APPLICATIONS OF GEOMETRIC MEASURE THEORY TO

COMPLEX AND QUASICONFORMAL ANALYSIS

By

Tyler Charles Bongers

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2018

ABSTRACT

APPLICATIONS OF GEOMETRIC MEASURE THEORY TO COMPLEX AND

QUASICONFORMAL ANALYSIS

By

Tyler Charles Bongers

There are many intersections between complex analysis, geometric measure theory, and

harmonic analysis; the interactions between these fields yield many important results and

applications.

In this work, we focus on two aspects of these connections: the regularity

theory of quasiconformal maps and the quantitative study of rectifiable sets.

Quasiconformal maps are orientation-preserving homeomorphisms that satisfy certain

distortion inequalities;

infinitesimally, they map circles to ellipses of uniformly bounded

eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible

and powerful generalization of conformal mappings. These maps arise naturally in the study

of elasticity, in complex dynamics, and in the analysis of partial differential equations. We

study the singularities of these maps; in particular, we consider the size and structure of the

sets where a quasiconformal map can exhibit given stretching and rotation behavior. We

improve the previously known results to give examples of stretching and rotation sets with

non-sigma-finite measure at the critical Hausdorff dimension. We further improve this to

give examples with positive Riesz capacity at the critical homogeneity, as well as positivity

of measure for a broad class of gauged Hausdorff measures at the critical dimension.

The local distortion properties of quasiconformal maps also give rise to a certain degree of

global regularity and H¨older continuity. We give new lower bounds for the H¨older continuity

of these maps, relating both the structure of the underlying partial differential equation for

the maps and the geometric distortion they can exhibit; the analysis is based on combining

the isoperimetric inequality with a study of the length of quasicircles. Furthermore, the

extremizers for H¨older continuity are characterized, and we give a natural application to

solutions of elliptic partial differential equations.

Finally, given a set in the plane, the average length of its projections in all directions

is called the Favard length of a set; it is closely related to the Buffon needle probability

of the set. This quantity measures the size and structure of a set, and is closely related

to metric and geometric properties of the set such as rectifiability, Hausdorff dimension,

and analytic capacity. We develop new geometrically motivated techniques for estimating

Favard length. We will give a new proof relating Hausdorff dimension to the decay rate of

the Favard length of neighborhoods of a set. We will also show that, for a large class of

self-similar one-dimensional sets, the sequence of Favard lengths of the generations of the set

is convex; this leads directly to lower bounds on Favard length for various fractal sets.

ACKNOWLEDGMENTS

Love. You can learn all the math in the ’Verse, but you take a boat in the air that you don’t

love, she’ll shake you off just as sure as the turning of the worlds. Love keeps her in the air

when she oughta fall down, tells you she’s hurtin’ ’fore she keens. Makes her a home.

- Malcolm Reynolds, Serenity

These years of study have been a wonderful and amazing experience, especially because of

the kindness and support of the people I have encountered. So many people have contributed

to my work, and I owe them all a great debt of gratitude; none of this would have been possi-

ble without them. First and foremost, I am thankful for my advisor, Ignacio Uriarte-Tuero.

Beginning with the little bit of geometric measure theory you showed us as qual students,

you have continually motivated, aided, and taught me. Your advice and counsel over the

years have been absolutely indispensable, and your decency and humility are inspiring.

Embarking on this journey as a graduate student would not have been possible without

the incredible education and opportunities I found at Colorado State University - Pueblo.

I am especially thankful to have met Darren Funk-Neubauer; taking your calculus class

and talking to you at math club was my original inspiration for changing my major to

math. Bruce Lundberg taught my first real and complex analysis classes and supervised

my undergraduate research, setting me on the path I’ve followed since; I learned so much

from you, and you were always happy to answer my questions. Your kindness, advice, and

hospitality whenever I am back in Pueblo have been wonderful. I had my first reading course

with James Louisell; seeing the beauty of advanced math for the first time had a huge impact

on me, and your advice to apply to Michigan State was invaluable. Janet Barnett’s historical

perspective and incredible passion for teaching inspired me as an educator. Jonathan Poritz’s

iv

reading course in number theory showed me some of the great beauty of complex analysis.

Learning chemistry from Rick Farrer was incredible; your energy and passion made for a

great experience.

I am also very thankful for the courses and discussions I was able to

have with Paul Chacon, Rick Kreminski, and Marta Wallin. My REU with Jorge Garcia at

CSUCI was an amazing experience, and I am happy I had the chance to work with Walter.

Finally, I owe a great deal to my time studying with Brena, Travis, Dan, and Gary; the

discussions we had and the help you gave me were incredibly useful, and your friendship

meant so much to me.

The faculty and staff of Michigan State have been a constant source of help for me over

the years.

I am deeply grateful to Tsveta Sendova, Andy Krause, Alec Drachman, and

Jane Zimmerman; without the support, advice, and opportunities you gave me, I would

not have developed into the teacher I am today. I am so happy that I had opportunities

to take courses with and learn from so many people, and I am especially thankful for the

work of my committee members Alexander Volberg, Dapeng Zhan, and Gabor Francsics.

Many other people have contributed to my growth as a graduate student, whether through

discussing math or careers or teaching: Jun Kitagawa, Russell Schwab, Sheldon Newhouse,

Jeanne Wald, James Dudziak, Vladimir Peller, Max Gilula, Jonathan Hall, Zhengfang Zhou,

and Shiv Karunakaran. I am also especially thankful for the help of the departmental staff,

especially Debie Lecato and Leslie Aitcheson.

This work was generously financially supported by several funding agencies. I am grateful

for the support of the Michigan State University College of Natural Science via fellowships,

as well as the National Science Foundation under grant DMS-1056965.

