characteristic properties of quasidisks · formal mappings. in this section we shall derive a...

SEMINAIRE DE MATHEMATIQUES SUPERIEURES

SEMINAIRE SCIENTIFIQUE OTAN (NATO ADVANCED STUDY INSTITUTE)

DEPARTEMENT DE MATHEMATIUQUES ET DE STATISTIQUE - UNIVERSITE DE MONTREAL

CHARACTERISTIC PROPERTIES

OF QUASIDISKS

FREDERICK W. GEHRING

University of Michigan

1982

LES PRESSES DE L’UNIVERSITE DE MONTREAL

C.P. 6128, succ. ≪A≫, Montreal (Quebec) Canada H3C 3J7

FOREWORD

These notes formed the basis of a short course of six lectures which

I gave at the NATO advanced Study Institute on Function Theory in

Montreal in August 1981. My object was to point out some of the

surprising connections which quasidisks have with various branches of

analysis. Unfortunately there was not time to treat all aspects of this

subject or to give more than a few proofs.

I should like to thank Professor B.G. Osgood for his help in writ-

ing up these notes, Miss B.A. Brown for making the figures, and the

Seminaire de mathematiques superieures of the University of Montreal

for arranging the typing.

This research was supported in part by grants from the National

Science Foundation, Grant MCS 70-01713, and from the Humboldt

Foundation.

F.W. Gehring

Ann Arbor, Michigan

3

Contents

FOREWORD 3

Chapter I. PRELIMINARIES 7

1. Plane quasiconformal mappings 7

2. Modulus estimates 11

3. Quasidisks 20

Chapter II. CHARACTERISITC PROPERTIES OF

QUASIDISKS 23

1. Introduction 23

2. Replection property 24

3. Local connectivity properties 24

4. Hyperbolic metric properties 27

5. Injectivity properties 34

6. Extension properties 39

7. Homogeneity property 42

8. Miscellaneous properties 43

Chapter III. SOME PROOFS OF THESE PROPERTIES 45

1. Table of implications 45

2. Quasidisks have the hyperbolic segment property 46

3. Hyperbolic segment property implies D is uniform 52

5

6 CONTENTS

4. Uniform domains are linearly locally connected 53

5. Linear local connectivity implies the three point property 55

6. Three point property implies D is a quasidisk 57

7. Uniform domains have the Schwarzian derivative property 64

8. Schwarzian derivative property implies D is linearly locally

connected 71

9. Quasidisks have then BMO extension property 75

10. BMO extension property implies hyperbolic bound

property 77

11. Hyperbolic bound property implies hyperbolic segment

property 82

Chapter IV. EPILOGUE 91

1. 91

REFERENCES 95

CHAPTER I

PRELIMINARIES

1. Plane quasiconformal mappings

In this section we shall collect some definitions and properties of

plane quasiconformal mappings. The basic reference is the book by

Lehto and Virtanen to which we refer for all details. In what follows

D and D′ will denote domains in the extended complex plane C =

C ∪ {∞}.

1.1. GEOMETRIC DEFINITION. (IV.4.1. and 4.2 in [19])

Suppose that f : D → D′ is a sence preserving homeomorphism. For

each z ∈ D \ {∞, f−1(∞)} let

H(z) = lim supr→0

L(z, r)

ℓ(z, r),

b bzf(z)

r f

Lℓ

where

L(z, r) = max|z−w|=r

|f(z) − f(w)| ,

ℓ(z, r) = min|z−w|=r

|f(z) − f(w)| .

7

8 I. PRELIMINARIES

We say that f is K−quasiconformal, abbreviated K−qc, 1 ≤ K <∞,

if H es bounded in D \ {∞, f−1(∞)} and if

H(z) ≤ K

a.e. in D.

1.2. Class ACL(D). A real function u is said to be absolutely

continuos on lines or ACL in D if, for each closed oriented rectangle

R = [a, b] × [c, d] ⊂ D ,

x

y

R

a b

c

d

D

u(x + iy) is absolutely continuos in x for almost all y ∈ [c, d] and

absolutely continuos in y for almost all x ∈ [a, b]. A complex valued

function f es said to be ACL in D if its real and imaginary parts are

ACL in D.

1.3. ANALYTIC DEFINITION. (I.V.2.3 in [19]). A sense pre-

serving homeomorphism f : D → D′ is K− qc if and only if f is ACL

en D and if

maxα

|∂αf(z)|2 ≤ KJ(z)

a.e. in D. Here ∂αf denotes the derivative of f in the direction α and

J(z) denotes the Jacobain of f at z.

1. PLANE QUASICONFORMAL MAPPINGS 9

1.4. REMARK. If a homeomorphism f : D → D′ is ACL en D,

then it has finite partial derivatives a.e. in D and hence has a dif-

ferential a.e. in D by a theorem of Gehring and Lehto (III.3.2. in

[19]).

1.5. Extremal Lengths. Suppose that Γ is a family of curves

γ ⊂ C. We want to assign a nummber, or modulus, which measures

the side of Γ and is conformally invariant. We say that a function ρ

is admissible for γ, written ρ ∈ adm Γ, if ρ es nonnegative and Borel

measurable in C an if∫

γ

ρ(z)|dz| ≥ 1

for each locally rectifiable curve γ ∈ Γ. We then define the modulus of

Γ to be

mod Γ = infρ

∫∫

C

ρ2(z)dxdy ,

where the infimum is taken over all ρ ∈ admΓ.

1.6. REMARK. There is a useful physical interpretation of this.

If the curves in Γ are disjoint arcs γ, we may think of each γ as a homo-

geneous wire. Then modΓ is a conformally invariant transconductance

for the family Γ and

λ(Γ) =1

mod Γ,

called the extremal length of Γ, is a measure of the total resistance of

the system. Thus mod Γ is large the curves γ are short and plentiful,

small if they are long and scarse.

10 I. PRELIMINARIES

1.7. EXTREMAL LENGTHS DEFINITION. (IV.3.3 in [19]).

A sense preserving homeomorphism f : D → D′ is K − qc if and only

if it satisfies the ine qualities

1

Kmod Γ ≤ mod Γ′ ≤ Kmod Γ

for each family of curves Γ in D where Γ′ = f(T ).

To conclude, we list several properties of quasiconformal mappings

that will be used in the sequel.

1.8. 1-qc mappings. (I.5.1 in [19]). A mapping f of D is 1 − qc

if and only if f is a conformal mapping, i.e., a homeomorphism which

is analytic as a function of a complex variable in D \ {∞, f−1(∞)}.

1.9. Composition and inverse. (I.3.2 in [19]). If f : D → D′ es

K1−qc and g : D′ → D′′ es K2−qc, then g ◦f : D → D′′ is K1K2−qc.

The inverse of a K − qc mappings is K − qc.

1.10. Extension theorem. (I.8.2 in [19]). If f : D → D′′ is

K − qc and if D and D′′ are Jordan domains, then f can be extended

to a homeomorphism mapping D onto D′.

1.11. Removable sets. (V.3.4 in [19]). Suppose a closed set E ⊂

D can be expressed as the enumerable union of rectifiable curves. If

f : D → D′ is a homeomorphism which is K-qc in each component of

D \ E, then f is K-qc in D.

2. MODULUS ESTIMATES 11

2. Modulus estimates

Estimates for the moduli of various curve families can be used very

effectively in the geometric theory of both conformal and quasicon-

formal mappings. In this section we shall derive a simple distortion

theorem for quasiconformal mappings of the plane. The first several

lemmas are typical of the type of arguments in this context.

2.1. LEMMA. If Γ is the family of arcs joining the horizontal

sides of the indicated rectangle R,

b

a

R

Γ

then mod Γ =b

a.

Proof. We may assume that R = [0, b]× [0, a]. For 0 < x < b the

segment

γ = {z = x+ iy : 0 < y < a}

is in the family Γ and hence for ρ ∈ adm Γ,

1 ≤

∫

γ

ρ(z)|dz| =

∫ a

0

ρ(x+ iy)dy ≤

(∫ a

0

ρ(x+ iy)2dy

)1

2(

∫ a

0

dy

)1

2

whence

∫∫

C

ρ2(z)dxdy ≥

∫ b

0

(∫ a

0

ρ(x+ iy)2dy

)

dx ≥

∫ b

0

1

adx =

b

a.

12 I. PRELIMINARIES

Thus

mod Γ = infρ

∫∫

C

ρ2(z)dxdy ≥b

a: .

On the other hand, the function

ρ(z) =

1

az ∈ R

0 z ∈ C \ R

is in adm Γ and∫∫

C

ρ2(z)dxdy =b

a.

�

2.2. COROLLARY. If Γ is a family of arcs which join (−∞, x1)

to (x2, x3) in the upper half plane H, then

mod Γ = m

(

x3 − x2

x2 − x1

)

,

where for 0 < t <∞, m(t) increases strictly from 0 to ∞ as t increases

from 0 to ∞ and m(1) = 1.

H

Γ

x1 x2 x4∞

Proof. By use of elliptic functions one can mapH onto a rectangle

R so that x1, x2, x3,∞ corespond to the vertices.


∞ x1 x2 x3

Γ

H

Γ′1

m

By conformal invariance and in the notation of the figure,

mod Γ = mod Γ′ = m

and m(t) can be written explicitly in terms of elliptic integrals of the

first kind. The above-mentioned properties of m(t) follow directly from

this representation. �

2.3. LEMMA. If Γ is a family of curves γ and is for each t with

a < t < b the circle |z − z1| = t contains a γ ∈ Γ, then

mod Γ ≥1

2πlog

b

a.

z1b

t

b

a

γ

14 I. PRELIMINARIES

Proof. Let ρ ∈ adm Γ. Then for a < t < b,

∫ 2π

0

ρ(teiθ)tdθ ≥

∫

γ

ρ(z)|dz| ≥ 1

whence

1 ≤

(∫ 2π

0

ρ(teiθ)tdθ

)2

≤ 2πt

∫ 2π

0

ρ2(teiθ)tdθ ,

and we obtain

∫∫

C

ρ2(z)dxdy ≥

∫ b

a

(∫ 2π

0

ρ2(teiθ)tdθ

)

dt ≥

∫ b

a

1

2πtdt =

1

2πlog

b

a.

Thus

mod Γ = infρ

∫∫

C

ρ2(z)dxdy ≥1

2πlog

b

a.

�

2.4. LEMMA. If α1 and α2 are disjoint arcs with

d(α1, α2) ≥ r, dia (α1) ≤ s ,

and if Γ is a family of arcs γ which join α1 and α2, then

mod Γ ≤ π(s

r+ 1

)2

.

Here d(α1, α2) is the euclidean distance from α1 to α2 and dia(α1) is

the euclidean diameter of α1.

α1

α2

Γ


Proof. Choose z1 ∈ α1 and z2 ∈ α2 so that

|z1 − z2| = d(α1, α2) ≥ r

and set

ρ(z) =

1

rz ∈ B(z1, r + s)

0 elsewhere

where B(z1, r + s) is the disk of radius r + s centered at z1.

Since α1 ⊂ B(z1, s), each γ ∈ Γ either joins α1 to α2 in B(z1, r+ s)

or joins ∂B(z1, s) to ∂B(z1, r + s).

bz1

α1 α2

γ

s

In either case γ contains a subarc δ of length r which lies in B(z1, r+s).

Hence∫

γ

ρ(z)|dz| ≥

∫

δ

ρ(z)|dz| =1

rℓ(δ) = 1 .

Thus ρ ∈ adm Γ and

mod Γ ≤

∫∫

C

ρ2(z)dxdy = π(s

r+ 1

)2

16 I. PRELIMINARIES

as desired. �

Next, for z ∈ C let p(z) denote the stereografic projection of z onto

the Riemann sphere S of radius 1. Then given z1, z2 ∈ C, we define

their spherical distance s(z1, z2) as the distance between p(z1) and p(z2)

measured on S. Thus

s(z1, z2) = infγ

∫

γ

2|dz|

1 + |z|2,

where the infinum is taken over all locally rectificable arcs γ which join

z1 and z2 in C.

b

b

b

b

b

b

z1z2

p(z1)

p(z2)

p

0

C S

0

2.5. LEMMA. If Γ is a family of closed curves γ each of which

separates z1, w1 from z2, w2 and if

s(z1, w1) ≥ ℓ, s(z2, w2) ≥ ℓ

then

mod Γ ≤π

ℓ2.


