SÉMINAIRE DE MATHÉMATIQUES SUPÉRIEURES
SÉMINAIRE SCIENTIFIQUE OTAN (NATO ADVANCED STUDY INSTITUTE)
DÉPARTEMENT DE MATHÉMATIUQUES ET DE STATISTIQUE - UNIVERSITÉ DE MONTRÉAL
CHARACTERISTIC PROPERTIES
OF QUASIDISKS
FREDERICK W. GEHRING
University of Michigan
1982
LES PRESSES DE L’UNIVERSITÉ DE MONTRÉAL
C.P. 6128, succ. ≪A≫, Montréal (Québec) 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 writing up these notes, Miss B.A. Brown for making the figures, and the
Séminaire de mathématiques supérieures 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
9. Quasidisks have then BMO extension property
71
75
10. BMO extension property implies hyperbolic bound
property
77
11. Hyperbolic bound property implies hyperbolic segment
property
Chapter IV. EPILOGUE
1.
REFERENCES
82
91
91
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 sup
r→0
z
b
r
f
L(z, r)
,
ℓ(z, r)
ℓ
f (z)
b
where
L(z, r) = max |f (z) − f (w)| ,
|z−w|=r
ℓ(z, r) = min |f (z) − f (w)| .
|z−w|=r
7
L
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 ,
y
d
R
c
a
D
b
x
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 preserving homeomorphism f : D → D ′ is K− qc if and only if f is ACL
en D and if
max |∂α f (z)|2 ≤ KJ(z)
α
a.e. in D. Here ∂α f 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 differential 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
Z
ρ(z)|dz| ≥ 1
γ
for each locally rectifiable curve γ ∈ Γ. We then define the modulus of
Γ to be
mod Γ = inf
ρ
ZZ
ρ2 (z)dxdy ,
C
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 homogeneous wire. Then mod Γ 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
mod Γ ≤ mod Γ′ ≤ Kmod Γ
K
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 K1 K2 − 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 quasiconformal 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,
a
Γ
R
b
b
then mod Γ = .
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≤
Z
ρ(z)|dz| =
Z
a
ρ(x + iy)dy ≤
0
γ
Z
a
2
ρ(x + iy) dy
0
12 Z
a
dy
0
whence
ZZ
C
2
ρ (z)dxdy ≥
Z b Z
0
0
a
2
ρ(x + iy) dy dx ≥
Z
0
b
1
b
dx = .
a
a
12
12
I. PRELIMINARIES
Thus
mod Γ = inf
ρ
ZZ
ρ2 (z)dxdy ≥
b
:.
a
C
On the other hand, the function
ρ(z) =

1




 a
z∈R




 0
is in adm Γ and
ZZ
z ∈C\R
ρ2 (z)dxdy =
b
.
a
C
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 map H onto a rectangle
R so that x1 , x2 , x3 , ∞ corespond to the vertices.
2. MODULUS ESTIMATES
13
H
Γ
∞
x1
1
x2 x3
Γ′
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
1
b
log .
2π
a
mod Γ ≥
γ
t
b
b
z1
a
14
I. PRELIMINARIES
Proof. Let ρ ∈ adm Γ. Then for a < t < b,
2π
Z
iθ
ρ(te )tdθ ≥
0
Z
ρ(z)|dz| ≥ 1
γ
whence
1≤
Z
2π
iθ
ρ(te )tdθ
0
2
≤ 2πt
iθ
Z
2π
ρ2 (teiθ )tdθ ,
0
and we obtain
ZZ
C
2
ρ (z)dxdy ≥
Z b Z
a
2π
2
ρ (te )tdθ dt ≥
0
Z
b
a
1
1
b
dt =
log .
2πt
2π
a
Thus
mod Γ = inf
ρ
ZZ
ρ2 (z)dxdy ≥
1
b
log .
2π
a
C
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
Γ
α2
2. MODULUS ESTIMATES
15
Proof. Choose z1 ∈ α1 and z2 ∈ α2 so that
|z1 − z2 | = d(α1 , α2 ) ≥ r
and set
ρ(z) =

1




 r
z ∈ 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).
z1
α1
b
α2
s
γ
In either case γ contains a subarc δ of length r which lies in B(z1 , r +s).
Hence
Z
ρ(z)|dz| ≥
γ
Z
1
ρ(z)|dz| = ℓ(δ) = 1 .
r
δ
Thus ρ ∈ adm Γ and
mod Γ ≤
ZZ
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 ) = inf
γ
Z
2|dz|
,
1 + |z|2
γ
where the infinum is taken over all locally rectificable arcs γ which join
z1 and z2 in C.
S
C
b
b
b
z1
0
b
z2
p(z1 )
p
b
0
b
p(z2 )
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
2. MODULUS ESTIMATES
17
Proof. For z ∈ C set
ρ(z) =
1
2
.
2ℓ 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
Z
ρ(z)|dz| =
1
ℓ(p(γ)) ≥ 1 ,
2ℓ
γ
ρ ∈ adm (Γ), and we find that
mod Γ ≤
ZZ
1
ρ (z)dxdy = 2
4ℓ
2
C
ZZ
4
1
π
dxdy = 2 m(S) = 2 .
2
2
(1 + |z| )
4ℓ
ℓ
C
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′ ≥
|f (z2 )|
1
log
.