My time in graduate school would not have been possible without the support of my

friends here. Especially to Ben and Lauren - those times of Beggars, haiku, Halo, and

v

climbing will stay with me forever, and your friendship changed my life. My path into

analysis would not have been the same without Guillermo, Jos´e, and Paata; your passion for

math and all the tools and ideas you showed me over the years were really wonderful. I owe

a great debt to all those people that I studied with throughout those first years; especially

Chris and Tom. I am glad I had the chance to share an office with Emily; your skill and

advice in teaching has been so helpful to me. My time here has also been brightened by

so many other wonderful people, especially Hitesh, Sami, Charlotte, Rami, Rani, Christos,

Michael, and Jingyun. I am so deeply thankful for Reshma; your companionship and loving

support mean more to me than you can know and you’ve brought so much joy into my life.

Meri zindagi mein aaye ho aur aise aaye ho tum, jo ghul gayaa hai saanson mein vo geet

laaye ho tum.

Finally, and most importantly, I am grateful for my family. Mom and Dad, you made me

who I am today. You didn’t just teach me in school, you taught me how to live; it was only

with your support that I was able to pursue my dreams. Gram, you’ve always been there

for me and that’s meant the world. Grandpa, even from those times making paper airplanes

over pancakes, you inspired me to pursue science and math. Grandma Betty and Grandpa

Ken and Uncle Frank, your loving support has been deeply important to me.

Tyler Bongers

East Lansing, Michigan

27 March 2018

vi

TABLE OF CONTENTS

KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Regularity of Quasiconformal Maps . . . . . . . . . . . . . . . . . . . . . . .
1.2 Favard Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2

Stretching and Rotation Sets of Quasiconformal Maps . . . . .
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Dimension Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Dimension Greater than Zero . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3

Improved H¨older Continuity of Quasiconformal Maps . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2 Estimate of H¨older Exponent
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Extremizers for H¨older Continuity . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Applications to Elliptic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Geometric Bounds for Favard Length . . . . . . . . . . . . . . .
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Dimension and Favard Length . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Self-similar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
6
12

16
16
19
20
32

54
54
59
64
71

75
75
77
81

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

vii

KEY TO SYMBOLS

The following is a list of some of the notation used throughout this paper.

• ˆC, the Riemann sphere
• D, the unit disk in the complex plane
• B(z, r), the closed disk of radius r centered at z
• ∂α, the directional derivative in direction α
• Df , the differential matrix of f = u + iv,

Df (z) =

(cid:21)

(cid:20) ux uy

vx

vy

2(∂x − i∂y)

• |Df|, the operator norm of the matrix Df
• ∂z, the complex derivative 1
• ∂z, the z−derivative 1
• |E|, Lebesgue measure of E; could be one- or two-dimensional according to context
• Hd, the d−dimensional Hausdorff measure
• Cα(Ω), the set of locally α-H¨older continuous functions on a domain Ω; we will also

2(∂x + i∂y)

refer to these functions as belonging to a Lipschitz class

• Jf , the Jacobian of a function f
• Lp(Ω), the space of functions f such that |f|p is integrable on Ω.
• Lp
loc(Ω), the space of functions f such that f p is integrable on each compact subset of
Ω

• A (cid:46) B if there exists a constant C (possibly depending on parameters, but not A and

B) such that A ≤ CB

• A ∼ B if A (cid:46) B and B (cid:46) A; that is, there exists a universal C such that 1

C B ≤ A ≤

CB

• W 1,p(Ω), the space of functions in Lp(Ω) with one weak derivative also in Lp(Ω)
• W 1,p

loc(Ω) with one weak derivative also in Lp

loc (Ω), the space of functions in Lp

loc(Ω)

viii

Chapter 1

Introduction

Quasiconformal maps were first introduced by Gr¨otzsch [32] in order to study the following

problem:

Problem 1.0.1. Given rectangles R and S in the plane, find a homeomorphism f : R → S

which is as close to a conformal map as possible.

This problem is closely related to modules of ring domains and extremal length [3]. The

Riemann mapping theorem shows a conformal map between a connected domain Ω with

smooth boundary is uniquely specified by its values at three boundary points. Therefore,

once the images under a conformal map of three vertices of a rectangle are given, the fourth

is constrained. This suggests that the geometry of the rectangle is important (that is, the

ratio of the side lengths), and that a different class of functions is needed. These functions

ought to allow more distortion than conformal maps can, and so we must have a more flexible

definition than conformality allows.

This leads to a natural definition of quasiconformality. In order to discuss the infinitesimal

distortion, it is useful to impose some degree of regularity on the maps; as such, as will assume

that they lie in the Sobolev space W 1,2

loc of functions with one (locally) square integrable weak

derivative.

Definition 1.0.2. Let f : Ω → Ω(cid:48) be an orientation-preserving homeomorphism of two
domains in the complex plane, and assume that f ∈ W 1,2

loc (Ω). We say that f is K-

1

quasiconformal if at almost every z ∈ Ω, we have

|∂αf| ≤ K min

α

|∂αf|

max

α

(1.0.1)

where ∂α is a directional derivative in direction α. We say that f is quasiconformal if there
exists a K ∈ [1,∞) such that f is K-quasiconformal.

This inequality can be interpreted geometrically: a K-quasiconformal map sends infinites-

imal circles to infinitesimal ellipses with uniformly bounded eccentricity; it is precisely this

change between circles and ellipses that provides the flexibility to study Gr¨otzsch’s problem.

Since 1-quasiconformal maps are conformal, we now have a useful generalization of the

idea of conformality. The freedom gained by relaxing the definition of conformality makes

quasiconformal maps appropriate for many applications. They arise naturally in complex

dynamics (due to their connections with holomorphic motions), in elasticity problems (from

their geometric distortion properties), in fluid dynamics and partial differential equations (by

generalizing harmonicity), and many other areas. For a general overview of quasiconformal

maps, see the books by Astala, Iwaniec and Martin [8] or Ahlfors [4].

Many of the analytic properties of quasiconformal maps follow from a differential in-

equality equivalent to (1.0.1), as discussed throughout [8]. It is possible to rewrite (1.0.1) as
|Df (z)|2 ≤ KJf (z). This may be rewritten further using the Wirtinger derivatives fz and
fz; combining the facts that |Df| = |fz| + |fz| and Jf = |fz|2 − |fz|2, we find that

|fz| ≤ K − 1

K + 1

|fz|.