Proof. For z ∈ C set

ρ(z) =1

2ℓ

2

1 + |z|2.

If γ ∈ Γ, then p(γ) is a closed curve on S which separates p(z1), p(w1)

from p(z2), p(w2). Since, by assumption, every arc on S which joins

p(z1) to p(w1) or p(z2) to p(w2) has length at least ℓ, one can show that

ℓ(p(γ)) ≥ 2ℓ .

Thus∫

γ

ρ(z)|dz| =1

2ℓℓ(p(γ)) ≥ 1 ,

ρ ∈ adm (Γ), and we find that

mod Γ ≤

∫∫

C

ρ2(z)dxdy =1

4ℓ2

∫∫

C

4

(1 + |z|2)2dxdy =

1

4ℓ2m(S) =

π

ℓ2.

�

We now prove the following distortion theorem.

2.6. THEOREM. If f : C → C is K-qc with f(∞) = ∞ and is

z0, z1, z2 ∈ C with

|z2 − z0| ≤ |z1 − z0| ,

then

|f(z2) − f(z0)| ≤ c|f(z1) − f(z0)| ,

where c = e8K .

18 I. PRELIMINARIES

Proof. By a change of variables we may first assume that z0 = 0

and f(z0) = 0. Second, we may also assume that

|f(z1)| < |f(z2)|

for otherwise there is nothing to prove. Let Γ′ be the family of circles

|w| = t for |f(z1)| < t < |f(z2)|. Then by Lemma 2.3,

mod Γ′ ≥1

2πlog

|f(z2)|

|f(z1)|.

Next, since each γ′ ∈ Γ′ separates the points f(z1), 0 from f(z2), ∞

each γ ∈ Γ = f−1(Γ′) separates the points z1, 0 from z2, ∞.

b

b

b

b

bb

b

b

0

z1

z2

0

f(z2)

f(z1)

∞∞

Γ

Γ′f

Now if |z1| = 1, then

s(z1, 0) =π

2, s(z2,∞) ≥

π

2,

and hence

mod Γ ≤π

(π/2)2=

4

π

by Lemma 2.5. If |z1| 6= 1 we take g(z) = z/z1, Γ′′ = g(γ), and the

above argument shows that

mod Γ = mod Γ′′ ≤4

π.


Fnally, since f is K-qc we have

1

2πlog

|f(z2)|

|f(z1)|≤ mod Γ′ ≤ Kmod Γ ≤

4K

π

or

|f(z2)|

|f(z1)|≤ e8K

as desired. �

2.7. COROLLARY. If f : C → C is K-qc with f(∞) = ∞, and

if z0, z1, z2 ∈ C with

|z2 − z0| ≤ 2k|z1 − z0| ,

where k is a nonnegative integer, then

|f(z2) − f(z0)| ≤ (2c)k+1|f(z2) − f(z0)|

where c = e8K .

Proof. We prove this by induction on k, the result holding for

k = 0 by Theorem 2.6. Assume that the conclusion holds for k = 1

and let z = 12(z2 + z0).

b

b

b

b

z2

z0

z

z1

Then

|z2 − z| = |z − z0| ≤ 2k−1|z1 − z0| ,

and hence

|f(z2) − f(z)| ≤ c|f(z) − f(z0)|

20 I. PRELIMINARIES

and

|f(z) − f(z0)| ≤ (2c)k|f(z1) − f(z0)|

by the induction hypothesis. Thus

|f(z2) − f(z0)| ≤ |f(z2) − f(z)| + |f(z) − f(z0)|

≤ (c+ 1)(2c)k|f(z1) − f(z0)|

≤ (2c)k+1|f(z1) − f(z0)|

proving the result for k and hence in general. �

2.8. REMARK. Corollary 2.7 can be used to derive the Holder

continuity and distortion properties of K-qc mappings f : C → C

which fix ∞. The exponents, however, will not be best possible.

3. Quasidisks

We now define the principal object of study in these lectures.

3.1. DEFINITION. D is a K-quasidisk if it is the image of an

open disk or half plane under a K-qc mapping f : C → C.

Thus if D is a quasidisk, then ∂D is a Jordan curve. Theorem

2.6 can be used to show that the boundary of a quasidisk has planar

measure zero, but the following example shows that such domains can

be quite wild.

3. QUASIDISKS 21

3.2. EJEMPLO. [12]. Choose four squares Qj and Q′j in the

square Q = [−1, 1] × [−1, 1] as indicated.

f0

f0

Q

Qj

Q′

j

Next choose a piecewise linear homeomorphism

f0 : Q \⋃

Qj → Q \⋃

Q′j

so that f0 is the identity on ∂Q and of the form

f0(z) = ajz + bj , aj > 0, on ∂Qj for j = 1, 2, 3, 4.

Then f0 is K-qc in Q \⋃

Qj , where K is a constant which depends

only on the size and relative positions of the squares Qj and Q′j .

Now choose squares Qjk in Qj and Q′jk in Q′

j in the same way as

Qj and Q′j were are chosen in Q. By scaling f0 we can extend it to a

homeomorphism

f1 : Q \⋃

Qjk → Q \⋃

Q′jk .

22 I. PRELIMINARIES

If we continue this way we obtain a homeomorphism f : Q \ E →

Q′ \ E ′, where E and E ′ are Cantor sets, which can then be extended

by continuity to give a K-qc mapping of Q onto Q. Set f(z) = z in

C \ Q. Then f is K-qc and maps the upper half plane H onto a K-

quasidisk D whose boundary is no-rectifiable. In fact by choosing the

Q′j properly, one can assure that the Hausdorff dimension of ∂D is at

least a for any prescribed constant a, 1 ≤ a < 2.

f

Although this example suggests that quasidisks are rather pathological,

they occur very naturally in many branches of analysis. In Chapter

II we shall list, with brief explanations, a number of characteristic

properties of quasidisks which generalize corresponding properties of

euclidean disks. Chapter III will be devoted to proofs of several of

these properties.

CHAPTER II

CHARACTERISITC PROPERTIES OF

QUASIDISKS

1. Introduction

We assume throughout this chapter that D is a simply connected

subdomain of the finite complex plane C and we let D∗ denote the

exterior of D. For zo ∈ C and 0 < r <∞ we let

B(z0, r) = {z : |z − z0| < r}

and B = B(0, 1). Finally we let H and H∗ denote the upper and lower

half planes.

We present now a list of seventeen characteristic properties of qua-

sidisk which divide into three different categories as follows:

1. Geometric properties:

Reflection property - 1

Local connectivity properties - 2

Hyperbolic metric properties - 3

2. Function theoretic properties:

Injectivity properties - 3

Extension properties - 3

23

24 II. CHARACTERISITC PROPERTIES OF QUASIDISKS

Homogeneity properties - 2

3. Miscellaneous properties

Limit set of a discontinuous group - 1

Dirichlet integral property - 1

Mapping property - 1

2. Replection property

2.1. Quasi-isometries in C. Suppose that f is a mapping of C

into C. We say that f is an L-quasi-isometry of C if f(∞) = ∞ and if

1

L|z1 − z2| ≤ |f(z1) − f(z2)| ≤ L|z1 − z2|

for all z1, z2 ∈ C.

2.2. REMARK. If D is a half plane, then reflection in ∂D is a

1-quasi-isometry of C which maps D onto D∗ and is the identity on

∂D.

2.3. Reflection property. We say thatD has the reflection prop-

erty if D is a Jordan domain with ∞ ∈ ∂D and if there exists an

L-quasi-isometry of C which maps D onto D∗ and is the identity on

∂D.

2.4. THEOREM. (Ahlfors [2]). If ∞ ∈ ∂D, then D is a qua-

sidisk if and only if it has the reflection property.

3. Local connectivity properties

A set E is locally connected at a point z0 if for each neighborhood

U of z0 there exists a second neighborhood V of z0 such that E ∩ V

3. LOCAL CONNECTIVITY PROPERTIES 25

lies in a component of E ∩ U .

We nwxt give two properties of quasidisks which specialize two cor-

responding properties of a Jordan domain D, namely that ∂D and D

are locally connected at each point of ∂D.

3.1. REMARK. If D is a disk or a half plane, then for each pair

of finite points z1, z2 ∈ ∂D we have

minj=1,2

dia(γj) = |z1 − z2|

where gamma1, γ2 are the components of ∂D \ {z1, z2}

z1

z2γ2

γ1

3.2. Three point property. We say that D has the three point

property if D is a Jordan domain and if there exists a constant d such

that for each pair of finite points z1, z2 ∈ ∂D,

minj=1,2

dia (γj) ≤ d|z1 − z2| ,

where γ1, γ2 are the components of ∂D \ {z1, z2}

This property derives its name from the fact that it implies

|z1 − z3| ≤ d|z1 − z2|


for each point z3 in the component of ∂D \ {z1, z2} with minimum

diameter. It says that ∂D is locally connected at each z0 ∈ ∂D ∩ C

and that when U is a disk about z0 we may choose V as a concentric

disk with

dia(V ) =1

ddia(U) .

3.3. THEOREM. (Ahlfors [2]). D is a quasidisk if and only if it

has the three point property.

3.4. Linear local connectivity property. We say that an arbi-

trary set E ⊂ C is linearly locally connected if there exists a constant

c such that for z0 ∈ C and 0 < r <∞,

(i) points in E ∩B(z0, r) can be joined in E ∩B(z0, cr),

(ii) points in E \B(z0r) can be joined in E \B(z0, r/c).

By “joined” we mean joined by an arc lying within the specified set.

Note that the condition is to hold for z0 ∈ C and not just for z0inE

or E. The two figures below illustrate the situations where (i) and (ii)

come into play.

z0r

cr

z0

rc

r

E

E

4. HYPERBOLIC METRIC PROPERTIES 27

Part (i) of this property says that D is locally connected at each

z0 ∈ C and that again when U is a disk about z0 we may choose V as

a concentric disk with

dia(V ) =1

cdia(U) .

Part (ii) is the counterpart of (i) when z0 = ∞.

3.5. REMARK. If D is a disk or a half plane, then D is linearly

locally connected with c = 1.

3.6. THEOREM. (Gehring [6]). D is a quasidisk if and only if

D is linearly locally connected.

3.7. REMARK. If D is a Jordain domain, then D has the three

point property if and only if ∂D is linearly locally connected.

4. Hyperbolic metric properties

We present here two properties relating euclidean and hyperbolic

geometry in a quasidisk D. The first says that the hyperbolic distance

between two points of D can be estimated in terms of the euclidean

distance between these points and the euclidean distances from these

points to ∂D. The second says that the euclidean length of a hyperbolic

geodesic in D is bounded above by a multiple of the euclidean distance

between is endpoints and by multiple of the distance its midpoints lies

from ∂D.


4.1. Hyperbolic metric en B. For the unit disk B the hyper-

bolic metric is defined by

ρB(z) =2

1 − |z|2.

If z1, z2inB, the hyperbolic distance between z1 and z2 is then defined

to be

hB(z1, z2) = infα

∫

α

ρB(z)|dz| ,

b

b

z1z2

α

where the infimum is taken over all rectifiable arcs α joining z1 and z2

in B. One can show that there is a unique arc α for which

hB(z1, z2) =

∫

α

ρB(z)|dz| ,

namely, that arc α between z1, z2 lying on the circular arc β through

z1, z2 orthogonal to the unit circle.

b

b

z2

z1

α


We shall call α the hyperbolic segment joining z1, z2 and β the hyper-

bolic line containing α. The distance is given explicitly by

hB(z1, z2) = log|1 − z1z2| + |z1 − z2|

|1 − z1z2| − |z1 − z2|.

4.2. Hyperbolic metric in D. More generally, the hyperbolic

metric ca be defined in a symple connected domain D by conformal

mapping. Let g be a conformal mapping of D onto B and define

ρD(z) = ρB(g(z))|g′(z)| .

One can show that this is independent of the choise of g. Next, for

z1, z2 ∈ D, the hyperbolic distance between z1, z2 is defined to be

hD(z1, z2) = infα

∫

α

ρD(z)|dz| ,

where the infimum is taken over all rectifiable arcs α joining z1 and z2

in D. Again there is a nique arc α for which

hD(z1, z2) =

∫

α

ρd(z)|dz| ,

and α will again be called the hyperbolic segment between z1, z2. Ob-

serve that

hD(z1, z2) = hB(g(z1), g(z2))

and that g preserves the class of hyperbolic segments. Finally, Schwarz’s

lemma and the Koebe distortion theorem applied to g give the inequal-

ities

1

2

1

d(z, ∂D)≤ ρD(z) ≤

2

d(z, ∂D)(1)


for all z ∈ D, where d(z, ∂D) denotes the euclidean distance from z to

∂D. Both inequalities are sharp.