2π
|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 , ∞.
z2
f (z2 )
Γ
b
b
b
b
f
z1
f (z1 )
b
0
Γ′
b
0
b
∞
b
∞
Now if |z1 | = 1, then
s(z1 , 0) =
π
,
2
s(z2 , ∞) ≥
π
,
2
and hence
mod Γ ≤
π
4
=
2
(π/2)
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
2. MODULUS ESTIMATES
19
Fnally, since f is K-qc we have
1
|f (z2 )|
4K
log
≤ mod Γ′ ≤ Kmod Γ ≤
2π
|f (z1 )|
π
or
|f (z2 )|
≤ e8K
|f (z1 )|
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 = 21 (z2 + z0 ).
b
z1
b
z0
b
z2
b
z
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 Hölder
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.
Q′j
Qj
f0
Q
f0
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) = aj z + bj ,
Then f0 is K-qc in Q \
S
aj > 0, on ∂Qj for j = 1, 2, 3, 4.
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 Kquasidisk 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 quasidisk 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
|z1 − z2 | ≤ |f (z1 ) − f (z2 )| ≤ L|z1 − z2 |
L
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 that D has the reflection property 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 quasidisk 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 corresponding 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
min dia(γj ) = |z1 − z2 |
j=1,2
where gamma1 , γ2 are the components of ∂D \ {z1 , z2 }
γ2
z1
z2
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,
min dia (γj ) ≤ d|z1 − z2 | ,
j=1,2
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 |
26
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
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
dia(U) .
d
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 arbitrary 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(z0 r) 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 z0 inE
or E. The two figures below illustrate the situations where (i) and (ii)
come into play.
E
z0
r
z0
cr
r
c
r
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
1
dia(V ) = dia(U) .
c
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.
28
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
4.1. Hyperbolic metric en B. For the unit disk B the hyperbolic metric is defined by
ρB (z) =
2
.
1 − |z|2
If z1 , z2 inB, the hyperbolic distance between z1 and z2 is then defined
to be
hB (z1 , z2 ) = inf
α
Z
ρB (z)|dz| ,
α
α
b
z1
z2
b
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 ) =
Z
ρB (z)|dz| ,
α
namely, that arc α between z1 , z2 lying on the circular arc β through
z1 , z2 orthogonal to the unit circle.
z2
b
α
b
z1
4. HYPERBOLIC METRIC PROPERTIES
29
We shall call α the hyperbolic segment joining z1 , z2 and β the hyperbolic line containing α. The distance is given explicitly by
hB (z1 , z2 ) = log
|1 − z1 z2 | + |z1 − z2 |
.
|1 − z1 z2 | − |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 ) = inf
α
Z
ρD (z)|dz| ,
α
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 ) =
Z
ρd (z)|dz| ,
α
and α will again be called the hyperbolic segment between z1 , z2 . Observe 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 inequalities
(1)
1
1
2
≤ ρD (z) ≤
2 d(z, ∂D)
d(z, ∂D)
30
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
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
1
hD (z1 , z2 ) ≥ jD (z1 , z2 ) ,
4
where
jD (z1 , z2 ) = log
|z1 − z2 |
+1
d(z1 , ∂D)
|z1 − z2 |
+1
d(z2 , ∂D)
.
Proof. Let α be the hyperbolic segment joining z1 , z2 in D. Then
by (1)
hD (z1 , z2 ) =
Z
α
1
ρD (z)|dz| ≥
2
Z
d(z, ∂D)−1 |dz| .
α
Next for each z ∈ α,
d(z, ∂D) ≤ d(z1 , ∂D) + |z − z1 |
4. HYPERBOLIC METRIC PROPERTIES
31
and thus
1
hD (z1 , z2 ) ≥
2
Z
α
1
d(|z − z1 |)
= log
d(z1 , ∂D) + |z − z1 |
2
|z1 − z2 |
+1
d(z1 , ∂D)
.
Interchanging the roles of z1 and z2 yields
1
hD (z1 , z2 ) ≥ log
2
|z1 − z2 |
+1
d(z2 , ∂D)
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 − z1 z2 | + |z1 − z2 |
n
= log .
|1 − z1 z2 | − |z1 − z2 |
d
Now
n = |1 − z1 z2 | + |z1 − z2 |
= |1 − |z2 |2 − z2 (z1 − z2 )| + |z1 − z2 |
≤ (1 − |z2 |2 ) + (1 + |z2 |)|z1 − z2 |
|z1 − z2 |
2
= (1 − |z2 | )
+1 ,
d(z2 , ∂D)
ans similarly
n ≤ (1 − |z1 |[2)
|z1 − z2 |
+1
d(z1 , ∂D)
.
32
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
Next
nd = |1 − z1 z2 |2 − |z1 − z2 |2
= (1 − |z1 |2 )(1 − |z2 |2 ) .
Thus
n
n2
=
≤
d
nd
|z1 − z2 |
+1
d(z1 , ∂D)
|z1 − z2 |
+1
d(z2 , ∂D)
and the result follows.
4.5. Hyperbolic bound property. We say that D has the hyperbolic 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 hyperbolic segment α in D and for each z ∈ α
ℓ(α) ≤
π
|z1 − z2 |
2
and
min ℓ(αj ) ≤
j=1,2
π
d(z, ∂D) ,
2
4. HYPERBOLIC METRIC PROPERTIES
33
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 hyperbolic 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
min ℓ(αj ) ≤ bd(z, ∂D) ,
j=1,2
where z1 , z2 are the endpoints of α and α1 , α2 are the components of
α \ {z}.
α1
b
z
α2
b
z1
b
z2
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.
34
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
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
min ℓ(αj ) ≤ bd(z, ∂D)
j=1,2
for z ∈ α, where α1 , α2 are the components of α \ {z}.