(1.0.2)

This inequality leads to a partial differential equation underlying any quasiconformal map.

2

We may define µ(z) = fz/fz whenever fz (cid:54)= 0 (and 0 otherwise, which can only occur on a

set of zero measure); therefore any K-quasiconformal map satisfies the Beltrami equation

∂zf = µ(z)∂zf with

(cid:107)µ(cid:107)L∞(C) ≤ K − 1

K + 1

.

(1.0.3)

The function µ is called the Beltrami coefficient, and represents the complex dilatation of

the map f . This representation allows the usage of the techniques of elliptic PDEs to study

quasiconformal maps.

Additionally, this equation gives a tool for constructing quasiconformal maps with given

dilatation. Given a compactly supported coefficient µ satisfying (cid:107)µ(cid:107)∞ ≤ K−1

K+1, there is a

(unique, once normalized appropriately) K-quasiconformal map satisfying ∂zf = µ∂zf ; this

is the measurable Riemann mapping theorem of Ahlfors and Bers [2]. The proof of this

unites harmonic analysis and complex analysis: the solution can be written explicitly with

the Beurling transform, which is a singular integral operator that intertwines ∂z and ∂z.

The regularity properties of quasiconformal maps (such as integrability of the derivatives)

are intimately related with the operator-theoretic properties of the Beurling transform and

are connected to important open problems about the Lp spaces where the Beurling transform

is bounded (see, e.g. [35]).

Many modern results about quasiconformal maps, including Astala’s area distortion the-

orem and the consequential distortion of Hausdorff dimension [7] (as well as the sharp inte-

grability and applications to elasticity of [13]) are proven by combining harmonic analysis

techniques with complex dynamics. Of fundamental importance is the fact that quasicon-

formal maps naturally embed within a global holomorphic motion (see, e.g.

[40, 58] and

Chapter 12 of [8] for some of the development of this theory). By using such an embedding,

3

we may introduce the techniques of classical complex analysis (such as growth estimates for

bounded analytic functions) to analyze the regularity of a map and the integrability of its

derivatives. This leads to many of the self-improvement properties of quasiconformal maps;

for example, a quasiconformal map lies within a Sobolev space with more integrability than

W 1,2

loc suggests.

In Chapters 2 and 3 of this work, we will consider some aspects of the regularity theory

of quasiconformal maps. In Chapter 2, we will consider the notion of stretching and rotation

for a quasiconformal map (giving a more precise notion of H¨older continuity and adapting

it to the argument of a map) from [10] and give examples of very irregular quasiconformal

maps. These maps will be irregular on a set of positive Riesz capacity and non-σ-finite

Hausdorff measure at the appropriate dimension, showing a fundamental limitation in the

techniques used to prove regularity. The constructions here are motivated by techniques of

geometric measure theory, and will involve highly non-self-similar Cantor sets. Chapter 3

will go on to demonstrate new bounds on the Lipschitz class that a quasiconformal map lies

in, based on its complex dilatation and the local geometric distortion of a map. We will also

give a classification of the quasiconformal extremizers for regularity.

Another aspect of the connection between complex analysis, harmonic analysis, and ge-
ometric measure theory is in the study of rectifiability. A set E ⊆ C is called rectifiable if
i=1 fi(R) has

there is a countable collection of Lipschitz maps fi : R → C such that E \(cid:83)∞

zero length. A set is called purely unrectifiable if it has no subset of positive length which

is rectifiable. Rectifiability is closely related to the geometry of a set: smooth curves are

rectifiable, while highly irregular sets such as Cantor sets are not. Morally, rectifiable sets

and Lipschitz maps play a role analogous to manifolds and smooth charts.

Unrectifiable sets can be detected using Favard length:

4

Definition 1.0.3. The Favard length of a set E ⊆ C is

(cid:90) 2π

0

Fav(E) =

|πθE| dθ

(1.0.4)

where πθ is projection onto a line through the origin at angle θ above the positive x−axis and
| · | is the length measure within this line.

Favard length gives a geometrically motivated measure of the size of a set, and carries

important metric and geometric information about the set. It is comparable to the Buffon

needle probability; that is, the likelihood that a long, thin needle dropped near a set will

interstect a given set. A theorem of Besicovitch [17] shows that a set with positive and finite

length is unrectifiable if and only if it has Favard length zero. Pure unrectifiability further

implies that a set cannot support a measure with positive and finite density with respect to

length [16, 54].

Favard length and unrectifiability are closely related to removability problems in complex

analysis. A classical question asks for which sets E in the complex plane there are no non-
constant, bounded, holomorphic functions on C \ E (in which case E is called removable).

It is a deep result that a compact subset of a rectifiable curve is removable if and only if it

has length zero; this follows from the L2-boundedness of the Cauchy transform on Lipschitz

graphs (which was first proved by Calder´on in [24] for graphs with small Lipschitz constant,

and later improved by many authors; see, for example, [26]).

As a consequence of Besicovitch’s theorem, the Favard lengths of the r-neighborhoods

of bounded unrectifiable sets of finite length must tend to zero as the scale r does. A more

quantitative understanding of this fact motivates the following problem:

Problem 1.0.4. How is the decay rate of the Favard length of the neighborhoods of a set

5

related to the metric and geometric properties of the set?

In Chapter 4, we consider several aspects of this problem. First, we give a new proof that

sufficiently fast decay leads to upper bounds on Hausdorff dimension. We also develop a new

geometrically motivated technique for estimating the Favard length decay for a large class of

self-similar unrectifiable sets in the plane; in particular, this work applies to the four-corner

Cantor set.

1.1 Regularity of Quasiconformal Maps

Before coming to the main problems considered in later chapters, we will outline the current

state of the art of regularity theory for quasiconformal maps. It is well known due to Mori
[46] that a K-quasiconformal map f is H¨older continuous with exponent α ≥ 1/K; that is,

for any compact set E in the domain Ω of f ,

|f (z) − f (w)|
|z − w|α < ∞.

sup
z,w∈E

Alternatively, this can be stated as the result that f lies in the Lipschitz class Cα(Ω).