If D′ is a simply connected subdomain of D, then Schwarz’s lemma

also implies that

ρD(z) ≤ ρD′(z)

for all z ∈ D′, and hence that

hD(z1, z2) ≤ hD′(z1, z2)

for all z1, z2 ∈ D′.

There is a natural lower bound for the hyperbolic distance:

4.3. LEMMA. (Gehring-Palka [11]). If z1, z2 ∈ D, then

hD(z1, z2) ≥1

4jD(z1, z2) ,

where

jD(z1, z2) = log

(

|z1 − z2|

d(z1, ∂D)+ 1

) (

|z1 − z2|

d(z2, ∂D)+ 1

)

.

Proof. Let α be the hyperbolic segment joining z1, z2 in D. Then

by (1)

hD(z1, z2) =

∫

α

ρD(z)|dz| ≥1

2

∫

α

d(z, ∂D)−1|dz| .

Next for each z ∈ α,

d(z, ∂D) ≤ d(z1, ∂D) + |z − z1|


and thus

hD(z1, z2) ≥1

2

∫

α

d(|z − z1|)

d(z1, ∂D) + |z − z1|=

1

2log

(

|z1 − z2|

d(z1, ∂D)+ 1

)

.

Interchanging the roles of z1 and z2 yields

hD(z1, z2) ≥1

2log

(

|z1 − z2|

d(z2, ∂D)+ 1

)

and adding both inequalities gives the desired conclusion. �

4.4. LEMMA. If D is a disk or a half plane, then

hD(z1, z2) ≤ jD(z1, z2)

for z1, z2 ∈ D.

Proof. It is sufficient to consider the case where D = B. Then

hD(z1, z2) = log|1 − z1z2| + |z1 − z2|

|1 − z1z2| − |z1 − z2|= log

n

d.

Now

n = |1 − z1z2| + |z1 − z2|

= |1 − |z2|2 − z2(z1 − z2)| + |z1 − z2|

≤ (1 − |z2|2) + (1 + |z2|)|z1 − z2|

= (1 − |z2|2)

(

|z1 − z2|

d(z2, ∂D)+ 1

)

,

ans similarly

n ≤ (1 − |z1|[2)

(

|z1 − z2|

d(z1, ∂D)+ 1

)

.


Next

nd = |1 − z1z2|2 − |z1 − z2|

2

= (1 − |z1|2)(1 − |z2|

2) .

Thus

n

d=n2

nd≤

(

|z1 − z2|

d(z1, ∂D)+ 1

) (

|z1 − z2|

d(z2, ∂D)+ 1

)

and the result follows. �

4.5. Hyperbolic bound property. We say that D has the hy-

perbolic bound property if there exist constants c and d such that

hD(z1, z2) ≤ cjD(z1, z2) + d

for z1, z2 ∈ D.

4.6. THEOREM. (Jones [17]). D is a quasidisk if and only if it

has the hyperbolic bound property.

The next property of a disk we wish to generalize for quasidisks

concerns the euclidean geometry of hyperbolic segments.

4.7. REMARK. If D is a disk or half plane, then for each hyper-

bolic segment α in D and for each z ∈ α

ℓ(α) ≤π

2|z1 − z2|

and

minj=1,2

ℓ(αj) ≤π

2d(z, ∂D) ,


where z1, z2 are the endpoints of α and α1, α2 are the components of

α \ {z}. Note that ℓ(α) is the euclidean length of α.

4.8. Hyperbolic segment propery. We say that D has the hy-

perbolic segment property if there exist constants a and b such that for

aech hyperbolic segment α in D and each z ∈ α,

ℓ(α) ≤ a|z1 − z2|

and

minj=1,2

ℓ(αj) ≤ bd(z, ∂D) ,

where z1, z2 are the endpoints of α and α1, α2 are the components of

α \ {z}.

b b

b

z1

z

z2

α1 α2

D

4.9. THEOREM. (Gehring-Osgood [10]). D is a quasidisk if and

only if it has the hyperbolic segment property.

We can considerer arcs other than hyperbolic segments and are thus

led to the notion of a uniform domain.


4.10. Uniform domains. We say that D is uniform is there exist

constants a and b such that each z1, z2 ∈ D can be joined by an arc α

in D with

ℓ(α) ≤ a|z1 − z2|

and

minj=1,2

ℓ(αj) ≤ bd(z, ∂D)

for z ∈ α, where α1, α2 are the components of α \ {z}.

bb

b

z1z2

z

α1

α2

∂D

5. Injectivity properties

If f is a local homeomorphism in C, then f is inyective in C. We

consider here analogues of this statement for two special classes of

local homeomorphisms f defined in a quasidisk D. In the first case f is

analytic with f ′ 6= 0 in D; in the second case f is local quasi-isometry

in D.

5. INJECTIVITY PROPERTIES 35

5.1. Schwarzian derivative. For f analytic and locally injective

in D define the Schwarzian derivative of f to be

Sf =

(

f ′′

f ′

)′

−1

2

(

f ′′

f ′

)2

.

Note that Sf ≡ 0 if and only if f is a Mobius transformation and that

Sg◦f = Sg(f)(f ′)2 + Sf .

In particular, Sg◦f = Sf if and only if g is a Mobius transformation.

There are both necessary and sufficient conditions for injectivity in

terms of the Schwarzian derivative and the hyperbolic metric ρD.

5.2. THEOREM. (Lehto [18]). If f is analytic and inyective in

D, then

|sf | ≤ 3ρ2D .

5.3. THEOREM. (Nehari [22]). If D is a disk or a half plane

and if f is anlytic in D with

|Sf | ≤1

2ρ2

D, f ′ 6= 0

in D, then f is injective.

5.4. Schwarzian derivative property. We say that D has the

Schwarzian derivative property if there exists a constant d > 0 such

that f is injective whenever f is anlytic with

|Sf | ≤ dρ2D , f ′ 6= 0


in D.

5.5. THEOREM. (Ahlfors [2] and Gehring [6]). D is a quasidisk

if and only if it has the Schwarzian derivative property.

There are similar results for the operator f ′′/f ′-

5.6. THEOREM. (Osgood [23]). If f is analytic and injective in

D, then∣

∣

∣

∣

f ′′

f ′

∣

∣

∣

∣

≤ 4ρD .

5.7. THEOREM. (Becker [3]). If D is a disk or a half plane and

if f is analytic in D with

∣

∣

∣

∣

f ′′

f ′

∣

∣

∣

∣

≤1

2ρD , f ′ 6= 0

in D, then f in injective.

5.8. Logarithmic derivative property. We say that D has the

logarithmic derivative property if there exists a constant d > 0 such

that f is injective whenever f is analytic with

∣

∣

∣

∣

f ′′

f ′

∣

∣

∣

∣

≤ dρD , f ′ 6= 0

5.9. THEOREM. (Ahlfors [2] and Gehring [8]) If D is a qua-

sidisk, then D has the logarithmic derivative property. Conversely, if

D∗ is a domain in C and if D and D∗ have this property, then D is a

quasidisk.

5. INJECTIVITY PROPERTIES 37

In view of the result for the Schwarzian derivative one may ask if

the hypothesis on D∗ in necessary in the second part of this theorem.

5.10. Local quasi-isometries. Suppose that f is a mapping of

D into C. We say that f is an L-quasi-isometry if

1

L|z1 − z2| ≤ |f(z1) − f(z2)| ≤ L|z1 − z2|(2)

for all z1, z2 ∈ D. We say that f is a local L-quasi-isometry if for each

M > L, each z0 ∈ D has a neighborhood U in which (2) holds with M

in place of L.

5.11. EXAMPLE. The mapping

f(z) =|z|

LeiL2 arg z

∞ 0D

is a local L-quasi.isometry in D but it is not injective for any L > 1.

5.12. THEOREM. (John [16]). If D is a disk or a half plane and

if f is a local L-quasi-isometry in D with L ≤ 21

4 , then f is injective.

Proof. It suffices to consider the case D = B. Suppose that f is

not injective. Then since f is a local homeomorphism we can choose a

disk U with U ⊂ B and points z,z2 ∈ ∂U such that f is injective in U

and

f(z1) = f(z2) .


Let α be the circular arc orthogonal to ∂U at z1, z2 and let E be the

component of U \ α whose image E ′ is enclosed by α′ = f(α). Then

ℓ(α′) ≤ Lℓ(α)

because f is local L-quasi-isometry.

z1

z2U

fαE

E ′U ′

α′

Next, the fact that f is injective in U implies that f−1 is a local L-

quasi-isometry in U ′ = f(U) and hence that

m(E) ≤ L2m(E ′) .

By elementary geometry and the isomperimetric inequality we conclude

that

ℓ(α)2

2π< m(E) ≤ L2m(E ′) ≤ L2 ℓ(α

′)2

4π≤ L2L

2ℓ(α)2

4π,

or that L4 > 2, a contradiction. �

5.13. Rigid domains. We say that D is rigid if there exists a

constant d > 1 such that f is injective whenever f is a local L-quasi-

isometry in D with L ≤ d.

6. EXTENSION PROPERTIES 39

5.14. THEOREM. (Gehring [7] and Martio-Sarvas [20]). D is a

quasidisk if and only if it is rigid

6. Extension properties

6.1. Class BMO(D). A real valued, locally integrable function u

is said to be of bounded mean oscillation in D, written u ∈ BMO(D),

if

‖u‖∗ = supB

1

m(B)

∫∫

B

|u− uB|dxdy <∞ ,

where the suprem is taken over all disks B with B ⊂ D, and where

uB =1

m(B)

∫∫

B

dxdy .

6.2. REMARK. If v ∈ BMO(D) then u = v|D ∈ BMO(D) with

‖u‖∗ ≤ ‖v‖∗.

6.3. THEOREM. (Reimann-Rychener [25]). If D is a disk or a

half plane and if u ∈ BMO(D), then u has an extension v ∈ BMO(D)

with

‖v‖∗ ≤ a‖u‖∗ ,

where a is an absolute constant.

Proof. Set

v(t) =

u(z) z ∈ D

(u ◦ ϕ)(z) z ∈ D∗ ∩ C


where ϕ denotes reflection in ∂D. An elementary but technical argu-

ment shows that v ∈ BMO(C) with

‖v‖∗ ≤ a‖u‖∗ ,

where a is an absolute constant. For the details we refer to [25, Chapter

1]. �

6.4. BMO extension property. We say that D has the BMO

extension property if there exists a constant a such that each u ∈

BMO(D) has an extension v ∈ BMO(C) with

‖v‖∗ ≤ a‖u‖∗ .

6.5. THEOREM. (Jones [17]). D is a quasidisk if and only if D

has the BMO extension property.

6.6. Class L21(D). We say that a function u ∈ L2

1(D =) if u ∈

L2

loc(D) and u has distributional derivatives v1, v2 ∈ L2(D). If this is

the case, set

E(u) =

∫∫

D

(v21 + v2

2)dxdy .

6.7. REMARK. If D is a disk or a half plane and if u ∈ L21(D),

then u has an extension v ∈ L21(C) with

E(v) ≤ 2E(u) .

6. EXTENSION PROPERTIES 41

6.8. L21 extension property. We say thatD has the L2

1 extension

property if there exists a constant a such that each u ∈ L21(D) has an

extension v ∈ L(C) with

E(v) ≤ aE(u) .

6.9. THEOREM. (Gol’dstein-Vodop’janov [13]). D is quasidisk

if and only if D has the L21 extension property.

6.10. Class Lip(k). Suppose 0 < k ≤ 1. We say that f ∈ Lip(k)

in D if f is analytic in D with

‖f‖Lip(k) = sup|f(z1) − f(z2)|

|z1 − z2|k<∞ ,

where the supremum is taken over all z1, z2 ∈ D.

6.11. REMARK. If f ∈ Lip(k) in D, then

|f ′(z)| ≤ ‖f‖Lip(k)d(z, ∂D)k−1 .

6.12. THEOREm. (hardy-Littlewood [14]). If D is a disk or a

half plane and if f is analytic with

|f ′(z)| ≤ ad(z, ∂D)k−1

in D, them f ∈ Lip(k) with

‖f‖Lip(k) ≤ca

k

where c is an absolute constant.