∂D
b
z
α1
α2
b
b
z1
z2
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
′′ ′
2
f
1 f ′′
Sf =
−
.
f′
2 f′
Note that Sf ≡ 0 if and only if f is a Möbius transformation and that
Sg◦f = Sg (f )(f ′ )2 + Sf .
In particular, Sg◦f = Sf if and only if g is a Möbius 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
1
|Sf | ≤ ρ2D ,
2
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
36
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
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 ≤ 4ρD .
f′ 5.7. THEOREM. (Becker [3]). If D is a disk or a half plane and
if f is analytic in D with
′′ f 1
≤ ρD ,
f′ 2
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 ≤ dρD , f ′ 6= 0
f′ 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
|z1 − z2 | ≤ |f (z1 ) − f (z2 )| ≤ L|z1 − z2 |
L
(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| iL2 arg z
e
L
D
0
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
1
if f is a local L-quasi-isometry in D with L ≤ 2 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 ) .
38
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
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
E
α
U
α′
f
z2
E′
U′
Next, the fact that f is injective in U implies that f −1 is a local Lquasi-isometry in U ′ = f (U) and hence that
m(E) ≤ L2 m(E ′ ) .
By elementary geometry and the isomperimetric inequality we conclude
that
ℓ(α)2
ℓ(α′ )2
L2 ℓ(α)2
< m(E) ≤ L2 m(E ′ ) ≤ L2
≤ L2
,
2π
4π
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-quasiisometry 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
1
kuk∗ = sup
B m(B)
ZZ
|u − uB |dxdy < ∞ ,
B
where the suprem is taken over all disks B with B ⊂ D, and where
1
uB =
m(B)
ZZ
dxdy .
B
6.2. REMARK. If v ∈ BMO(D) then u = v|D ∈ BMO(D) with
kuk∗ ≤ kvk∗ .
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
kvk∗ ≤ akuk∗ ,
where a is an absolute constant.
Proof. Set
v(t) =


u(z)
z∈D
 (u ◦ ϕ)(z) z ∈ D ∗ ∩ C
40
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
where ϕ denotes reflection in ∂D. An elementary but technical argument shows that v ∈ BMO(C) with
kvk∗ ≤ akuk∗ ,
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
kvk∗ ≤ akuk∗ .
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 ∈ L21 (D =) if u ∈
L2loc (D) and u has distributional derivatives v1 , v2 ∈ L2 (D). If this is
the case, set
ZZ
E(u) =
(v12 + v22 )dxdy .
D
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 that D has the L21 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
kf kLip(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)| ≤ kf kLip(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
kf kLip(k) ≤
where c is an absolute constant.
ca
k
42
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
6.13. Hardy-Littlewood property. We say that D has the HardyLittlewood property if for 0 < k ≤ 1 there exists a constant c such that
f ∈ Lip(k) with
kf kLip(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 Möb, the family of Möbius transformations 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) =Möb.
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 homeomorphisms 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 = lim gj (z1 ) .
j→∞
8.2. REMARK. If D is a disk or a half plane then there exists a
finitely generated group G of mappings in Möb 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
ZZ
D
2
|∇u| dxdy =
ZZ
D∗
|∇u⋆ |2 dxdy .
44
II. CHARACTERISITC PROPERTIES OF QUASIDISKS
8.5. THEOREM. (Ahlfors [1] and Springer [27]). D is a quasidisk if and only if D is a Jordan domain and there exists a constant
K such that
ZZ
ZZ
ZZ
1
2
⋆ 2
|∇u| dxdy ≤
|∇u | dxdy ≤ K
|∇u|2dxdy
K
D
D⋆
D
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 B 3 in R3 .
8.7. THEOREM. (Gehring [5]). D is a quasidisk if and only if
G = R3 \ D can be mapped quasiconformally onto B 3 .
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.
L21 extension property
Dirichket integral property
BMO extension property
III.9
III.10
Quasidisk
III.6
III.2
Three point property
Hyperbolic segment
property
III.11
III.5
Linear local connectivity
III.3
III.4
Uniform domain
III.8
III.7
Hyperbolic bound
property
Schwarzian derivative
property
Rigid domain
This chapter will be devoted to establishing several of these implications.
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 Kquasidisk and that f : D → D ′ is conformal. Then f has an extension
which is K 2 -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 K 2 -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
(1)
Z
y
|f ′ (it)|dt ≤ c1 d(f (iy), ∂D)
0
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 K 2 -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)|
2
for j = 1, 2, . . . , where c = e8K .
H
iyj
b
b
iy
b
iyj+1
b
f
b
b
b
b
0
0
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)| ≤ 4
d(f (iy), ∂D)
|f (iy0 )|
≤ 4c−j+1
d(iy, ∂H)
y
by the Koebe distortion theorem. Thus
Z yj
yj
(2)
|f ′ (iy)|dy ≤ 4c−j+1|f (iy0 )| log
.
yj+1
yj+1
D
48
III. SOME PROOFS OF THESE PROPERTIES
yj
. For this, let k denote
yj+1
the smallest positive integer for which c ≤ 2k . Then
It remains to find an upper bound for log
|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
log
yj
≤ (k + 1) log 2c = c2 ,
yj+1
and which (2) we obtain
Z
y0
′
|f (iy)|dy =
0
∞ Z
X
j=0
yj
|f ′ (iy)|dy
yj+1
≤ 4c2 |f (iy0 )|
∞
X
c−j+1
j=0
= 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. QUASIDISKS HAVE THE HYPERBOLIC SEGMENT PROPERTY
49
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
(3)
ℓ(α) ≤ a|z1 − z2 |
and
(4)
min ℓ(αj ) ≤ bd(z, ∂D)
j=1,2
for each z ∈ α, where α1 , α2 are the components of α \ {z}. By Lemma
2.1 there exists a K 2 -qc mapping f : C → C which maps D conformally onto B. By employing an auxiliary Möbius 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 K 2 -quasidisk and α is a hyperbolic line in D ′ .