The degree of smoothness of a quasiconformal map is closely related to the integrability

of fz and Jf . If the Jacobian Jf lies in an Lp
loc space, then it is quite direct to show that f
is H¨older continuous with exponent at least 1 − 1/p. To see this, note that applications of

quasisymmetry and Jensen’s inequality allow us to estimate that

|f (z + r) − f (z)|2 ∼K

(cid:90)
(cid:12)(cid:12)f(cid:0)B(z, r)(cid:1)(cid:12)(cid:12) = πr2
(cid:33)1/p
(cid:32)(cid:90)

J p
f

dA
πr2

Jf

dA
πr2

B(z,r)

≤ πr2

B(z,r)

6

(cid:46) r2−2/p(cid:107)Jf(cid:107)Lp(B(z,1))
(cid:46) r2−2/p

for r < 1. Taking a square root gives the desired result.

Following this idea, it is important to understand the optimal Sobolev class within which

quasiconformal maps lie. Improved integrability beyond W 1,2

loc was proven by Boyarski˘ı[23],

and the sharp Sobolev exponents were conjectured by Gehring and Reich [31]. Integrability
up to the sharp exponent 2K/(K − 1) was proved by Astala in [7] and the question of inte-

grability at the borderline was studied in [13]. In the converse direction, it was conjectured

(e.g. in [35]) that H¨older continuity above the exponent 1/K implies improved integrability
above exponent 2K/(K − 1); however, this was disproved by Koskela in [38] and further

studied by Clop in [25].

Furthermore, recalling that the Beltrami equation (1.0.3) may be solved rather explicitly

with the Beurling transform, many authors have studied the Lp bounds for the Beurling

transform

(Sϕ) (z) = − 1
π

(cid:90)

ϕ(τ )
(z − τ )2 dτ.

C

The integral here may be understood in the principal value sense, as the Beurling transform

is a Calder´on-Zygmund convolution operator with a nonintegrable kernel. The Lp properties
of quasiconformal maps are closely tied to the sharp Lp bounds of S; it is conjectured that
(cid:107)S(cid:107)Lp→Lp = max{p− 1, 1
p−1}. The correct asymptotic behavior (that the bound is O(p) as
p → ∞) is known, but the sharp bounds are still open; see, for example, [34, 35, 36, 12, 50,

29, 14] for some of the progression of this conjecture. Furthermore, the sharp bounds are

closely connected to questions about rank-one convexity and quasiconvexity [11].

7

Many recent results about regularity hinge upon the fact that quasiconformal maps can

be embedded within a holomorphic motion. Given a set A in the Riemann sphere, we say
that a map Φ : D × A → ˆC is a holomorphic motion if Φ is holomorphic in the first variable
for any fixed a ∈ A, injective in the second variable for any fixed λ, and Φ(0, a) = a for all
a ∈ A. A fundamental result is the following application of S(cid:32)lodkowski’s extended λ-lemma

[58] by Astala [7]:

Theorem 1.1.1. If Φ : D × A → ˆC is a holomorphic motion of A ⊂ C, then Φ has an
extension to ˜Φ : D × ˆC → ˆC such that each ˜Φλ is a K-quasiconformal self-homeomorphism
of ˆC with

K ≤ 1 + |λ|
1 − |λ|.
(cid:17)

Conversely, if f : C → C is K-quasiconformal, then there is a holomorphic motion Φ :
D × C → C such that f (z) = F

for all z ∈ C.

(cid:16) K−1

K+1 , z

The utility of this theorem is that it enables the usage of classical techniques from complex

and harmonic functions to give estimates for quasiconformal maps. The key idea of the proof

of the second statement is to embed a quasiconformal map f with Beltrami coefficient µ into

a flow via solving

∂zf λ = λ

µ

(cid:107)µ(cid:107)∞ ∂zf λ.

(1.1.1)

Varying λ allows us to transfer information about the functions f λ from the origin (λ = 0,

in which case f 0 is the identity) out to other points on the disk, using classical growth

estimates of bounded analytic functions on the disk. As a particular usage, consider the

following self-improvement property from [7], which has deep consequences:

Theorem 1.1.2. If f : C → C is K-quasiconformal, then f ∈ W 1,p

loc (C) for all p < 2K
K−1.

8

The key estimate leading to this result is based on embedding f within a holomorphic
flow f λ(z) as indicated in (1.1.1), noting that λ (cid:55)→ |(f λ)(cid:48)(z)|2 is harmonic for each fixed z,

and applying Harnack’s inequality in λ. In a similar vein, sharpening Harnack’s inequality

leads to results on quasicircles and quasilines [59, 53].

Complementary to these H¨older regularity results, it is also possible to study more refined

notions of smoothness. To this end, we define the stretching and rotation sets of a function,

first introduced in [10]:

Definition 1.1.3. Given z ∈ C, we say that f stretches with exponent α and rotates with
exponent γ at z if there exist scales rn → 0 with

log |f (z + rn) − f (z)|

log rn

lim
n→∞

= α and lim
n→∞

arg(f (z + rn) − f (z))
log |f (z + rn) − f (z)| = γ.

The set of points z where f stretches with exponent α and rotates with exponent γ simulta-

neously is denoted Ef (α, γ).

Here, the argument is understood in terms of the winding number:

it is the number
of times that the image of the ray [z + rn → ∞) wraps around f (z) (up to a small, but

fixed, term depending on the choice of branch of the argument). Note that by applying the
classical H¨older regularity results to f and f−1, we are only interested in α ∈ [K−1, K].

Given Definition 1.1.3, we can now state the main question considered in Chapter 2 of

this work:

Problem 1.1.4. How large can the stretching and rotation set Ef (α, γ) of a quasiconformal

map be?