6.13. Hardy-Littlewood property. We say thatD has the Hardy-

Littlewood property if for 0 < k ≤ 1 there exists a constant c such that

f ∈ Lip(k) with

‖f‖Lip(k) ≤ca

k

whenever f is analytic with

|f ′(z)| ≤ ad(z.∂D)k−1

in D.

6.14. THEOREM. (Gehring-Martio [9]). If D is a quasidisk,

then D has the Hardy-Littlewood property. If D∗ is a domain in C and

if D and D∗ have this property, then D is a quasidisk.

7. Homogeneity property

7.1. Homegeneous set. We say that a set E ⊂ C is homogeneous

with respect to a family of mappings G if for each z1, z2 ∈ B there exists

a g ∈ G with

g(E) = E , g(z1) = z2 .

7.2. REMARK. If D is a disk or a half plane, then ∂D and D are

homogeneous with respect to Mob, the family of Mobius transforma-

tions in C.

7.3. Class QC(K). The class QC(K) is the family of all K-qc

mappings g : C → C. Note that QC(1) =Mob.

8. MISCELLANEOUS PROPERTIES 43

7.4. THEOREM. (Erkama [4]). D is a quasidisk if and only if

∂D is a Jordan curve which is homogeneous with respect to QC(K) for

same K.

7.5. THEOREM. (Sarvas [26]). D is a quasidisk if and only if

D is a Jordan domain which is homogeneous with respect to QC(K)

for same K.

8. Miscellaneous properties

8.1. Limit set of a group. Suppose that G is a group of home-

omorphisms g : C → C. We say that z0 is in the limit set L(G) of G if

there exist distinct gj ∈ G and a z1 ∈ C such that

z0 = limj→∞

gj(z1) .

8.2. REMARK. If D is a disk or a half plane then there exists a

finitely generated group G of mappings in Mob with ∂D as its limi set.

8.3. THEOREM. (Maskit [21], Sullivan [28], Tukia [29]). D is a

quasidisk if and only if ∂D is a Jordan curve which is the limit set of

a finitely generated group G of mappings in QC(K) for same K.

8.4. REMARK. If D is a disk or a half plane and if u and u⋆ are

harmonic in D and D⋆ with equal and continuous boundary values,

then∫∫

D

|∇u|2dxdy =

∫∫

D∗

|∇u⋆|2dxdy .


8.5. THEOREM. (Ahlfors [1] and Springer [27]). D is a qua-

sidisk if and only if D is a Jordan domain and there exists a constant

K such that

1

K

∫∫

D

|∇u|2dxdy ≤

∫∫

D⋆

|∇u⋆|2dxdy ≤ K

∫∫

D

|∇u|2dxdy

for each pair of functions u and u⋆ harmonic in D and D⋆ with equal

continuous boundary values.

8.6. REMARK. If D is a disk or a half plane, then G = R3 \ D

can be mapped 2-quasiconformally to the unit ball B3 in R3.

8.7. THEOREM. (Gehring [5]). D is a quasidisk if and only if

G = R3 \D can be mapped quasiconformally onto B3.

CHAPTER III

SOME PROOFS OF THESE PROPERTIES

1. Table of implications

The sollowing diagram indicates some of the routes one can take in

establishing the equivalence of the properties introduced in Chapter 2.

Quasidisk

L21 extension property

Rigid domain

Three point property

Linear local connectivity

Hyperbolic boundproperty

BMO extension property Dirichket integral property

Schwarzian derivativeproperty

Uniform domain

Hyperbolic segmentproperty

III.5

III.8III.11

III.10

III.9

III.2III.6

III.3III.4

III.7

This chapter will be devoted to establishing several of these implica-

tions.

45

46 III. SOME PROOFS OF THESE PROPERTIES

2. Quasidisks have the hyperbolic segment property

2.1. We begin with the following useful observation.

LEMMA. Suppose that D is a disk or half plane, that D′ is a K-

quasidisk and that f : D → D′ is conformal. Then f has an extension

which is K2-qc in C.

Proof. By hypothesis there exists a K-qc mapping g : C → C

with g(D′) = B. Then h = g ◦f is a K-qc mapping of D onto B which

extends as a homeomorphism of D onto B. Let ϕ and ψ denote the

reflections in ∂D, ∂B, respectively, and define

h(z) = (ψ ◦ g ◦ f ◦ ϕ−1)(z)

for z ∈ D⋆. Then h is K-qc in D, in D⋆, and hence in C. If we now set

f(z) = (g−1 ◦ h)(z)

for z ∈ D⋆, then f is K2-qc in D⋆, in D, and hence in C. �

2.2. The next lemma will enable us to estimate the arclength of a

hyperbolic segment. The proof is based on an argument due to Jerison

and Kenig [15].

LEMMA. Suppose that D is a K-quasidisk with ∞ ∈ ∂D and

that f : H → D is a homeomorphism which is conformal in H with

f(∞) = ∞. Then

∫ y

0

|f ′(it)|dt ≤ c1d(f(iy), ∂D)(1)

2. QUASIDISKS HAVE THE HYPERBOLIC SEGMENT PROPERTY 47

for 0 < y <∞, where c1 = c1(K).

Proof. By Lemma 2.1, f has an extension which is K2-qc in C.

By a change of variables we may assume that f(0) = 0. Fix 0 < y0 <∞

and choose a sequence {yj} so that

0 < yj+1 < yj ≤ y0

and

|f(iyj)| = c−j |f(iy0)|

for j = 1, 2, . . . , where c = e8K2

.

b

b

b

b

b

b

b

b

0

iyj+1

iy

iyj

f

0

D

H

Fix a subscript j. Then for yj+1 ≤ y ≤ yj,

d(f(iy), ∂D) ≤ |f(iy) − f(0)|

≤ c|f(iyj) − f(0)| = c−j+1|f(iy0)|

by Theorem I.2.6, while

|f ′(iy)| ≤ 4d(f(iy), ∂D)

d(iy, ∂H)≤ 4c−j+1 |f(iy0)|

y

by the Koebe distortion theorem. Thus

∫ yj

yj+1

|f ′(iy)|dy ≤ 4c−j+1|f(iy0)| logyj

yj+1.(2)


It remains to find an upper bound for logyj

yj+1

. For this, let k denote

the smallest positive integer for which c ≤ 2k. Then

|f(iyj) − f(0)| ≤ 2k|f(iyj+1 − f(0)| ,

and hence

yj = |iyj − 0| ≤ (2c)k+1|iyj+1 − 0| = (2c)k+1yj+1

by Corollary I.2.7. This gives

logyj

yj+1≤ (k + 1) log 2c = c2 ,

and which (2) we obtain

∫ y0

0

|f ′(iy)|dy =

∞∑

j=0

∫ yj

yj+1

|f ′(iy)|dy

≤ 4c2|f(iy0)|

∞∑

j=0

c−j+1

= c3|f(iy0)| .

Finally, if x ∈ ∂H , then

|f(iy0)| ≤ c|f(iy0) − f(x)|

by Theorem I.2.6 again and thus

|f(iy0)| ≤ cd(f(iy0), ∂D) .

This complete the proof of (1) with c1 = cc3, a constant depending

only on K. �


2.3. THEOREM. (Gehrinh-Osgood [10]). If D is a K-quasidisk,

then D has the hyperbolic segment property with a = a(K) and b =

b(K).

Proof. Fix a hyperbolic segment α in D with endpoints z1, z2. We

want to exhibit constants a and b such that

ℓ(α) ≤ a|z1 − z2|(3)

and

minj=1,2

ℓ(αj) ≤ bd(z, ∂D)(4)

for each z ∈ α, where α1, α2 are the components of α\{z}. By Lemma

2.1 there exists a K2-qc mapping f : C → C which maps D confor-

mally onto B. By employing an auxiliary Mobius transformation of

the disk we may assume that f(z1) and f(z2) are real. Let B′ denote

the open disk in B with f(z1) and f(z2) as diametral points and let

D′ = f−1(B′). Then D′ is a bounded K2-quasidisk and α is a hyper-

bolic line in D′.

b

b

b

f

B

B′

D

z1

z2

α


Since

d(z, ∂D′) ≤ d(z, ∂D)

for z ∈ α, we see that it is sufficient to establish (3) and (4) for the

case where α is a hyperbolic line in D and D is bounded.

Assume then that this is the case and let D′ and α′ be the images

of D and α under

w = g(z) =z − z1z − z2

.

Let f map H conformally onto D′. Again f extends to a homeomor-

phism of H onto D′and we may assume that f(0) = 0, f(∞) = ∞.

bb

b

b b

b

fg

hz1

z2α

D

0

w

α′

D′

∞

iy

∞

H

Now D′ is a K-quasidisk and α′ is the image of the positive imag-

inary axis under f . Hence if w ∈ α′, then w = f(iy) for some

0 < y <∞, and

s =

∫ y

0

|f ′(it)|dt ≤ c1d(f(iy), ∂D′)(5)

= c1d(w, ∂D′)


by Lemma 2.2, where s is the arclength of α′ between 0 and w. Let

h = g−1. Then

ℓ(α) =

∫

α′

|h′(w)||dw| = |z1 − z2|

∫

α′

|dw|

|w − 1|2

= |z1 − z2|

∫ ∞

0

ds

|w(s) − 1|2,

where w = w(s) is the arclength representation for α′. If we let

s0 =c1

c1 + 1,

then for 0 < s ≤ s0,

|w(s) − 1| ≥ 1 − |w(s)| ≥ 1 − s ≥1

c1 + 1,

while for s0 ≤ s <∞,

|w(s) − 1| ≥ d(w(s), ∂D′) ≥s

c1

by (5) since D ⊂ C implies that

1 = g(∞) /∈ D′ .

Thus

∫ ∞

0

ds

|w(s) − 1|2≤

∫ s0

0

(c1 + 1)2ds+

∫ ∞

s0

c21s2ds = 2c1(c1 + 1)

and we obtain (3) with a = 2c1(c1 + 1).

Finally sinceD is bounded, we can find aK2-qc mapping f : C → C

which maps D conformally onto B with f(∞) = ∞. Fix z ∈ α, choose

z0 ∈ ∂D so that

|z − z0| = d(z, ∂D)


and let w1, w2, w and w0 be the images of z1, z2, z and z0 under f .

b

b b

b

b

b

b

bz1

zz2

z0 w1w

w2

w0

α

D B

f

Since f(α) is hyperbolic line in B it is easy to check that

minj=1,2

|w − wj| ≤ 2d(w, ∂B) ≤ 2|w − w0|

and hence

minj=1,2

|zj − z| ≤ 4c2|z − z0| = 4c2d(z0, ∂D)

by Corollary I.2.7, where c = e8K2

, If αj is the component of α \ {z}

which has zj as an endpoints, then

ℓ(αj) ≤ a|zj − z|

and we obtain (4) with b = 4c2a. �

3. Hyperbolic segment property implies D is uniform

Fix z1, z2 ∈ D. Then there exists a unique hyperbolic segment α in

D with z1, z2 as its endpoints. The hypothesis is exactly the condition

that this arc satisfies the properties required to show that D is uniform.

4. UNIFORM DOMAINS ARE LINEARLY LOCALLY CONNECTED 53

4. Uniform domains are linearly locally connected

4.1. THEOREM. If D is uniform, then D is linearly locally con-

nected with c = 2 max(a, b) + 1.

Proof. Fix z0 ∈ C, 0 < r < ∞, and suppose that z1, z2 ∈ D ∩

B(z0, r). We must show that z1 and z2 can be joined in D ∩B(z0, cr).

Since D is uniform there exists an arc α joining z1 and z2 in D with

ℓ(α) ≤ a|z1 − z2| ≤ 2ar .

b

b

b

bz

α z1

z2

z0r

If z ∈ α, then

|z − z0| ≤ |z − z1| + |z1 − z0| ≤ ℓ(α) + r

≤ (2a+ 1)r ≤ cr ,

and hence α joins z1 and z2 in D ∩ B(z0, cr) as required.