D
z1
B
B′
b
f
α
b
b
z2
50
III. SOME PROOFS OF THESE PROPERTIES
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 − z1
.
z − 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 (∞) = ∞.
∞
∞
H
D′
D
α
b
b
g
w
α′
z2
iy
b
f
b
z1
h
b
b
0
Now D ′ is a K-quasidisk and α′ is the image of the positive imagHence if w ∈ α′ , then w = f (iy) for some
inary axis under f .
0 < y < ∞, and
(5)
s=
Z
y
|f ′(it)|dt ≤ c1 d(f (iy), ∂D ′)
0
= c1 d(w, ∂D′ )
2. QUASIDISKS HAVE THE HYPERBOLIC SEGMENT PROPERTY
51
by Lemma 2.2, where s is the arclength of α′ between 0 and w. Let
h = g −1 . Then
Z
ℓ(α) =
′
|h (w)||dw| = |z1 − z2 |
α′
Z
|dw|
|w − 1|2
α′
= |z1 − z2 |
Z
∞
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
Z
∞
0
ds
≤
|w(s) − 1|2
Z
s0
2
(c1 + 1) ds +
0
Z
∞
s0
c21
ds = 2c1 (c1 + 1)
s2
and we obtain (3) with a = 2c1 (c1 + 1).
Finally since D is bounded, we can find a K 2 -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)
52
III. SOME PROOFS OF THESE PROPERTIES
and let w1 , w2 , w and w0 be the images of z1 , z2 , z and z0 under f .
D
α
b
B
z2
b
b
z
f
b
z1
b
b
w1
z0
w
b
b
w2
w0
Since f (α) is hyperbolic line in B it is easy to check that
min |w − wj | ≤ 2d(w, ∂B) ≤ 2|w − w0 |
j=1,2
and hence
min |zj − z| ≤ 4c2 |z − z0 | = 4c2 d(z0 , ∂D)
j=1,2
2
by Corollary I.2.7, where c = e8K , 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 = 4c2 a.
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 connected 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 .
α
z1
b
r
z0
z
b
b
b
z2
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
min ℓ(βj ) ≤ bd(z, ∂D)
j=1,2
54
III. SOME PROOFS OF THESE PROPERTIES
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
r
c
z
b
z2
b
z0
r
b
z1
Then there exists a point z ∈ β with
|z − z0 | <
r
r
≤
,
c
2b + 1
and for j = 1, 2 we have
ℓ(βj ) ≥ |zj − z|
≥ |zj − z0 | − |z − z0 | ≥
2b
r.
2b + 1
Thus
d(z, ∂D) ≥
1
2
r
min ℓ(βj ) ≥
r > |z − z0 | +
j=1,2
b
2b + 1
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 PROPERTY
55
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
min |z − zj | ≤ bd(z, ∂D)
j=1,2
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
r
C \ B z0 ,
⊂U
c
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.
56
III. SOME PROOFS OF THESE PROPERTIES
Now fix finite points z1 , z2 ∈ ∂D. We shall show that
(1)
d = c2 ,
min dia(γj ) ≤ d|z1 − z2 | ,
j=1,2
where γ1 , γ2 are the components of ∂D \ {z1 , z2 }. For this, set
1
z0 = (z1 + z2 ) ,
2
1
r = |z1 − z2 |
2
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 , c2 t)
for j = 1, 2.
b
c2 t
w2
t
z1
b
b
z0
s z2
D
b
b
w1
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
zj′ ∈ D and which lies in D ∩ B(z0 , s). Next, since D is linearly locally
connected, we can find an arc α3 joining z1′ and z2′ 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).
z1
b
b
cs
z1′
b
z0
z2
b
b
z2′
α3
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 ,
58
III. SOME PROOFS OF THESE PROPERTIES
where c = π(2d2 e2π + 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 ≤ 2d2 e2π r .
(1)
For this we may clearly assume that
s
> dr ,
2d
sinceotherwise (1) would follow trivially. Because D has the rhree point
property,
min dia(γj ) ≤ d|z1 − z2 | = dr ,
j=1,2
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) .
6. THREE POINT PROPERTY IMPLIES D IS A QUASIDISK
59
Choose z0 ∈ β2 and let δ1 , δ2 denote the components of ∂D \ {z0 , z1 }
labeled so that α2 ⊂ δ2 .
b
α1
β2
b
z1
z0
b
b
b
β1
z2
b
b
α2
Then again by the three point property,
min dia(δj ) ≤ d|z1 − z0 | ,
j=1,2
while obviously
dia(δ2 ) ≥ dia(α2 ) ≥ s .
Choose w1 , w2 ∈ α1 so that
|w1 − w2 | = dia(α1 ) = s ,
60
III. SOME PROOFS OF THESE PROPERTIES
b
b
w1
β2
α1
z1
b
z0
b
w2
b
b
b
z2
β1
α2
b
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
< min dia(δj ) ≤ d|z1 − z2 | ,
2 j=1,2
and we conclude that
s
β2 ∩ B z1 ,
=∅.
