Namely, p-harmonic mappings minimize the Lp norm of the gradient similar to the situation of L2 norm under standard harmonic mappings. This provides possibilities to extend our research to quasiconformal aspects of potential analysis, a very active area of current research. The characterization of boundary values can also be approached in terms of intrinsic metric properties of the domain. In particular, hyperbolic and quasihyperbolic metrics, which depend on the geometric properties of the domain in Rn , capture some of the most essential properties relevant for boundary behavior. A map f : X → Y can then be analyzed as a map between corresponding metric spaces (X, dX ) and (Y, dY ). We say that f is a (C, D) quasi-isometry if 1 dY (f (x), f (y)) − D ≤ dX (x, y) ≤ CdY (f (x), f (y)) + D. C A hyperbolic space Hn can be modeled as a unit ball Bn , equipped with a hyperbolic metric. It has a boundary S n−1 at infinity. Each quasi-isometry f : H3 → H3 extends continuously to S 2 , and the restriction of this extension to S 2 is quasiconformal. The question about existence of harmonic quasiconformal extension of a quasiconformal (quasisymmetric for dimension n = 2) boundary condition was a long-standing open problem. The famous Schoen conjecture (formulated in the Introduction) was proved by V. Markovi´c in [108]. The analogue of this conjecture in dimension 3 is also proved by V. Markovi´c [107] while analogues for dimensions n > 3 are proved by L. Marius and V. Markovi´c [106]. Theorem 7.4 (Generalized Schoen Conjecture [106–108]) Suppose that f : S n−1 → S n−1 is a quasiconformal map. Then there exists a unique harmonic (with respect to the hyperbolic metric) and quasi-isometric map fˆ : Hn → Hn that extends f .

7.2 QC and HQC Mappings on Non-smooth Domains

149

The two-dimensional case is important because of the links with the Teichmüller theory. Because of the uniqueness part of this conjecture, proved by P. Li and L. F. Tam in [96], harmonic quasiconformal hyperbolic mappings can serve as representatives of a class in the Universal Teichmüller space. Using quasi-isometries, which are a natural and more flexible generalization of quasiconformal mappings, it seems possible that there is another possible approach leading us to the following question: Problem 7.5 Characterize boundary mappings f : S n−1 → S n−1 for which the Euclidean harmonic extension fˆ : Bn → Bn is quasi-isometric with respect to the hyperbolic metric. For general domains, the quasihyperbolic metric is comparable to the usual hyperbolic metric in a simply connected plane domain by the Koebe distortion theorem, and the quasihyperbolic metric continues to be a useful tool in dimensions n ≥ 3, whereas the hyperbolic metric is less useful. Also, it seems natural to consider quasi-isometries with respect to quasihyperbolic metric as a condition to replace quasiconformality. Thus, we propose a more general question in higher dimensions: Problem 7.6 Characterize boundary mappings f : ∂D → ∂D  for which there exists a unique Euclidean harmonic map fˆ : D → D  , quasi-isometric with respect to the quasihyperbolic metrics, that extends f , where D and D  are proper domains in Rn .

7.2 QC and HQC Mappings on Non-smooth Domains The geometry of domains has been an important topic in Geometric Function Theory. Martio and Sarvas [112] were the first to consider and introduce the socalled uniform domains in the late 1970s. We recall that these are domains in which every two points can be joined by an arc of length bounded by a constant times the Euclidean distance between them such that each point on that arc divides the arc into two parts, at least one of them of length smaller than a constant times the distance of dividing point to the boundary of the domain (i.e., the arc does not come too close to the boundary of the domain). Around the same time, Jones [72] characterized domains which have the property that bounded mean oscillation (BMO) functions can be extended from the domain to the whole space. Gehring and Osgood in [53] have proved that Jones domains are precisely uniform domains. Uniform domains equipped with the quasihyperbolic metric have the property of being Gromov hyperbolic. In the eighties the Russian mathematician M. Gromov introduced the notion of hyperbolic space, which has thereafter been studied and developed further by many authors. For a long time the research was centered at Hyperbolic Group Theory, but lately many researchers in Geometric Function Theory have developed interests towards the theory of Gromov hyperbolic spaces.

150

7 Possible Research Directions

In particular, Bonk, Heinonen, and Koskela [29] have established a canonical correspondence between geodesic Gromov hyperbolic spaces and non-complete uniform domains. The non-geodesic case has been considered by Bonk and Schramm [28] and by Väisälä [156]. The main result of the author’s paper [91] with P. Koskela and P. Lammi provides characterization of Gromov hyperbolicity of the quasihyperbolic metric spaces (Ω, k) by geometric properties of the Ahlfors regular length metric measure space (Ω, d, μ). The characterizing properties are called the Gehring–Hayman condition and ball-separation condition. The investigation of an Ahlfors regular domain in the context of the HQC mappings can be deepened. An Ahlfors regular domain D is a domain D ⊂ C which is bounded by an Ahlfors regular curve, , i.e., a rectifiable curve γ such that any small disk of radius r covers part of γ of length at most Cr for some C > 0. Consider mappings f : D −→ Ω, where D = {z ∈ C : |z| < 1} and Ω is a simply connected domain in the complex plane. Theorem 7.7 Suppose f : D → Ω is conformal and Ω is an arbitrary simply connected domain. Then log f  (z) ∈ B, where B denotes the Bloch space, consisting of analytic functions g : D → C with gB := sup (1 − |z|2 )|g  (z)| < ∞. |z|