Next suppose that z1, z2 ∈ D \ B(z0, r). Again we obtain an arc β

joining z1 and z2 in D with

minj=1,2

ℓ(βj) ≤ bd(z, ∂D)


for each z ∈ β, where β1, β2 are the components of β \ {z}. Suppose

that β does not join z1 and z2 in D \B(

z0,rc

)

.

b

b

b

b

z1

z2

z0

z

r

rc

Then there exists a point z ∈ β with

|z − z0| <r

c≤

r

2b+ 1,

and for j = 1, 2 we have

ℓ(βj) ≥ |zj − z|

≥ |zj − z0| − |z − z0| ≥2b

2b+ 1r .

Thus

d(z, ∂D) ≥1

bminj=1,2

ℓ(βj) ≥2

2b+ 1r > |z − z0| +

r

c

and hence B(

z0,rc

)

⊂ D. But this implies that D \ B(

z0,rc

)

is con-

nected and hence that z1, z2 can be joined by an arc in this set. Thus

z1, z2 can always be joined in D \B(

z0,rc

)

. �

5. LINEAR LOCAL CONNECTIVITY IMPLIES THE THREE POINT PROPERTY55

4.2. REMARK. The above proof actually used only the fact that

for each z1, z2 ∈ D there exist arcs α, β joining these points in D such

that

dia(α) ≤ a|z1 − z2|

and

minj=1,2

|z − zj | ≤ bd(z, ∂D)

for each z ∈ β. This is a substantially weaker hypothesis than requiring

that D be uniform.

5. Linear local connectivity implies the three point property

5.1. THEOREM. (Gehring [6]). If D is linearly locally con-

nected, then D has the three point property with d = c2.

Proof. Let z0 ∈ ∂D. With each neighborhood U of z0 we associate

a second neighborhood V of z0 as follows. If zo ∈ C choose 0 < r <∞

that

B(z0, cr) ⊂ U

and let V = B(z0, r). If z0 = ∞, choose 0 < r <∞ so that

C \B(

z0,r

c

)

⊂ U

and let V = C\B(z0, r). In each case the fact that D is linearly locally

connected implies that D ∩ V lies in a component of D ∩ U . Thus D

is locally connected at each point of its boundary, and hence D is a

Jordan domain by a converse of the Jordan curve theorem.


Now fix finite points z1, z2 ∈ ∂D. We shall show that

minj=1,2

dia(γj) ≤ d|z1 − z2| , d = c2 ,(1)

where γ1, γ2 are the components of ∂D \ {z1, z2}. For this, set

z0 =1

2(z1 + z2) , r =

1

2|z1 − z2|

and suppose that (1) does not hold. Then there must exist t with

r < t <∞, and finite points w1, w2 such that

wj ∈ γj \B(z0, c2t)

for j = 1, 2.

b

b

b

b

b

z1

w2

w1

z2z0D

tc2t

s

Choose s with r < s < t. Since

z1, z2 ∈ ∂D ∩B(z0, s)

we can find for j = 1, 2 an endout αj of D which joins zj to a point

z′j ∈ D and which lies in D ∩B(z0, s). Next, since D is linearly locally

connected, we can find an arc α3 joining z′1 and z′2 in D ∩ B(z0, cs).

6. THREE POINT PROPERTY IMPLIES D IS A QUASIDISK 57

Then α1 ∪ α2 ∪ α3 contains a crosscut α of D which joins z1, z2 in

B(z0, cs).

bb

b

b

b

z0

z1z′1 z2

z′2α3

cs

The same argument applied to w1, w2 yields a crosscut β of D which

joins w1, w2 in C \B(z0, ct). But now since s < t,

α ∩ β ⊂ B(z0, cs) \B(z0, ct) = ∅ ,

while the fact that z1, z2 separate w1, w2 in ∂D implies that

α ∩ β 6= ∅ .

This contradiction establishes (1) and completes the proof. �

6. Three point property implies D is a quasidisk

6.1. We first require an estimate for the modulus of a path family.

LEMMA. Suppose that D has the three point property, that α1 and

α2 are disjoint arcs in ∂D and that Γ and Γ⋆ are the families of arcs

which join α1 and α2 in D and D⋆m respectively. If modΓ = 1, then

modΓ⋆ ≤ c ,


where c = π(2d2e2π + 1)2.

Proof. We employ a modification, suggested by J. Sarvas, of a

well-known argument due to Ahlfors [2]. Choose z1 ∈ α1, z2 ∈ α2 so

that

|z1 − z2| = d(α1, α2) = r .

Since α1 and α2 are disjoint, one is bounded and by relabeling we may

assume that

s = dia(α1) ≤ dia(α2) , s <∞ .

We shall establish the upper bound

s ≤ 2d2e2πr .(1)

For this we may clearly assume that

s

2d> dr ,

sinceotherwise (1) would follow trivially. Because D has the rhree point

property,

minj=1,2

dia(γj) ≤ d|z1 − z2| = dr ,

where γ1, γ2 are the components of ∂D \ {z1, z2}. Again by relabeling

we may assume that

dia(γ1) ≤ dr .

Let β1, β2 denote the components of ∂D \ (α1 ∪ α2), labeled so that

βj ⊂ γj. Then

β2 ⊂ γ1 ⊂ B(z1, dr) .


Choose z0 ∈ β2 and let δ1, δ2 denote the components of ∂D \ {z0, z1}

labeled so that α2 ⊂ δ2.

b

b

b

b

b b

b

z0z1

z2

β2

β1

α1

α2

Then again by the three point property,

minj=1,2

dia(δj) ≤ d|z1 − z0| ,

while obviously

dia(δ2) ≥ dia(α2) ≥ s .

Choose w1, w2 ∈ α1 so that

|w1 − w2| = dia(α1) = s ,


b

bb

b

b

b

b

b

z0

w1

z1w2

z2

β2 α1

β1

α2

Then w1, w2 ∈ γ1 ∪ δ1, and the fact that

dia(γ1) ≤ dr < s

implies that not both of these points can lie in γ1. If one, say w1 lies

in γ1, then

dia(δ1) ≥ |w2 − w1| ≥ |w1 − w2| − |z1 − w1|

≥ s− dia(γ1) ≥ s− dr >s

2.

If both lie in δ1, then

dia(δ1) ≥ |w1 − w2| = s .

Thus

s

2< min

j=1,2dia(δj) ≤ d|z1 − z2| ,

and we conclude that

β2 ∩B(

z1,s

2d

)

= ∅ .


In particular we see that for dr < t < s/2d, the circle |z − z1| = t

separates β1 and β2, and hence must contain an arc γ which joins α1

and α2 in D, i.e., γ ∈ Γ.

b

b

b

b

b

b

z1

z2

γβ2

α2

α1

t

Now by Lemma I.2.3,

1 = modΓ ≥1

2πlog

s

2d2r

from which (1) follows.

Finally by (1) and Lemma I.2.4,

modΓ⋆ ≤ π(s

r+ 1

)2

≤ π(2d2e2π + 1)2

as desired. �


6.2. THEOREM. (Ahlfors [2]). If D satisfies the three point

property, then D is a K-quasidisk, where K = K(d).

Proof. Choose conformal mappings h1, h2 which map D,D⋆ onto

H,H⋆, respectively. Since D and D⋆ are Jordan domains, h1 and h2

extend to homeomorphisms of D,D⋆

onto H,H⋆

and by composing one

of these with a Mobius transformation we may assume that h−11 (∞) =

h−12 (∞). Next, for x ∈ ∂H let

ϕ(x) = (h2 ◦ h−11 )(x) .

Then ϕ is an increasing function of x with ϕ(∞) = ∞. Fix x ∈ ∂H ,

t > 0, and let Γ and Γ⋆ be the curve families indicated in the figure

below.

∞ x− t x x+ t

H

Γ

∞

ϕ(x−t) ϕ(x) ϕ(x+t)

H⋆

Γ⋆

b bb b b b

b

b

b

bΓ⋆

1D⋆

Γ1

D

h−11

h1


By Corollary I.2.2 and Lemma 6.1,

modΓ1 = modΓ = 1

and

m

(

ϕ(x+ t) − ϕ(x)

ϕ(x) − ϕ(x− t)

)

≤ m−1(c) = b .

Reversing the roles of the solid and hatched intervals in the above figure

yields with the above

1

b≤ϕ(x+ t) − ϕ(x)

ϕ(x) − ϕ(x− t)≤ b .

Now define

g(z) =1

2

∫ 1

0

(ϕ(x+ ty) +ϕ(x− ty))dt+1

2

∫ 1

0

(ϕ(x+ ty)−ϕ(x− ty))dt

for z = x+ iy ∈ H \{∞} and g(∞) = ∞. Then g is a homeomorphism

of H onto H which is K-qc in H with g = ϕ on ∂H , where

K = 8b(b+ 1)2 = K(d) .

(see II.6.5. in [19] for this calculation.) Therefore

f(z) =

(g ◦ h1)(z) , z ∈ D

h2(z) , z ∈ D⋆

is a K-qc mapping of C onto itself which maps D onto H . �

6.3. REMARK. With the proof of Theorem 6.2 we have com-

pleted the innermost loop in the table of implications.


7. Uniform domains have the Schwarzian derivative property

The arguments in this section are based on the ideas of Martio and

Sarvas in [20]. We begin with the following lemma on the size of f ′′/f ′.

7.1. LEMMA. Suppose that z1, z2 ∈ D, that α is a rectifiable

open arc joining z1, z2 in D with midpoint z0 and that 0 < c < 1. If f

is meromorphic in D and if

∣

∣

∣

∣

f ′′

f ′(z)

∣

∣

∣

∣

≤c

min(s, ℓ(α) − s), f ′(z) 6= 0(1)

for z ∈ α, where s is the arclength of α from z1 to z, then

∣

∣

∣

∣

f(z1) − f(z2)

f ′(z0)− (z1 − z2)

∣

∣

∣

∣

≤c

1 − cℓ(α) .

Proof. By the triangle inequality it is sufficient to prove that

∣

∣

∣

∣

f(zj) − f(z0)

f ′(z0)− (zj − z0)

∣

∣

∣

∣

≤1

2

c

1 − cℓ(α)(2)

for j = 1, 2; by symmetry we need only consider the case when j = 1.

Now (1) implies that f ′′/f ′ is finite, and hence that f is analytic,

at each z ∈ α. For such z let

g(z) =

∫ z

z0

f ′′

f ′(ζ)dζ ,

b

b

b

b

z1

zz0

z2α

7. UNIFORM DOMAINS HAVE THE SCHWARZIAN DERIVATIVE PROPERTY65

where the integral is taken along α. Then

eg(z) =f ′(z)

f ′(z0)

and we have

f(z) − f(z0)

f ′(z0)− (z − z0) =

∫ z

z0

(eg(ζ) − 1)dζ

for z ∈ α. If z ∈ α is between z1 and z0, the from (1)

∣

∣

∣

∣

f ′′

f ′(z)

∣

∣

∣

∣

≤c

s,

whence

|g(z)| ≤

∫ z

z0

∣

∣

∣

∣

f ′′

f ′(ζ)

∣

∣

∣

∣

|dζ | ≤

∫ 1

2ℓ(α)

s

c

σdσ = c log

ℓ(α)

2s.

Therefore

|eg(z) − 1| ≤ e|g(z)| − 1 ≤

(

ℓ(α)

2s

)c

− 1 ,

and we obtain

∣

∣

∣

∣

f(z) − f(z0)

f ′(z0)− (z − z0)

∣

∣

∣

∣

≤

∫ z

z0

|eg(ζ) − 1||dζ

≤

∫ 1

2ℓ(α)

0

((

ℓ(α)

2σ

)c

− 1

)

dσ =1

2

c

1 − cℓ(α)

for z ∈ α between z1 and z0. This inequality then implies that f is

bounded near z1 and hence analytic at z1. Thus we can let z → z1

along α to get (2). �

With this lemma at hand we can easily deduce our first injectivity

result.


7.2. THEOREM. (Martio-Sarvas [20]). If D is uniform, then D

has the logarithmic derivative property for all d satisfying

0 < d <1

2(a+ 1)b.