2d
6. THREE POINT PROPERTY IMPLIES D IS A QUASIDISK
61
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
α1
t
β2
z1
γ
b
b
α2
b
b
z2
b
Now by Lemma I.2.3,
1 = modΓ ≥
1
s
log 2
2π
2d r
from which (1) follows.
Finally by (1) and Lemma I.2.4,
modΓ⋆ ≤ π
as desired.
s
r
+1
2
≤ π(2d2 e2π + 1)2
62
III. SOME PROOFS OF THESE PROPERTIES
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 Möbius transformation we may assume that h−1
1 (∞) =
h−1
2 (∞). Next, for x ∈ ∂H let
ϕ(x) = (h2 ◦ h−1
1 )(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.
H
Γ
ϕ(x−t) ϕ(x)
∞
b
b
b
x−t x
h−1
1
b
∞
x+t
b
⋆
h1
b
Γ1
D
⋆
D
b
b
b
H⋆
Γ
b
ϕ(x+t)
Γ⋆1
6. THREE POINT PROPERTY IMPLIES D IS A QUASIDISK
63
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
ϕ(x + t) − ϕ(x)
1
≤
≤b.
b
ϕ(x) − ϕ(x − t)
Now define
1
g(z) =
2
Z
0
1
1
(ϕ(x + ty) + ϕ(x − ty))dt +
2
Z
1
(ϕ(x + ty) − ϕ(x − ty))dt
0
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

 (g ◦ h1 )(z) , z ∈ D
f (z) =

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 completed the innermost loop in the table of implications.
64
III. SOME PROOFS OF THESE PROPERTIES
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
c
(z) ≤
(1)
f ′ min(s, ℓ(α) − s) ,
f ′ (z) 6= 0
for z ∈ α, where s is the arclength of α from z1 to z, then
f (z1 ) − f (z2 )
≤ c ℓ(α) .
−
(z
−
z
)
1
2
1−c
′
f (z0 )
Proof. By the triangle inequality it is sufficient to prove that
f (zj ) − f (z0 )
1 c
≤
(2)
−
(z
−
z
)
j
0
2 1 − c ℓ(α)
f ′ (z0 )
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
z
z0
f ′′
(ζ)dζ ,
f′
b
b
z
b
z1
α
b
z0
z2
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 )
− (z − z0 ) =
f ′ (z0 )
Z
z
(eg(ζ) − 1)dζ
z0
for z ∈ α. If z ∈ α is between z1 and z0 , the from (1)
′′ f
(z) ≤ c ,
f′ s
whence
Z
|g(z)| ≤
z
z0
Therefore
g(z)
|e
′′ Z 1 ℓ(α)
f
2
c
ℓ(α)
(ζ) |dζ| ≤
dσ = c log
.
f′
σ
2s
s
|g(z)|
− 1| ≤ e
−1≤
ℓ(α)
2s
c
−1,
and we obtain
Z z
f (z) − f (z0 )
− (z − z0 ) ≤
|eg(ζ) − 1||dζ
f ′ (z0 )
z0
c
Z 1 ℓ(α) 2
ℓ(α)
1 c
≤
− 1 dσ =
ℓ(α)
2σ
21−c
0
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.
66
III. SOME PROOFS OF THESE PROPERTIES
7.2. THEOREM. (Martio-Sarvas [20]). If D is uniform, then D
has the logarithmic derivative property for all d satisfying
1
.
2(a + 1)b
0<d<
Proof. Suppose that f is analytic with
′′ f ≤ dρD
f′ 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
(3)
min ℓ(αj ) ≤ bd(z, ∂D)
j=1,2
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
min(s, ℓ(α)s) ,
b
and hence
′′ f
2d
2bd
(z) ≤ dρD (z) ≤
≤
f′ d(z, ∂D)
min(s, ℓ(α) − s)
for z ∈ α. But now, with z0 as the midpoint of α,
f (z1 ) − f (z2 )
2bd
≤
−
(z
−
z
)
ℓ(α)
1
2
′
f (z0 )
1 − 2bd
≤
2abd
|z1 − z2 | < |z1 − z2 |
1 − 2bd
7. UNIFORM DOMAINS HAVE THE SCHWARZIAN DERIVATIVE PROPERTY67
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
Z t
w = u exp −
v(s)ds .
a
Then w is absolutely continuous if each closed subinterval of [1, b),
Z t
′
w = exp −
v(s)ds (u′ − uv) ≤ 0
a
a.e. in [a, b) and hence
w ≤ w(a) = 0 ,
u = w exp
Z
t
v(s)ds
a
≤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 < 21 . 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
68
III. SOME PROOFS OF THESE PROPERTIES
for z ∈ α, where s is the arclength of α from z1 to z, then
f (z1 ) − f (z2 )
≤ 2c ℓ(α) .
(4)
−
(z
−
z
)
1
2
1 − 2c
′
f (z0 )
Proof. By Lemma 7.1 it is sufficient to show that
′′ f
2c
(z) ≤
(5)
f ′ min(s, ℓ(α) − s)
α
b
b
b
z2
z
z0
b
z1
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)) ,
1
a = ℓ(α) .