The question is necessarily vague, as there are many (not necessarily compatible) notions

9

of the size of a set. Motivated by [5], we will address several different interpretations of size

in Chapter 2.

This question was introduced and studied by Astala, Iwaniec, Prause, and Saksman in

[10], in which they gave an upper bound on the size in terms of Hausdorff dimension (as a

function of α, γ and K); they also showed this bound to be optimal at the level of dimension.

The main idea in giving the upper bound was to use a holomorphic motion first studied in
[9] to give efficient bounds on the integrability of |f β
z | for complex exponents β, while the

lower bound was proven by a direct construction. Hitruhin sharpened this lower bound in

[33] to the level of Hausdorff measure by constructing quasiconformal maps whose stretching

and rotation sets have positive and finite Hausdorff measure at the critical dimension.

We improve these results by constructing far larger Cantor sets that are stretching and

rotation sets; this is done not only in the sense of Hausdorff measure (where we extend the

previous results to both the non-σ-finite setting but also gauged Hausdorff measures), but

also in terms of Riesz capacity. By analogy with many results in area distortion (e.g. the

studies of distortion of Hausdorff measure and Riesz capacity of [39, 5]), it was expected

that the stretching and rotation set could not be larger than σ-finite with respect to the

Hausdorff measure at the critical dimension. Thus our result is rather surprising and shows

a limitation of the techniques used to prove regularity. The details of this work are the

content of Chapter 2.

The constructions in Chapter 2 are heavily based on summing or modifying maps like
z (cid:55)→ z|z|α(1+iγ). This is the prototypical map for studying stretches and rotations, since
it can be easily computed to have Ef (α, γ) = {0} and Ef (1, 0) = C \ {0}; it also has a

simple Beltrami coefficient. As a complementary question to the work in Chapter 2, we are

interested in determining which maps can exhibit the worst-case stretching behavior:

10

Problem 1.1.5. Which K-quasiconformal maps are no better than 1/K-H¨older continuous,

and what can we say about the structure of such maps? In an appropriate sense, do they
need to be “close” to the pure radial stretch z|z|1/K−1?

We address this question in Chapter 3.

It turns out that the appropriate notion of

“closeness” is in terms of the Beltrami coefficient. Using the isoperimetric inequality together

with computation of the length of quasicircles, we are able to give a new formula for the

H¨older exponent of the solutions to ∂zf = µ∂zf . The new idea here is to vary a technique

of Morrey [47]: we may write

(cid:90)

Dt

Jf dL2 = |f (Dt)| =

|f (Dt)|
H1(St)2H1(St)2.

(1.1.2)

The first term is bounded above by the isoperimetric inequality, while the second term can

be explicitly computed in terms of the Jacobian Jf and the Beltrami coefficient µ. Having
replaced the integral over Dt with an integral over its boundary St, we can find an inequality

Jf in terms of its t-derivative, leading eventually to pointwise estimates in t. The

for (cid:82)Dt

details of this are provided in Section 2 of Chapter 3.

As a further application of these ideas, we gain substantial new information about the

H¨older continuity extremizers. The key realization is that if a K-quasiconformal map is no

more regular than C1/K , all the inequalities arising from (1.1.2) must be sharp. We may use

this to not only constrain the structure of the Beltrami coefficient, but also understand how

much distortion such a map can introduce to small circles. This gives an answer to Problem

1.1.5: the Beltrami coefficient of such a map must be a small perturbation of the coefficient
of z|z|1/K−1. This is explored in detail in Section 3 of Chapter 3.

The techniques introduced in Chapter 3 are not limited to quasiconformal maps, but

11

have consequences for solutions to elliptic partial differential equations as well. Following

the classical correspondence

∂zf = 0 ⇐⇒ ∆u = 0

between the Cauchy-Riemann equations (encoding analyticity of a function f ) and Laplace’s

equation (encoding harmonicity of the real part u = Re f ), there is a generalization to

∂zf = µ∂zf ⇐⇒ div(A∇u) = 0

(1.1.3)

between quasiconformality and solving a divergence-form elliptic PDE where the matrix A

is positive definite, symmetric, measurable, has determinant 1, and satisfies an ellipticity

bound.

Regularity theory of elliptic PDEs has a rich history. Explicit bounds for the H¨older

exponents were given by Morrey [47], improved by Piccinini and Spagnolo [52], and improved

further by Ricciardi [56, 57]. Following these results, the correspondence (1.1.3) is explored

in detail in Section 4 of Chapter 3. The main result is to give estimates of the Lipschitz

class that solutions lie in. Furthermore, the Stoilow factorization theorem allows us to give

a classification of the H¨older continuity extremizers. Finally, we state a generalization to a

large class of non-linear elliptic operators.

1.2 Favard Length

Before discussing the new techniques and results introduced in Chapter 4, we will give a

brief background and motivation of Favard length. We are interested in an old problem of

Painlev´e: give a metric or geometric characterization of those sets in the complex plane which

12

are removable for analytic functions. It is well known that sets of length zero are removable,

and that sets of Hausdorff dimension greater than 1 are not; as such, dimension 1 is the

critical point where the problem is most interesting. Vitushkin conjectured that a compact

set with positive and finite length is removable if and only if it is purely unrectifiable; this

was eventually proved by David in [27].

Just as Favard length gives a quantitative measurement associated to rectifiability, it is

useful to introduce the analytic capacity as a quantitative object associated to removability.
Given a compact E ⊆ C, its analytic capacity is

γ(E) = sup(cid:8)|f(cid:48)(∞)| : f is analytic on C \ E and (cid:107)f(cid:107)∞ ≤ 1(cid:9)

(1.2.1)

where f(cid:48)(∞) = limz→∞ z(f (z) − f (∞)).

It is immediate that a set is removable if and

only if it has analytic capacity zero, and the capacity gives another measure of the size and

distribution of sets. Connecting analytic capacity zero and Favard length zero would give

a great deal of information about Painlev´e’s problem. In particular, we are interested in
whether γ(E) ∼ Fav(E) (or similar inequalities) can hold for a large class of sets E.