Proof. Suppose that f is analytic with

∣

∣

∣

∣

f ′′

f ′

∣

∣

∣

∣

≤ dρD

in D, and fix z1, z2 ∈ D. Because D is uniform we can find an open

arc α joining z1, z2 in D such that

ℓ(α) ≤ a|z1 − z2|

and

minj=1,2

ℓ(αj) ≤ bd(z, ∂D)(3)

for all z ∈ α, where α1, α2 are the components of α \ {z}. For z ∈ α

let s denote the arclength of α between z1 and z. Then (3) becomes

d(z, ∂D) ≥1

bmin(s, ℓ(α)s) ,

and hence

∣

∣

∣

∣

f ′′

f ′(z)

∣

∣

∣

∣

≤ dρD(z) ≤2d

d(z, ∂D)≤

2bd

min(s, ℓ(α) − s)

for z ∈ α. But now, with z0 as the midpoint of α,

∣

∣

∣

∣

f(z1) − f(z2)

f ′(z0)− (z1 − z2)

∣

∣

∣

∣

≤2bd

1 − 2bdℓ(α)

≤2abd

1 − 2bd|z1 − z2| < |z1 − z2|


by Lemma 7.1 and our choise of d. Therefore f(z1) 6= f(z2) and f is

inyective in D. �

We shall use a similar argument to deduce the Schwarzian derivative

property. For this we require the following simple lemma.

7.3. LEMMA. If u and v are absolutely continuous in each closed

subinterval of [a, b), if u(a) = 0 and if

u′ ≤ uv

a.e. in [a, b), then u ≤ 0 in [a, b).

Proof. Let

w = u exp

(

−

∫ t

a

v(s)ds

)

.

Then w is absolutely continuous if each closed subinterval of [1, b),

w′ = exp

(

−

∫ t

a

v(s)ds

)

(u′ − uv) ≤ 0

a.e. in [a, b) and hence

w ≤ w(a) = 0 , u = w exp

(∫ t

a

v(s)ds

)

≤ 0

in [a, b). �

Corresponding to Lemma 7.1 we have

7.4. LEMMA. Suppose that z1, z2 ∈ D that α is a rectifiable open

arc joining z1, z2 in D with mid point z0, and that 0 < c < 12. If f is

meromorphic in D with f ′′(z0) = 0 and if

|Sf(z)| ≤c

min(s, ℓ(α) − s)2, f ′(z) 6= 0


for z ∈ α, where s is the arclength of α from z1 to z, then

∣

∣

∣

∣

f(z1) − f(z2)

f ′(z0)− (z1 − z2)

∣

∣

∣

∣

≤2c

1 − 2cℓ(α) .(4)

Proof. By Lemma 7.1 it is sufficient to show that

∣

∣

∣

∣

f ′′

f ′(z)

∣

∣

∣

∣

≤2c

min(s, ℓ(α) − s)(5)

bb

b

b

z1z0

z

z2α

for each z ∈ α. By symmetry we need only prove (5) for the case where

z lies between z0 and z2, i.e., for

s ∈ [a, ℓ(α)) , a =1

2ℓ(α) .

Since f ′′(z0) = 0, f is analytic at z0, and so there exists a t ∈ (a, ℓ(α))

such that f is analytic at z(s) for s ∈ [a, t), where z(s) is the arclength

representation of α. Let b denote the supremum of all such numbers t

and for s ∈ [a, b) let

ϕ(s) =2c

ℓ(α) − s, ψ(s) =

∣

∣

∣

∣

f ′′

f ′(z(s))

∣

∣

∣

∣

+ ϕ(a) .

Then ϕ and ψ are sbsolutely continuous in each closed subinterval of

[a, b) with

ϕ′(s) −1

2ϕ(s)2 =

2c(1 − c)

(ℓ(α) − 2)2≥

c

(ℓ(s) − s)2


and

ψ′(s) −1

2ψ(s)2 <

∣

∣

∣

∣

(

f ′′

f ′(z)

)′

z′(s)

∣

∣

∣

∣

−1

2

∣

∣

∣

∣

f ′′

f ′(z)

∣

∣

∣

∣

2

≤

∣

∣

∣

∣

∣

(

f ′′

f ′(z)

)′

−1

2

(

f ′′

f ′(z)

)2∣

∣

∣

∣

∣

= |Sf(z)| ≤c

(ℓ(α) − s)2

a.e. in [a, b). Thus

ψ′ − ϕ′ <1

2(ψ2 − ϕ2) = (ψ − ϕ)

(

ψ + ϕ

2

)

a.e. in [a, b) and we can apply Lemma 7.3 with u = ψ − ϕ and v =

12(ψ + ϕ) to conclude that

∣

∣

∣

∣

f ′′

f ′(z)

∣

∣

∣

∣

< ψ(s) ≤ ϕ(s) =2c

min(s, ℓ(α) − s)(6)

for s ∈ [a, b). Finally we claim that b = ℓ(α). For otherwise, (6) would

imply that f ′′/f ′ is bounded near z(b) and hence that f is analytic at

z(b). In particular, we would then get a t ∈ (b, ℓ(α)) such that f is

analytic at z(s) for s ∈ [a, t) contradicting the way b was chosen. This

completes the proof of (5) and hence of (4). �

We can now show that uniforme domains have the Schwarzian de-

rivative property.

7.5. THEOREM. (Martio-Sarvas [20]). If D is uniform, then D

has the Schwarzian derivative property for all d satisfying

0 < d <1

8(a + 1)b2.


Proof. Suppose that f is analytic in D with

|Sf | ≤ dρ2D , f ′ 6= 0 ,

and fix z1, z2 ∈ D. Because D is uniform we get an open arc α joining

z1, z2 in D such that

ℓ(α) ≤ a|z1 − z2|

and

minj=1,2

ℓ(αj) ≤ bd(z, ∂D)(7)

for each z ∈ α, where α1, α2 are the components of α \ {z}. Then (7)

implies that

|Sf(z)| ≤ dρD(z)2 ≤4d

d(z, ∂D)≤

4b2d

min(s, ℓ(α) − s)2

for z ∈ α, where s is the arclength of α from z1 to z. Let z0 be the nid

point of α. If f ′′(z0) = 0 then

∣

∣

∣

∣

f(z1) − f(z2)

f ′(z0)− (z1 − z2)

∣

∣

∣

∣

≤8b2d

| − 8b2dℓ(α)

≤8ab2d

1 − 8b2d|z1 − z2|

< |z1 − z2|

by Lemma 7.4 and our choice of d; thus f(z1) 6= f(z2). If f ′′(z0) 6= 0

then we choose a Mobius transformation g so that h′′(z0) = 0 where

h = g ◦ f . Then since Sh = Sf , we can apply the above argument to

h to conclude that h(z1) 6= h(z2) whence f(z1) 6= f(z2), and we have

shown that f is injective in D. �

8. SCHWARZIAN DERIVATIVE PROPERTY IMPLIES D IS LINEARLY LOCALLY CONNECTED71

8. Schwarzian derivative property implies D is linearly

locally connected

8.1. The proof of this implication requires several preliminary lem-

mas.

LEMMA. Suppose that c > 1 and that there exist two points in

D ∩B(z0, r) which cannot be joined in D ∩B(z0, cr). Then there exist

points z1, z2 ∈ D and w1, w2 /∈ D such that

|h(z1) − h(z2) − 2πi| ≤4

c− 1,(1)

where

h(z) = logz − w1

z − w2

.

Proof. Let z′1, z′2 be two points in D ∩ B(z0, r) which cannot be

joined in D ∩ B(z0, cr), and let α′ be the segment and β ′ a rectifiable

arc which join z′1, z′2 in C and D, respectively.

b b b b

z′1 α′

z1 z2

z′2

β ′

D1D2

We may choose β ′ it intersects α′ in a finite set of points E. Then

in E there exist two adjacent points z1, z2 which cannot be joined in

D ∩ B(z0, cr). Let α and β denote the parts of α′ and β ′ between z1

and z2. Then γ = α∪β is a Jordan curve and we denote by D1 and D2,


respectively, the bounded and unbounded components of C\γ. The fact

that z1, z2 cannot be joined in D ∩ B(z0, cr) and a simple topological

argument based on Kerekjarto’s theorem imply the existence of points

w1, w2 such that

wj ∈ (C \D) ∩ ∂B(z0, cr) ∩Dj

for j = 1, 2. (For the details see [6].)

bb

b

b

w1

w2

z1z2

α

β D1

Since D is simple connected, we can choose an analytic branch of

h(z) = logz − w1

z − w2

,

8. SCHWARZIAN DERIVATIVE PROPERTY IMPLIES D IS LINEARLY LOCALLY CONNECTED73

and

h(z1) − h(z2) =

∫

β

h′(z)dz =

∫

γ

h′(z)dz −

∫

α

h′(z)dz

= 2πi(n(γ, w1) − n(γ, w2)) −

∫

α

dz

z − w1

+

∫

α

dz

z − w2

,

where n(γ, wj) is the winding number of γ with respect to wj . Now

n(γ, w1) = n = ±1 , n(γ, w2) = 0 ,

and hence

|h(z1) − h(z2) − 2πni| ≤

∫

α

|dz|

|z − w1|+

∫

α

|dz|

|z − w2|.(2)

Since α ⊂ B(z0, r) and wj ∈ ∂B(z0, cr),

∫

α

|dz|

|z − wj|≤

ℓ(α)

(c− 1)r≤

2

c− 1

and (1) follows from (2) if n = 1. If n = −1, the result follows by

interchanging z1 and z2. �

8.2. LEMMA. Suppose that c > 1 and that there exist two points

in D \ B(z0, r) which cannot be joined in D \ B(

z0,rc

)

. Then the

conclusion of Lemma 8.1 again holds.

Proof. Let D′ be the image of D under

f(z) =r2

z − z0+ z0 .

Then there are two points in D′ ∩ B(z0, r) which cannot be joined in

D′ ∩ B(z0, cr). The result is now obtained by applying Lemma 8.1 to

D′ and mapping back to D. Again, for the details we refer to [6]. �


8.3. Finally we need a simple lower bound for the hyperbolic met-

ric.

LEMMA. If w1, w2 ∈ C \D, then

ρD(z) ≥1

2

|w1 − w2|

|z − w1||z − w2|

for all z ∈ D.

Proof. Let D′ be the image of D under

f(z) =z − w1

z − w2.

Then D′ is a simply connected subdomain of C which omits 0. Thus

by the Koebe theorem

ρD(z) = ρD′(f(z))|f ′(z)|

≥1

2d(f(z), ∂D′)

|w1 − w2|

|z − w2|2

≥1

2|f(z)|

|w1 − w2|

|z − w2|2

=1

2

|w1 − w2|

|z − w1||z − w2|

as desired. �

8.4. THEOREM. (Gehring [6]). If D has the Schwarzian deriv-

ative property, then D is linearly locally connected with

c = max

(

5

d+ 1, 3

)

.(3)

9. QUASIDISKS HAVE THEN BMO EXTENSION PROPERTY 75

Proof. Suppose otherwise. Then there exists a z0 ∈ C and 0 <

r < ∞ for which the hypotheses of either Lemma 8.1 or Lemma 8.2

hold. In either case we get points z1, z2 ∈ D and w1, w2 /∈ D such that

|h(z1) − h(z2) − 2πi| ≤4

c− 1,(4)

where

h(z) = logz − w1

z − w2.

Set

f(z) = ebh(z) , b =2πi

h(z1) − h(z2).

Then f is analytic in D with

Sf (z) =1 − b2

2

(

w1 − w2

(z − w1)(z − w2)

)2

,

and hence

|Sf(z)| ≤ 2|1 − b2|ρD(z)2

by Lemma 8.3. Next (3) and (4) imply that

2|1 − b2| ≤5

c− 1≤ d ,

and thus f is injective inD sinceD was assumed to have the Schwarzian

derivative property. But

f(z1)

f(z2)= eb(h(z1)−h(z2)) = e2πi = 1

and we have a contradiction. We conclude that D is linearly locally

connected with c as in (3). �

9. Quasidisks have then BMO extension property

We shall need a result due to H.M. Reimann.


9.1. LEMMA. Suppose that D1, D2 ⊂ C and that f : D1 → D2

is K-qc. If u2 ∈ BMO(D2), then u1 = u2 ◦ f ∈ BMO(D1) and

‖u1‖⋆ ≤ b‖u2‖⋆ ,

where b = b(K).

The proof is based on the integrabikity properties of quasiconformal

mappings and may be found in [24].

9.2. THEOREM. (Jones [17]). If D is a K-quasidisk, then D

has the BMO extension property with a = a(K).

Proof. By hypothesis there is a K-qc mapping f : C → C which

maps D onto a disk or half plane D′. By following f with a Mobius

transformation we may assume that f(∞) = ∞.