2
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
2c
ϕ(s) =
,
ℓ(α) − s
′′
f
ψ(s) = ′ (z(s)) + ϕ(a) .
f
Then ϕ and ψ are sbsolutely continuous in each closed subinterval of
[a, b) with
2c(1 − c)
c
1
ϕ′ (s) − ϕ(s)2 =
≥
2
2
(ℓ(α) − 2)
(ℓ(s) − s)2
7. UNIFORM DOMAINS HAVE THE SCHWARZIAN DERIVATIVE PROPERTY69
and
′′ ′
f
1 f ′′ 2
1
2
′
ψ (s) − ψ(s) < (z) z (s) − ′ (z)
2
f′
2 f
f ′′ ′ 1 f ′′ 2 ≤ (z) −
(z) f′
2 f′
′
= |Sf (z)| ≤
c
(ℓ(α) − s)2
a.e. in [a, b). Thus
1
ψ − ϕ < (ψ 2 − ϕ2 ) = (ψ − ϕ)
2
′
′
ψ+ϕ
2
a.e. in [a, b) and we can apply Lemma 7.3 with u = ψ − ϕ and v =
1
(ψ
2
+ ϕ) to conclude that
′′ f
2c
(z) < ψ(s) ≤ ϕ(s) =
(6)
f′ min(s, ℓ(α) − s)
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 derivative 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
70
III. SOME PROOFS OF THESE PROPERTIES
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
(7)
min ℓ(αj ) ≤ bd(z, ∂D)
j=1,2
for each z ∈ α, where α1 , α2 are the components of α \ {z}. Then (7)
implies that
|Sf (z)| ≤ dρD (z)2 ≤
4d
4b2 d
≤
d(z, ∂D)
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 )
8b2 d
− (z1 − z2 ) ≤
ℓ(α)
f ′ (z0 )
| − 8b2 d
8ab2 d
≤
|z1 − z2 |
1 − 8b2 d
< |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 Möbius 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 CONNECTED
71
8. Schwarzian derivative property implies D is linearly
locally connected
8.1. The proof of this implication requires several preliminary lemmas.
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| ≤
(1)
4
,
c−1
where
h(z) = log
z − w1
.
z − w2
Proof. Let z1′ , z2′ 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 z1′ , z2′ in C and D, respectively.
β′
z1
b
z1′
b
α
z2
′
D2
D1
b
b
z2′
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 ,
72
III. SOME PROOFS OF THESE PROPERTIES
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 Kerékjártó’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].)
w2
b
z1
b
β
D1
b
α
b
z2
w1
Since D is simple connected, we can choose an analytic branch of
h(z) = log
z − w1
,
z − w2
8. SCHWARZIAN DERIVATIVE PROPERTY IMPLIES D IS LINEARLY LOCALLY CONNECTED
73
and
h(z1 ) − h(z2 ) =
Z
′
h (z)dz =
Z
′
h (z)dz −
γ
β
Z
h′ (z)dz
α
= 2πi(n(γ, w1 ) − n(γ, w2 )) −
Z
α
dz
+
z − w1
Z
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
(2)
|h(z1 ) − h(z2 ) − 2πni| ≤
Z
α
|dz|
+
|z − w1 |
Z
α
|dz|
.
|z − w2 |
Since α ⊂ B(z0 , r) and wj ∈ ∂B(z0 , cr),
Z
|dz|
ℓ(α)
2
≤
≤
|z − wj |
(c − 1)r
c−1
α
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
+ z0 .
z − 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].
74
III. SOME PROOFS OF THESE PROPERTIES
8.3. Finally we need a simple lower bound for the hyperbolic metric.
LEMMA. If w1 , w2 ∈ C \ D, then
ρD (z) ≥
|w1 − w2 |
1
2 |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
|w1 − w2 |
2d(f (z), ∂D ′ ) |z − w2 |2
≥
1 |w1 − w2 |
2|f (z)| |z − w2 |2
=
1
|w1 − w2 |
2 |z − w1 ||z − w2 |
as desired.
8.4. THEOREM. (Gehring [6]). If D has the Schwarzian derivative property, then D is linearly locally connected with
(3)
c = max
5
+ 1, 3
d
.
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
(4)
|h(z1 ) − h(z2 ) − 2πi| ≤
4
,
c−1
where
h(z) = log
z − w1
.
z − w2
Set
f (z) = ebh(z) ,
b=
2πi
.
h(z1 ) − h(z2 )
Then f is analytic in D with
1 − b2
Sf (z) =
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
≤d,
c−1
and thus f is injective in D since D was assumed to have the Schwarzian
derivative property. But
f (z1 )
= eb(h(z1 )−h(z2 )) = e2πi = 1
f (z2 )
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.
76
III. SOME PROOFS OF THESE PROPERTIES
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
ku1 k⋆ ≤ bku2 k⋆ ,
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 Möbius
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
ku′ k⋆ ≤ bkuk⋆ ,
where b = b(K). Next by Theorem II.6.3, u′ has an extension v ′ ∈
BMO(C) with
kv ′ k⋆ ≤ cku′ k⋆ ,
where c is an absolute constant. Then again by Lemma 9.1, v = v ′ ◦f ∈
BMO(C) with
kvk⋆ ≤ bkv ′ k⋆ .
10. BMO EXTENSION PROPERTY IMPLIES HYPERBOLIC BOUND PROPERTY
77
Thus v is the desired BMO extension of u and
kvk⋆ ≤ akuk⋆ ,
where a = b(K)2 c = 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 hyperbolic 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
r0
|uB1 − uB0 | ≤ c1 log + 1 kuk⋆ ,
r1
where c1 = e2 .
Proof. Suppose first that r0 ≤ er1 . Then
ZZ
1
|uB1 − uB0 | = (u − uB0 )dxdy m(B1 )
B1
ZZ
1
≤
|u − uB0 |dxdy
m(B1 )
B1
m(B0 ) 1
≤
m(B1 ) m(B0 )
2
ZZ
|u − uB0 |dxdy
B0
≤ e kuk⋆
r0
≤ c1 log + 1 kuk⋆ .
r1
78
III. SOME PROOFS OF THESE PROPERTIES
Suppose next that r0 > er1 and let k be the smallest integer for
which
r0 ≤ ek r1 .