In [30], Garnett used the four-corner Cantor set to give an example of a removable set
with positive length. The generations of this set are denoted Kn, and each consist of 4n
squares of side-length 4−n; see Chapter 4, Section 3 for the full definition. This showed that

the geometry of the underlying set is an important factor in removability, and not just its

metric properties. Later, Calder´on’s proof [24] of the weak (1, 1) and strong (p, p) bounds

of the Cauchy operator on Lipschitz graphs of small Lipschitz constant (and the extensions

of this result to arbitrary Lipschitz constant, such as in [26]) was used to prove that sets of

positive length contained in a rectifiable curve are non-removable; this is proved by a direct

13

construction of non-constant bounded holomorphic functions via the Cauchy transform. It is

worth mentioning that there are now numerous proofs with a variety of different techniques

for the boundedness of the Cauchy transform (e.g. [26, 6] and many others).

Mattila [43] showed that for any f : R2 → R2 with two continuous derivatives, there

is a set E such that Fav(E) = 0 < Fav(f E). Thus zero Favard length is not a conformal
invariant, while zero analytic capacity clearly is; this shows that Fav(E) ∼ γ(E) must be

false. Jones and Murai [37] gave an example of a set with positive capacity and zero Favard
length, so γ(E) (cid:46) Fav(E) cannot hold. The other inequality is still open, and we state it as

a problem:

Problem 1.2.1. Given a compact set E, is it true that Fav(E) (cid:46) γ(E)?

Cantor sets are frequently used to test this problem, since they have many geometric

properties that simplify estimates for analytic capacity. The sharp estimate of analytic

capacity for a broad class of Cantor sets (including the four-corner Cantor set) was given by

Tolsa [61] in terms of a quantity called positive capacity (later shown to be comparable to

analytic capacity in [62]). Mateu, Tolsa, and Verdera [42] also classified the Cantor sets of

analytic capacity zero and gave upper bounds on analytic capacity in general. These papers

generally rely on finding estimates of the curvature of measures supported on an unrectifiable
set. Combining the results, it is known that the n-th generation Kn of the four-corner Cantor
set has analytic capacity γ(Kn) ∼ n−1/2.

We are also interested in estimating the Favard lengths of the generations of Cantor

sets.
It was shown by Mattila [44] that the Favard length of the r-neighborhood of any
1-dimensional set decays no faster than (log 1/r)−1, regardless of the geometric structure;
in particular, this implies that Fav(Kn) (cid:38) n−1. Upper bounds on Fav(Kn) were found by

14

Murai [48] and Peres and Solomyak [51]; this latter paper also showed that the expected
Favard length of the n-th generation of certain random cantor sets is of order n−1. Tao [60]

gave a quantitative version of the Besicovitch projection theorem that gave upper bounds

on the Favard lengths of a broad class of fractal sets.

More recently, there have been a number of improved bounds for the four-corner Cantor

set, as well as other fractal sets with similar arithmetic and product structure. Bateman and
Volberg [15] used a square-counting technique to improve the lower bound to Fav(Kn) (cid:38)

(ln n)/n. Nazarov, Peres, and Volberg [49] used Fourier analytic techniques based on the
structure of Kn to improve the upper bound to Fav(Kn) (cid:46) nδ−1/6 for any δ > 0. Bond,

(cid:32)Laba and Volberg [19] generalized this technique to a large class of self-similar product

sets. These are currently the best-known bounds; as such, it is not possible to say whether
Fav(Kn) (cid:46) γ(Kn).

In Chapter 4, we introduce new techniques for estimating Favard length. In Section 2,

we use a new geometrically motivated argument to study sets with Hausdorff dimension less

than 1. This new, simpler technique recovers a result of [44] to compute the power law for

decay of Favard length in terms of the Hausdorff dimension. In Section 3, we develop a new

method for studying self-similar sets. In particular, we will show that the sequence of Favard

lengths of the generations of a 1-dimensional set generated by similitudes is convex; this has

powerful consequences in terms of the asymptotic behavior of the Favard length.

15

Chapter 2

Stretching and Rotation Sets of

Quasiconformal Maps

The work in this chapter first appeared in [21].

2.1

Introduction

We say that a map f ∈ W 1,2
loc (C) is K-quasiconformal if it is an orientation preserving
homeomorphism and satisfies the distortion inequality maxα |∂αf| ≤ K minα |∂αf| almost

everywhere, where ∂α is a directional derivative. Geometrically, f maps infinitesimal circles

to infinitesimal ellipses; these can be viewed as perturbations of conformal maps, which are

1-quasiconformal. Such maps are also realized as solutions to the Beltrami equation

∂zf = µ(z)∂zf

where the coefficient µ satisfies (cid:107)µ(cid:107)∞ ≤ K−1

K+1 < 1.

We are interested in geometric distortion properties of these maps. Given z ∈ C, we

say that f stretches with exponent α and rotates with exponent γ at z if there exist scales

16

rn → 0 with

log |f (z + rn) − f (z)|

log rn

lim
n→∞

= α and lim
n→∞

arg(f (z + rn) − f (z))
log |f (z + rn) − f (z)| = γ.

Here, the argument is interpreted as the total angular change with respect to f (z) along the
image of the ray [z + rn,∞); see section 2 or [10] for the full definitions.

A classical theorem of Mori (see [46]) states that every K-quasiconformal map is locally
1/K-H¨older continuous, which implies that 1/K ≤ α ≤ K. In the more recent [10], Astala,

Iwaniec, Prause and Saksman improved this substantially to give the exact range of both

stretching and rotation exponents which can be realized by a K-quasiconformal map f : if
we let BK ⊂ C be the open disk centered at 1
K ), then f can
stretch like α and rotate like γ if and only if α(1+iγ) ∈ BK . As a particular application, this

K ) with radius 1

2(K + 1

2(K − 1

gives the precise rotation behavior that a bilipschitz map can exhibit. Moreover, this work

gave the precise multifractal spectrum FK (α, γ) - that is, the maximal possible Hausdorff

dimension of the simultaneous stretching and rotation set of such maps; the sharp result was

the following theorem.