Suppose now that u ∈ BMO(D) and let u′ = u ◦ f ′. Then by

Lemma 9.1, u′ ∈ BMO(D′) and

‖u′‖⋆ ≤ b‖u‖⋆ ,

where b = b(K). Next by Theorem II.6.3, u′ has an extension v′ ∈

BMO(C) with

‖v′‖⋆ ≤ c‖u′‖⋆ ,

where c is an absolute constant. Then again by Lemma 9.1, v = v′◦f ∈

BMO(C) with

‖v‖⋆ ≤ b‖v′‖⋆ .

10. BMO EXTENSION PROPERTY IMPLIES HYPERBOLIC BOUND PROPERTY77

Thus v is the desired BMO extension of u and

‖v‖⋆ ≤ a‖u‖⋆ ,

where a = b(K)2c = a(K). �

10. BMO extension property implies hyperbolic bound

property

The initial estimates in this section will enable us to deduce that

a naturally defined function, namely hD(z, z1), where hD is the hyper-

bolic distance in D and z1 is a given point in D, is actually BMO in

D.

10.1. LEMMA. If u ∈ BMO(C) and if B1 = B(z1, r1) and B0 =

B(z0, r0) with B1 ⊂ B0, then

|uB1− uB0

| ≤ c1

(

logr0r1

+ 1

)

‖u‖⋆ ,

where c1 = e2.

Proof. Suppose first that r0 ≤ er1. Then

|uB1− uB0

| =

∣

∣

∣

∣

∣

∣

1

m(B1)

∫∫

B1

(u− uB0)dxdy

∣

∣

∣

∣

∣

∣

≤1

m(B1)

∫∫

B1

|u− uB0|dxdy

≤m(B0)

m(B1)

1

m(B0)

∫∫

B0

|u− uB0|dxdy

≤ e2‖u‖⋆

≤ c1

(

logr0r1

+ 1

)

‖u‖⋆ .


Suppose next that r0 > er1 and let k be the smallest integer for

which

r0 ≤ ekr1 .

It is not difficult to see that we can choose disks Bj = B(zj , rj) such

that

B1 ⊂ B2 ⊂ · · · ⊂ Bk+1 = B0

and

rj ≤ erj+1

for j = 1, 2, . . . , k. Then

|uB1− uB0

| ≤k

∑

j=1

|uBj− uBj+1

|

≤ e2k‖u‖⋆

≤ e2(

r0r1

+ 1

)

‖u‖⋆

by what was proved above and the choice of k. �

More generally we have the following

10.2. LEMMA. If u ∈ BMO(C) and if B1 = B(z1, r1) and B2 =

B(z2, r2) with

|r1 − r2| ≤ |z1 − z2| ,

then

|uB1− uB2

| ≤

(

c2 log

(

|z1 − z2|

r1+ 1

) (

|z1 − z2|

r2+ 1

)

+ d2

)

‖u‖⋆ ,

where c2 = e2 and d2 = 2e2.


Proof. By relabeling if necessary we may assume that r1 ≤ r2.

Let

B0 = B(z2, r0) , r0 = |z1 − z2| + r1 .

Then if z ∈ B1,

|z − z2| ≤ |z1 − z2| + |z − z1| < r0

and B1 ⊂ B0 as well. Hence by Lemma 10.1

|uB1− uB0

| ≤ c1

(

logr0r1

+ 1

)

‖u‖⋆

= c1

(

log

(

|z1 − z2|

r2+ 1

)

+ 1

)

‖u‖⋆ ,

and similarly

|uB2− uB0

| ≤ c1

(

log

(

|z1 − z2|

r2+ 1

)

+ 1

)

‖u‖⋆ ,

where c1 = e2. The desired conclusion now follows from the triangle

inequality. �

As before, let hD(z1, z2) denote the hyperbolic distance in D be-

tween z1 and z2.

10.3. LEMMA. If B0 = B(z0, r) ⊂ D, then

∫∫

B0

hD(z, z0)dxdy ≤ 2m(B0) .

Proof. Suppose first that B0 = D = B. Then

hB(z, 0) = log1 + |z|

1 − |z|


and hence

∫∫

B

hB(z, 0)dxdy =

∫ 2π

0

∫ 1

0

log1 + r

1 − rrdrdθ = 2π = 2m(B) .

For the general case, the mapping

w = f(z) =1

r(z − z0)

maps B0 conformally onto B and since

hD(z, z0) ≤ hB0(z, z0)

for z ∈ B0, we obtain

∫∫

B0

hD(z, z0)dxdy ≤

∫∫

B0

hB0(z, z0)dxdy

=

∫∫

B

hB(w, 0)

∣

∣

∣

∣

dz

dw

∣

∣

∣

∣

2

dudv

= 2πr2 = 2m(B0)

as desired. �

10.4. LEMA. If z1 ∈ D and if

u(z) = hD(z, z1)

for z ∈ D, then

|u(z0) − uB0| ≤ 2(1)

for each disk B0 = B(z0, r) ⊂ D. En particular, u ∈ BMO(D) with

‖u‖⋆ ≤ 4 .


Proof. By the triangle inequality

|u(z) − u(z0)| = |hD(z, z1) − hD(z0, z1)| ≤ hD(z, z0)

in D, and so by Lemma 10.3,

|u(z0) − uB0| ≤

1

m(B0)

∫∫

B0

|u(z0) − u(z)|dxdy

≤1

m(B0)

∫∫

B0

hD(z, z0)dxdy ≤ 2

which proves (1). Then since

|u(z) − uB0| ≤ |u(z) − u(z0)| + 2

we find that

1

m(B0)

∫∫

B0

|u(z) − uB0|dxdy ≤

1

m(B0)

∫∫

B0

|u(z) − u(z0)|dxdy + 2 ≤ 4

for each B0 ⊂ D. Thus u ∈ BMO(D) with ‖u‖⋆ ≤ 4. �

10.5. THEOREM. (Jones [17]). If D has the BMO extension

property, then D has the hyperbolic bound property with c = 4ae2,

d = 2c+ 4.

Proof. Fix z1, z2 ∈ D and let

u(z) = hD(z, z1)

in D. By Lemma 10.4, u ∈ BMO(D) with ‖u‖⋆ ≤ 4. Hence by

hypothesis u has an extension v ∈ BMO(C) with

‖v‖⋆ ≤ a‖u‖⋆ ≤ 4a .


Now for j = 1, 2 let Bj = B(zj, rj), where rj = d(zj, ∂D). Then

|r1 − r2| = |d(z1, ∂D) − d(z2, ∂D)| ≤ |z1 − z2|

and hence by Lemma 10.2,

|VB1− VB2

| ≤

(

c2 log

(

|z1 − z2|

r1+ 1

)(

|z1 − z2|

r2+ 1

)

+ d2

)

‖v‖⋆

≤ c3jD(z1, z2) + d3 ,

where c3 = 4ae2, d3 = 8ae2. Finally

hD(z1, z2) = u(z2) = |u(z2) − u(z1)|

≤ |u(z2) − uB2| + |uB2

− uB1| + |u(z1) − uB1

|

≤ |vB1− vB2

| + 4 ≤ cjD(z1, z2) + d ,

where c = c3 and d = d3 + 4, completing the proof. �

11. Hyperbolic bound property implies hyperbolic segment

property

In this section we shall complete the left loop in the table of impli-

cations by establishing the following theorem.

11.1. THEOREN. (Gehring-Osgood [10]). If D has the hyper-

bolic bound property, then D has the hyperbolic segment property with

a = b = 4b′e4b′ , b′ = 128c2e2d .

11. HYPERBOLIC BOUND PROPERTY IMPLIES HYPERBOLIC SEGMENT PROPERTY83

For the proof we require a lower bound for the hyperbolic metric

slightly different from that given in Lemma II.4.3. As before, if z1, z2 ∈

D, then hD(z1, z2) denotes their hyperbolic distance in D.

11.2. LEMMA. (Gehring-Palka [11]). If z1, z2 ∈ D, then

hD(z1, z2) ≥1

2

∣

∣

∣

∣

logd(z1, ∂D)

d(z2, ∂D)

∣

∣

∣

∣

.

Proof. As in the proof of Lemma II.4.3

hD(z1, z2) ≥1

2log

(

|z1 − z2| + d(z1, ∂D)

d(z1, ∂D)

)

≥1

2log

(

d(z2, ∂D)

d(z1, ∂D)

)

.

Interchanging the roles of z1, z2 gives

hD(z1, z2) ≥1

2log

(

d(z1, ∂D)

d(z2, ∂D)

)

,

and the result follows. �

11.3. PROOF of Theorem 11.1. By hypothesis there exist con-

stants c, d such that

hD(z1, z2) ≤ cjD(z1, z2) + d

for all z1, z2 ∈ D. By Lemma II.4.3,

1

4jD(z1, z2) ≤ hD(z1, z2)

for all z1, z2 ∈ D, and hence we see from first letting z1 → z2 and then

letting z1 → ∂D that d ≥ 0 and c ≥ 14.


Fix z1, z2 ∈ D and let α be the hyperbolic segment joining z1 to z2

in D. We must show that for each z ∈ α that

ℓ(α ≤ a|z1 − z2| ,

minj=1,2 ℓ(αj) ≤ ad(z, ∂D) ,(1)

where α1, α2 are the components of α\{z}. If z, w ∈ α, we shall denote

by α(z, w) the subarc of α with endpoints z and w.

Define

r = min

(

supz∈α

d(z, ∂D), 2|z1 − z2|

)

.

We shall consider the cases

r < maxj=1,2

d(zj, ∂D)

and

r ≥ maxj=1,2

d(zj , ∂D)(2)

separately.

Suppose first that r < d(z1, ∂D). Then r = 2|z1 − z2|. For any z

on the euclidean line segment β joining z1 to z2 we clearly have

d(z, ∂D) ≥1

2d(z1, ∂D) ≥ |z1 − z2| ,

and hence

hD(z1, z2) ≤

∫

β

ρD(z)|dz| ≤

∫

β

2

d(z, ∂D)|dz|

≤4|z1 − z2|

d(z1, ∂D)≤ 2 .


Since hD(z, z1) ≤ hD(z1, z2) for z ∈ α, Lemma 11.2 yields the estimate

e−4d(z1, ∂D) ≤ d(z, ∂D) ≤ e4d(z1, ∂D)

for z ∈ α. Thus

ℓ(α ≤ e4d(z1, ∂D)

∫

α

|dz|

d(z, ∂D)

≤ 2e4d(z1, ∂D)hD(z1, z2)

≤ 8e4|z1 − z2| < a|z1 − z2|

and for z ∈ α,

ℓ(α(z1, z)) ≤ ℓ(α) ≤ 4e4d(z1, ∂D)

≤ 4e8d(z, ∂D) ≤ bd(z, ∂D) .

This establishes (1) in the case where r < d(z1, ∂D). Similarly we ob-

tain (1) in the case where r < d(z2, ∂D) by reversing the roles of z1

and z2 in the above argument.

Suppose next that (2) holds. By compactness there exists a point

z0 ∈ α with

r ≤ supz∈α

d(z∂D) = d(z0, ∂D) .

For j = 1, 2 let mj be the largest integer for which

2mjd(zj , ∂D) ≤ r .

Let wj be the first point of α(zj, zo) with

d(wj, ∂D) = 2mjd(zj , ∂D) .


as we traverse α from zj towards z0. Obviously

d(wj, ∂D) ≤ r < 2d(wj, ∂D) .(3)

We shall show that for j = 1, 2,

ℓ(α(zj, wj)) ≤ b′d(wj, ∂D) ,

ℓ(α(zj, z)) ≤ b′e2b′d(z, ∂D) for z ∈ α(zj, wj) .(4)

Clearly we need only consider the case where j = 1 and m1 ≥ 1. To

establish (4) choose points

ζ1, ζ2, . . . , ζm1+1 ∈ α(z1, w1)

so that ζ1 = z1 and ζk is the first point of α(z1, w1) with

d(ζk, ∂D) = 2k−1d(z1, ∂D)(5)

as we traverse α from z1 towards w1. Then ζm1+1 = w1. Fix k and set

t =ℓ(α(ζk, ζk+1))

d(ζk, ∂D).