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
X
|uBj − uBj+1 |
j=1
≤ e2 kkuk⋆
r0
2
≤ e
+ 1 kuk⋆
r1
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
|z1 − z2 |
|z1 − z2 |
+1
+ 1 + d2 kuk⋆ ,
|uB1 − uB2 | ≤ c2 log
r1
r2
where c2 = e2 and d2 = 2e2 .
10. BMO EXTENSION PROPERTY IMPLIES HYPERBOLIC BOUND PROPERTY
79
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
r0
|uB1 − uB0 | ≤ c1 log + 1 kuk⋆
r1
|z1 − z2 |
= c1 log
+ 1 + 1 kuk⋆ ,
r2
and similarly
|z1 − z2 |
+ 1 + 1 kuk⋆ ,
|uB2 − uB0 | ≤ c1 log
r2
where c1 = e2 . The desired conclusion now follows from the triangle
inequality.
As before, let hD (z1 , z2 ) denote the hyperbolic distance in D between z1 and z2 .
10.3. LEMMA. If B0 = B(z0 , r) ⊂ D, then
ZZ
hD (z, z0 )dxdy ≤ 2m(B0 ) .
B0
Proof. Suppose first that B0 = D = B. Then
hB (z, 0) = log
1 + |z|
1 − |z|
80
III. SOME PROOFS OF THESE PROPERTIES
and hence
ZZ
Z
hB (z, 0)dxdy =
2π
0
B
Z
1
log
0
1+r
rdrdθ = 2π = 2m(B) .
1−r
For the general case, the mapping
1
w = f (z) = (z − z0 )
r
maps B0 conformally onto B and since
hD (z, z0 ) ≤ hB0 (z, z0 )
for z ∈ B0 , we obtain
ZZ
ZZ
hD (z, z0 )dxdy ≤
hB0 (z, z0 )dxdy
B0
B0
=
ZZ
B
2
dz hB (w, 0) dudv
dw
= 2πr 2 = 2m(B0 )
as desired.
10.4. LEMA. If z1 ∈ D and if
u(z) = hD (z, z1 )
for z ∈ D, then
(1)
|u(z0 ) − uB0 | ≤ 2
for each disk B0 = B(z0 , r) ⊂ D. En particular, u ∈ BMO(D) with
kuk⋆ ≤ 4 .
10. BMO EXTENSION PROPERTY IMPLIES HYPERBOLIC BOUND PROPERTY
81
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,
1
|u(z0 ) − uB0 | ≤
m(B0 )
ZZ
1
m(B0 )
ZZ
≤
|u(z0 ) − u(z)|dxdy
B0
hD (z, z0 )dxdy ≤ 2
B0
which proves (1). Then since
|u(z) − uB0 | ≤ |u(z) − u(z0 )| + 2
we find that
1
m(B0 )
ZZ
B0
1
|u(z) − uB0 |dxdy ≤
m(B0 )
ZZ
|u(z) − u(z0 )|dxdy + 2 ≤ 4
B0
for each B0 ⊂ D. Thus u ∈ BMO(D) with kuk⋆ ≤ 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 kuk⋆ ≤ 4. Hence by
hypothesis u has an extension v ∈ BMO(C) with
kvk⋆ ≤ akuk⋆ ≤ 4a .
82
III. SOME PROOFS OF THESE PROPERTIES
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
|z1 − z2 |
|z1 − z2 |
− VB2 | ≤
c2 log
+1
+ 1 + d2 kvk⋆
r1
r2
≤ c3 jD (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 implications by establishing the following theorem.
11.1. THEOREN. (Gehring-Osgood [10]). If D has the hyperbolic bound property, then D has the hyperbolic segment property with
′
a = b = 4b′ e4b ,
b′ = 128c2e2d .
11. HYPERBOLIC BOUND PROPERTY IMPLIES HYPERBOLIC SEGMENT PROPERTY
83
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
d(z1 , ∂D) 1 hD (z1 , z2 ) ≥ log
.
2
d(z2 , ∂D) Proof. As in the proof of Lemma II.4.3
1
|z1 − z2 | + d(z1 , ∂D)
hD (z1 , z2 ) ≥
log
2
d(z1 , ∂D)
1
d(z2 , ∂D)
≥
log
.
2
d(z1 , ∂D)
Interchanging the roles of z1 , z2 gives
d(z1 , ∂D)
1
,
hD (z1 , z2 ) ≥ log
2
d(z2 , ∂D)
and the result follows.
11.3. PROOF of Theorem 11.1. By hypothesis there exist constants c, d such that
hD (z1 , z2 ) ≤ cjD (z1 , z2 ) + d
for all z1 , z2 ∈ D. By Lemma II.4.3,
1
jD (z1 , z2 ) ≤ hD (z1 , z2 )
4
for all z1 , z2 ∈ D, and hence we see from first letting z1 → z2 and then
letting z1 → ∂D that d ≥ 0 and c ≥ 41 .