Theorem 2.1.1. If f : C → C is a K-quasiconformal mapping with K > 1, and α(1 + iγ) ∈

BK , then the Hausdorff dimension of the stretching and rotation set Ef of f is bounded by

dimH Ef ≤ FK (α, γ) := 1 + α − K + 1
K − 1

and this result is sharp at the level of dimension.

(cid:115)

(1 − α)2 +

4Kα2γ2
(K + 1)2

The techniques used to prove this theorem mainly involved improved integrability esti-

mates for complex powers of the derivatives of f . There is very substantial overlap with the

17

techniques used in studying area distortion, and as such it is a natural conjecture that the

Hausdorff measure at the critical dimension should be finite, in analogy with Theorem 1.2

in [5]. However, we will show that this is not the case.

In the direction of lower bounds, that paper gives constructions to attain all dimensions

below the bound FK (α, γ). Hitruhin improved this in [33] to give examples of quasiconformal

maps whose stretching and rotation sets have positive and finite Hausdorff measure at the

critical dimension. That paper used a Cantor set construction from [63] to prove this; the

work gives a construction of a quasiconformal map whose distortion of a family of disks is

completely understood.

In this work, we improve the above results beyond finite measure, showing that the

stretching and rotation set can actually have positive measure with respect to many gauged
Hausdorff measures which are much smaller than the typical Hd. Our main theorem is

Theorem 2.1.2. Let Λ be a gauge function of the form Λ(r) = rdh(r) where h is a non-
negative, nondecreasing function satisfying the growth condition h(r)/h(s) ≥ C
(r/s)
 for
all 
 > 0 and 0 < r ≤ s sufficiently small. Select parameters α < 1 and γ such that d > 0

is the maximum allowed Hausdorff dimension of the corresponding stretching and rotation

set; that is, d = FK (α, γ). Then there is a K-quasiconformal mapping f and a set E with
HΛ(E) > 0 such that E is the stretching and rotation set for f .

We have a generalization to stretching exponents α > 1 under an additional constraint on

the gauge function Λ. Furthermore, as a corollary, there is an application to an interesting

class of gauge functions:

Corollary 2.1.3. There are positive measure stretching and rotation sets associated to the

gauges Λ(r) = rd(cid:16)

for every β > 0.

18

(cid:17)−β

log 1
r

As an interesting second corollary, we can extend this to positive Riesz capacity ˙Cβ,p

for all parameter choices (β, p) with homogeneity matching the dimension d. Again, this

is a surprising result: by analogy with the work of [5], it is reasonable to expect that at

the critical homogeneity, the ˙Cβ,p Riesz capacity would be zero for some range of indices.

However, this conjecture is also false, and we have the following corollary:

Corollary 2.1.4. Fix any parameter τ = α(1 + iγ) ∈ BK with α ∈ (1/K, 1), and a pair
(β, p) with 1 < p < ∞ and 2 − βp = FK (α, γ). There is a K-quasiconformal map f and a

set E such that f stretches with exponent α and rotates with exponent γ at every point in
E, and E has positive (β, p)−Riesz capacity.

The paper is organized as follows.

In Section 2, we give a brief recollection of some

notions involving quasiconformal mappings, and a more precise definition of the rotation.

In Section 3, we analyze the Hausdorff dimension zero case; our main results here will be

a construction of a quasiconformal mapping that stretches on any given countable set, as
well as a first construction of a map with Hd non-σ-finite stretching and rotation set, where

d = FK (α, γ). In Section 4, we will prove the main theorem and indicate applications to

particular gauges and Riesz capacities.

2.2 Prerequisites

Following [10], given a quasiconformal map f , we will say that it stretches like α at a point

z0 if there exists a sequence of scales rn decreasing to zero for which

lim
n→∞

log |f (z0 + rn) − f (z0)|

log rn

19

= α.

Rotation is similar, but a little more subtle. For a principal quasiconformal map f , that is
as |z| → ∞, we can
a map whose domain and codomain are both C and f (z) = z + O

(cid:17)

(cid:16) 1

z

select a branch of log f . We can find a corresponding choice of argument, and using this we
can understand arg(f (z0 + r) − f (z0)) as the total rotation around the point f (z0) of the
image of the ray [z0 + r,∞) under f . Using this interpretation, we will say that f rotates

like γ at a point z0 if

arg(f (z0 + rn) − f (z0))
log |f (z0 + rn) − f (z0)| = γ

lim
n→∞

for a sequence rn → 0. It is worth noting that the stretch and rotation at a point are not
uniquely defined; it is possible that a quasiconformal map stretches like α and α(cid:48) at a point
with α (cid:54)= α(cid:48) (or rotates with two different behaviors); this is due to the dependence on the

particular choice of sequence rn.

Given a quasiconformal mapping f , we set Ef (α, γ) to be its simultaneous rotation-like-

γ and stretching-like-α set; when it is clear from context, this will be abbreviated as Ef .

Finally, we have the multifractal spectrum

FK (α, γ) = sup(cid:8)dimH(Ef (α, γ)) : f is K-quasiconformal(cid:9)

where this FK (α, γ) is that of Theorem 2.1.1, as proved in [10].

2.3 Dimension Zero

There are two complementary senses in which we will improve upon results with positive

measure. The first is to give particular examples of stretching and rotation sets with very

large measure, perhaps uncountable or having positive measure with respect to some gauged

20

Hausdorff measure. The second is to give a broader class of examples of sets, in particular

including that every countable set can appear as a stretching set. Before the constructions,

we will start with a useful lemma that will allow us to simplify some of the subsequent

computations involving stretching. Although it was not stated as a separate result, the

computation here is more or less contained in [33].

Lemma 2.3.1. Suppose that z is a point with the following property: there is a sequence of
balls Bn = B(zn, rn) such that z ∈ Bn for each n, rn → 0, and

log |f (Bn)|
log |Bn| = α + 
n

with error 
n → 0. Then f stretches like α at z.