Suppose that z ∈ α(ζk, ζk+1). Then

d(z, ∂D) ≤ d(ζk+1, ∂D) = 2d(ζk, ∂D) ,

the first inequality holding by definition of ζk+1. Therefore

t ≤ 2

∫

αk

1

d(z, ∂D)|dz| ≤ 4hD(ζk, ζk+1) ,

where αk = α(ζk, ζk+1). Now

jD(ζk, ζk+1) < 2 log

(

|ζk − ζk+1|

d(ζk, ∂D)+ 1

)

≤ 2 log(t+ 1) ,


whence

t

4≤ hD(ζk, ζk+1) ≤ cjD(ζk, ζk+1) + d

≤ 2c log(e2d(t+ 1))

≤ 2c(e2d(t+ 1))1

2

since log x ≤ x1

2 for x > 0. If t ≥ 1, then

t ≤ 8c(2te2d)1

2

or

t ≤ 128c2e2d = b′ ,(6)

and hence

hD(ζk, ζk+1) ≤ 2c(2b′e2d)1

2 < b′ .(7)

It t < 1, then t < b′ and again we obtain (7). Next if z ∈ α(ζk, ζk+1),

then form Lemma 11.2,

0 < logd(ζk+1, ∂D)

d(z, ∂D)≤ 2hd(z, ζk+1) ≤ 2hD(ζk, ζk+1) ,

and with (6) and (7) we conclude that

ℓ(α(ζk, ζk+1) ≤ b′d(ζk, ∂D) ,

d(ζk+1, ∂D) ≤ e2b′d(z, ∂D) for z ∈ α(ζk, ζk+1)(8)


for k = 1, 2, . . . , m1. Hence

ℓ(α(z1, w1) =

m1∑

k=1

ℓ(α(ζk, ζk+1)

≤ b′m1∑

k=1

d(ζk, ∂D)

= b′(2m1 − 1)d(z1, ∂D)

< b′d(w1, ∂D)

by (5) and (8). This prove the first inequality in (4). Now let z ∈

α(z1, w1). Then z ∈ α(ζk, ζk+1) for some k and so

ℓ(α(z1, z) ≤k

∑

i=1

ℓ(α(ζi, ζi+1)

≤ b′k

∑

i=1

d(ζi, ∂D)

= b′(2k − 1)d(z1, ∂D)

< b′d(ζk+1, ∂D) ≤ b′e2b′d(z, ∂D)

again by (5) and (8). This completes the proof of (4).

We show next that if d(w1, ∂D) ≤ d(w2, ∂D), then

ℓ(α(w1, w2) ≤ 2b′e2b′d(w1, ∂D)

d(w2, ∂D) ≤ e2b′d(z, ∂D) ,(9)

for all z ∈ α(w1, w2). We may assume that w1 6= w2 since otherwise

there is nothing to prove. Again we consider two cases. Suppose first

that

r = supz∈α

d(z, ∂D)


and set

t =ℓ(α(w1, w2))

d(w1, ∂D).

If z ∈ α(w1, w2), then by (3)

d(z, ∂D) ≤ r < 2d(w1, ∂D) ,

and we can repeat the proof of (8) with ζk replaced by w1 and ζk+1 by

w2 to obtain (9) in this case. Suppose next that r = 2|z1 − z2|. Then

using the triangle inequality, (3) and (4) we find that

|w1 − w2| ≤ ℓ(α(z1, w1)) + ℓ(α(z2, w2)) + |z1 − z2|

≤ b′d(w1, ∂D) + b′d(w2, ∂D) +r

2

≤ 4b′d(w1, ∂D) .

Therefore jD(w1, w2) ≤ 2 log 5b′ and

hD(w1, w2) ≤ 2c log(5b′e2d)

≤ 2c(5b′e2d)1

2 < b′ .

Now if z ∈ α(w1, w2), then by Lemma 11.2,

e−2b′d(w2, ∂D) ≤ d(z, ∂D) ≤ e2b′d(w1, ∂D)

and from this

ℓ(α(w1, w2) ≤ 22b′d(w1, ∂D)

∫

α(w1,w2)

|dz|

d(z, ∂D)

≤ 2e2b′d(w1, ∂D)hD(w1, w2)

≤ 2b′e2b′d(w1, ∂D)


proving (9) in this case as well.

We are now in a position to complete the proof of the theorem. By

relabeling we may assume that d(w1, ∂D) ≤ d(w2, ∂D). Then

ℓ(α) ≤ ℓ(α(z1, w1)) + ℓ(α(z2, w2)) + ℓ(α(w1, w2))

≤ 4b′e2b′d(w2, ∂D) ≤ 4b′e2b′r

leq 8b′e2b′ |z1 − z2| < a|z1 − z2|

by (3), (4) and (9). This establishes the first inequality in (1). Next,

if z ∈ α, then either z ∈ α(zj, wj) = and

minj=1,2

ℓ(α(zj, z)) ≤ ℓ(α(zj , z)) ≤ b′e2b′d(z, ∂D) ≤ bd(z, ∂D)

by (4), or z ∈ α(w1, w2) and

minj=1,2

ℓ(α(zj, z)) ≤1

2ℓ(α)

≤ 2b′e2b′d(w2, ∂D)

≤ 2b′e4b′d(z, ∂D)

≤ bd(z, ∂D)

by (9). In each case we obtain the second inequality in (1) and the

proof of Theorem 11.1 is complete.

CHAPTER IV

EPILOGUE

1

1.1. These lecture have been devoted to examining the many ways

quasidisks appear naturally in analysis. We take this opportunity to

discuss one application of some of these ideas.

1.2. Constant L(D). Given a simply connected domain D ⊂ C

we let L(D) denote the supremum of the numbers d ≥ 1 such that f is

injective in D whenever f is local L-quasi-isometry in D with L ≤ d.

Note that by Theorem II.5.14, L(D) > 1 if and only if D is a quasidisk.

1.3. THEOREM. (Gehring [7]). If D is a quasidisk and if f is

a local L-quasi-isometry in D with L < L(D), then f is injective in

D and has an extension to C which is an L′-quasi-isometry, where L′

depends only on L and L(D).

We give an idea of the proof.

Let D′ = f(D) and suppose that g is a local M-quasi-isometry in

D′ with

1 ≤M <L(D)

L.

91

92 IV. EPILOGUE

Then h = g ◦ f is a local LM-quasi-isometry in D. But since

LM < L(D) ,

h is injective in D, g is injective in D′ and so

L(D′) ≥L(D)

L> 1 .

This implies that D′ is also a qusidisk. Now tha fact that D and D′ are

both uniform allows one to conclude that f is an L′′-quasi-isometry in

D and hence has an extension as a homeomorphism mapping D onto

D′.

Suppose ∞ ∈ ∂D1. Then ∞ ∈ ∂D2 and f(∞) = ∞. By Theorem

II.2.4 we get M and M ′-quasi-isometries ϕ and ϕ′ of C such that

ϕ(D) = D⋆ , ϕ′(D′) = (D′)⋆

and such that ϕ and ϕ′ are the identity when restricted to ∂D and ∂D′,

respectively. Moreover,

M = M(L(D)) , M ′ = M ′(L(D′)) , L′′ = L′′(L(D), L(D′)) .

Let L′ = L′′MM ′ and set

F (z) =

f(z) , z ∈ D

(ϕ′ ◦ f ◦ ϕ−1)(z) , z ∈ D⋆ .

Then F is a homeomorphism of C onto C, F (∞) = ∞ and F is an

L′-quasi-isometry in D and D⋆. It then follows that F is an L′-quasi-

isometry in C.

1 93

Finally, the case where ∞ /∈ ∂D can be reduced to the above case by

using auxiliary Mobius transformations and an extension of Theorem

II.2.4. (For details see [7].)

1.4. There is a physical interpretation of this theorem. Think of

D as an elastic plane body, let f denote the deformation of D under a

force field, and let

L(z) = lim suph→0

max

(

|f(z + h) − f(z)|

|h|,

|h|

|f(z + h) − f(z)|

)

denote the strain in D at z caused by the force field. Then f is a local

quasi-isometry if and only if the strain in D is bouded, and L(D) is

the supremum of the allowable strains before D collapses. Theorem 1.3

thus says that if

supL(z) < L(D) ,

then the shape of the deformed D is roughly the same as that of the

original body.

REFERENCES

[1] AHLFORS, L.V., Remarks on the Neumann-Poincare integral equation. Pa-

cific J. Math. 2 (1952), 271-280.

[2] AHLFORS, L.V., Quasiconformal reflections. Acta Math. 109 (1963), 291-

301.

[3] BECKER, J., Lownersche Differentialgleichung und quasikonform fortsetz-

bate schlichte funktionen. J. Reine Angew. Math. 225 (1972), 23-43.

[4] ERKAMA, T., Quasiconformally homogeneous curves. Michigan Math. J.

24 (1977), 157-159.

[5] GEHRING, F.W., Quasiconformal mappings of slit domains in three space.

J. Math. Mech. 18 (1969), 689-703.

[6] GEHRING, F.W., Univalent functions and the Schwarzian derivative. Com-

ment. Math. Helv. 52 (1977), 561-572.

[7] GEHRING, F.W., Injectivity of local quasi-isometries. Comment. Math.

Helv. (to appear).

[8] GEHRING, F.W., (unpublished).

[9] GEHRING, F.W., MARTIO, O., Quasidisks and the Hardy-Littlewood prop-

erty. Complex Variables. (To appear.)

[10] GEHRING, F.W., OSGOOD, B.G., Uniform domains and the quasi-

hyperbolic metric. J. Analyse Math. 36 (1979), 50-74.

[11] GEHRING, F.W., PALKA, B.P., Quasiconformally homogeneous domains.

J. Analyse Math. 30 (1976), 172-199.

[12] GEHRING, F.W., VAISALA, J., Hausdorff dimension and quasiconformal

mappings. J. London Math. Soc. 6 (1973), 504-512.

95

96 REFERENCES

[13] GOL’DSTEIN, V.M., VODOP’JANOV, S.K., Prolongement des fonctions

de classe L1

pet applications quasi conformes. C.R. Acad. Sci. Paris 290

(1980), 453-456.

[14] HARDY, G.H., LITTLEWOOD, J.E., Some properties of fractional inte-

grals II. Math. Z. 34 (1932), 403-439.

[15] JERISON, D.S., KENIG, C.E., Hardy spaces, A∞, and singular integrals

on chord-arc domains. Math. Scand. (to appear).

[16] JOHN, F., On quasi-isometric mappings II. Comm. Pure Appl. Math. J. 29

(1980), 41-66.

[17] JONES, P.M., Extension theorems for BMO. Indiana Univ. Math. J. 29

(1980), 41-66.

[18] LEHTO, O., Domain constants associated with Schwarzian derivative. Com-

ment. Math. Helv. 52 (1977), 603-610.

[19] LEHTO, O., VIRTANEN, K.I., Quasiconformal Mappings in the plane.

Springer-Verlag, New York, 1973.

[20] MARTIO, O., SARVAS, J., Injectivity theorems in plane and space. Ann.

Acad. Sci. Fenn. Ser. AI Math. 4 (1978-79), 383-401.

[21] MASKIT, B., On boundaries of Teichmuller spaces and on kleinian groups:

II. Ann. of Math. 91 (1970), 607-639.

[22] NEHARI, Z., The Schwarzian derivative and schlicht functions. Bull. Amer.

Math. Soc. 55 (1949), 545-551.

[23] OSGOOD, B.G., Some properties of f ′′/f ′ and the Poincare metric. Indiana

Univ. Math. J. (to appear)

[24] REIMANN, H.M., Functions of bounded mean oscillation and quasiconfor-

mal mappings. Comment. Math. Helv. 49 (1974), 269-276.

[25] REIMANN, H.M., RYCHENER, T., Funktionen beschrankter mittlerer Os-

zillation. Lecture Notes Math. 487, Springer-Verlag, Berlin, 1975.

[26] SARVAS, J., Boundary of a homogeneous Jordan domain. (Unoublished.)

REFERENCES 97

[27] SPRINGER, G., Fredholm eigenvalues and quasiconformal mapping. Acta

Math. 111 (1964) 121-142.

[28] SULLIVAN, D., On the ergodic theory at infinity of an arbitrary discrete

group if hyperbolic motions, in: Riemann Surfaces and Related Topics: Pro-

ceedings of the 1978 Stony Brook Conference. Annals Math. Studies 97,

Princeton Univ. Press, 1981.

[29] TUKIA, P., On two-dimensional quasiconformal groups. Ann. Acad. Sci.

Fenn. Ser. AI Math. 5 (1980), 73-78.

characteristic properties of quasidisks · formal mappings. in this section we shall derive a...

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