84
III. SOME PROOFS OF THESE PROPERTIES
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 | ,
(1)
 min
ℓ(α ) ≤ ad(z, ∂D) ,
j=1,2
j
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 sup d(z, ∂D), 2|z1 − z2 | .
z∈α
We shall consider the cases
r < max d(zj , ∂D)
j=1,2
and
(2)
r ≥ max d(zj , ∂D)
j=1,2
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
1
d(z, ∂D) ≥ d(z1 , ∂D) ≥ |z1 − z2 | ,
2
and hence
hD (z1 , z2 ) ≤
Z
ρD (z)|dz| ≤
Z
β
≤
4|z1 − z2 |
≤2.
d(z1 , ∂D)
β
2
|dz|
d(z, ∂D)
11. HYPERBOLIC BOUND PROPERTY IMPLIES HYPERBOLIC SEGMENT PROPERTY
85
Since hD (z, z1 ) ≤ hD (z1 , z2 ) for z ∈ α, Lemma 11.2 yields the estimate
e−4 d(z1 , ∂D) ≤ d(z, ∂D) ≤ e4 d(z1 , ∂D)
for z ∈ α. Thus
4
ℓ(α ≤ e d(z1 , ∂D)
Z
|dz|
d(z, ∂D)
α
4
≤ 2e d(z1 , ∂D)hD (z1 , z2 )
≤ 8e4 |z1 − z2 | < a|z1 − z2 |
and for z ∈ α,
ℓ(α(z1 , z)) ≤ ℓ(α) ≤ 4e4 d(z1 , ∂D)
≤ 4e8 d(z, ∂D) ≤ bd(z, ∂D) .
This establishes (1) in the case where r < d(z1 , ∂D). Similarly we obtain (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 ≤ sup d(z∂ D) = d(z0 , ∂D) .
z∈α
For j = 1, 2 let mj be the largest integer for which
2mj d(zj , ∂D) ≤ r .
Let wj be the first point of α(zj , zo ) with
d(wj , ∂D) = 2mj d(zj , ∂D) .
86
III. SOME PROOFS OF THESE PROPERTIES
as we traverse α from zj towards z0 . Obviously
(3)
d(wj , ∂D) ≤ r < 2d(wj , ∂D) .
We shall show that for j = 1, 2,

 ℓ(α(zj , wj )) ≤ b′ d(wj , ∂D) ,
(4)
 ℓ(α(z , z)) ≤ b′ e2b′ d(z, ∂D) for z ∈ α(z , w ) .
j
j
j
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
(5)
d(ζk , ∂D) = 2k−1 d(z1 , ∂D)
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
Z
1
|dz| ≤ 4hD (ζk , ζk+1) ,
t≤2
d(z, ∂D)
αk
where αk = α(ζk , ζk+1). Now
jD (ζk , ζk+1) < 2 log
|ζk − ζk+1 |
+1
d(ζk , ∂D)
≤ 2 log(t + 1) ,
11. HYPERBOLIC BOUND PROPERTY IMPLIES HYPERBOLIC SEGMENT PROPERTY
87
whence
t
≤ hD (ζk , ζk+1) ≤ cjD (ζk , ζk+1) + d
4
≤ 2c log(e2d (t + 1))
1
≤ 2c(e2d (t + 1)) 2
1
since log x ≤ x 2 for x > 0. If t ≥ 1, then
1
t ≤ 8c(2te2d ) 2
or
t ≤ 128c2 e2d = b′ ,
(6)
and hence
1
hD (ζk , ζk+1) ≤ 2c(2b′ e2d ) 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 < log
d(ζk+1, ∂D)
≤ 2hd (z, ζk+1) ≤ 2hD (ζk , ζk+1) ,
d(z, ∂D)
and with (6) and (7) we conclude that
(8)

 ℓ(α(ζk , ζk+1) ≤ b′ d(ζk , ∂D) ,
 d(ζ , ∂D) ≤ e2b′ d(z, ∂D) for z ∈ α(ζ , ζ )
k+1
k k+1
88
III. SOME PROOFS OF THESE PROPERTIES
for k = 1, 2, . . . , m1 . Hence
m1
X
ℓ(α(z1 , w1) =
ℓ(α(ζk , ζk+1)
k=1
′
≤ b
m1
X
d(ζk , ∂D)
k=1
= 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
X
ℓ(α(ζi, ζi+1 )
i=1
′
≤ b
k
X
d(ζi , ∂D)
i=1
= 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)
(9)
 d(w , ∂D) ≤ e2b′ d(z, ∂D) ,
2
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 = sup d(z, ∂D)
z∈α
11. HYPERBOLIC BOUND PROPERTY IMPLIES HYPERBOLIC SEGMENT PROPERTY
89
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 )
1
≤ 2c(5b′ e2d ) 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
2b′
ℓ(α(w1 , w2 ) ≤ 2 d(w1 , ∂D)
Z
|dz|
d(z, ∂D)
α(w1 ,w2 )
′
≤ 2e2b d(w1 , ∂D)hD (w1 , w2 )
′
≤ 2b′ e2b d(w1 , ∂D)
90
III. SOME PROOFS OF THESE PROPERTIES
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
′
min ℓ(α(zj , z)) ≤ ℓ(α(zj , z)) ≤ b′ e2b d(z, ∂D) ≤ bd(z, ∂D)
j=1,2
by (4), or z ∈ α(w1 , w2 ) and
min ℓ(α(zj , z)) ≤
j=1,2
1
ℓ(α)
2
′
≤ 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 <
91
L(D)
.
L
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)
> 1.
L
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)
,z ∈ D
F (z) =
 (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′ -quasiisometry in C.
1
93
Finally, the case where ∞ ∈
/ ∂D can be reduced to the above case by
using auxiliary Möbius 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 sup max
h→0
|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
sup L(z) < L(D) ,
then the shape of the deformed D is roughly the same as that of the
original body.
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