The utility of this lemma is that we can transfer stretching information at a central point

not only to points at difference r away, but to all nearby points. As an idea of an application,

it is frequently possible to get stretching at exponent α on an entire Cantor set just by taking

a quasiconformal map that stretches like α at each of the points used at successive scales to

generate the Cantor set.

Proof. Fix n. We can rotate using quasisymmetry. Fix a point w ∈ ∂D(zn, rn) that is

equidistant with z and zn (e.g. an intersection point of the perpendicular bisector of zzn

with the boundary of the circle). Then

log |f (z + rn) − f (z)| = log |f (z + rneiθ) − f (z)| + CK

= log |f (w) − f (z)| + CK
= log |f (w) − f (w + |z − w|eiν)| + CK

21

K

= log |f (w) − f (zn)| + C(cid:48)
= log |f (zn + |w − zn|) − f (zn)| + C(cid:48)(cid:48)
= log |f (zn + rn) − f (zn)| + C(cid:48)(cid:48)

K

K

given appropriate choices of ν and θ; the constants CK , C(cid:48)
K are unimportant except
in that they are bounded in terms of K only. Dividing by log rn and letting n → ∞, we find

K and C(cid:48)(cid:48)

that

log |f (z + rn) − f (z)|

log rn

=

=

= α + 
n +

1

1

K

log rn

log |f (zn + rn)| + C(cid:48)(cid:48)
2 log |f (Bn)| + C(cid:48)(cid:48)(cid:48)
K
2 log |Bn| − 1
2 log π
2C(cid:48)(cid:48)(cid:48)
log |Bn| + o(1)

K

following a final application of quasisymmetry. The result follows.

Note that we can replace the measures of the balls with their radii. We can actually
extract a little more information: if C is a fixed constant, and z is a point for which |z−zn| ≤

Crn, the same result holds. To see this, notice that there is a polygonal path connecting z

to zn where each segment has length rn, and the number of segments is uniformly bounded

by a constant only involving C. Repeating the double-rotation idea of the proof, we now

lose a constant several times (but a uniformly bounded number), which does not impact the

result.

Moreover, the same result holds for rotations:

Lemma 2.3.2. Suppose that z is a point with the following property: there is a sequence of

22

balls Bn = B(zn, rn) such that z ∈ Bn for each n, rn → 0, and

arg(f (zn + rn) − f (z))
log |f (zn + rn) − f (zn)| = γ + 
n

with error 
n → 0. Then f rotates like γ at z.

Proof. We will only give a brief description of the technique of the proof, as it is rather
similar to the previous one. Fix n, and consider the rays [zn,∞) and [z,∞) parallel to the

positive x-axis. By a rotation, which changes the cumulate argument by an O(1) factor, we
may assume that z lies on the ray [zn,∞). Now reusing the double rotation argument of the

previous lemma, the denominators of the rotation are the same up to an O(1) error, which

is enough.

Our first result will be a large dimension zero set which has the most extreme stretching

and rotation allowed by the multifractal spectrum bounds of [10]. The construction will

be a sort of Cantor set built from disks, within which we can explicitly keep track of the

stretching and rotation.

Theorem 2.3.3. For any pair (α, γ) for which z|z|α(1+iγ)−1 is K-quasiconformal, there is a

K-quasiconformal map f and an uncountable set Ef for which f stretches like α and rotates

like γ at every point in Ef .

Proof. Start with B0,1 = D and f (z) = z on all of C. Now assume that Bn,i has been defined

and has radius rn, and that there are complex numbers βn,i, wn,i for which f (z) = βn,iz+wn,i

in a neighborhood of Bn,i. Choose a number ˜rn (which will be substantially smaller than

rn); take a concentric ball An,i within Bn,i of radius ˜rn, and place two disjoint balls Bn+1,j

within An,i each with radius 1

4 ˜rn. We now modify the construction of f ; without loss of

23

generality, we may assume that wn,i = 0 and f (wni) = 0 - otherwise, pre- and post-compose

with an appropriate translation (this only simplifies the notation). Now modify the definition

of f to become



f (z) =

(cid:12)(cid:12)(cid:12) z
(cid:16) ˜rn

rn

rn

(cid:12)(cid:12)(cid:12)α(1+iγ)−1
(cid:17)α−1

eiθz

βn,iz

βn,iz

βn,i

near Bn,i but in Bc
n,i
z ∈ Bn,i \ An,i
z ∈ An,i

where eiθ is chosen so that f is continuous across ∂An,i, and

(cid:19)α−1

(cid:18) ˜rn

rn

βn+1,j = βn,i

eiθ.

Note that the original function f is injective; on the other hand, the construction only

carries out a local modification by stretching and rotating the ball An,i, and remains injec-

tive. Moreover, the limiting function of the construction is K-quasiconformal as long as the

parameters (α, γ) are chosen to allow this. In particular, following [33], we can choose α, γ

to be any pair for which FK (α, γ) = 0.

We just need to compute the change in argument induced by crossing the annulus between

Bn,i and An,i, find the corresponding stretching on scale ˜rn with respect to the center point,

and choose the sequence of radii carefully. Since

(cid:12)(cid:12)(cid:12)z

r

(cid:12)(cid:12)(cid:12)α(1+iγ)

(cid:12)(cid:12)(cid:12) z

r

(cid:12)(cid:12)(cid:12)α

=

eiαγ log |z/r|

it is immediate that the change in argument across the annulus is αγ log ˜rn
rn

+ O(1). The

24

numerator of the stretching with respect to the center point of Bn,i on scale ˜rn is

(cid:19)α−1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)βn,i

(cid:18) ˜rn

rn

log

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = α log ˜rn + log |βn,i| − (α − 1) log rn.

eiθ ˜rn

As a consequence, we see that the overall stretching of f with respect to the center point is

log |f (˜rn) − f (0)|

log ˜rn

= α +

log |βn,i|
log ˜rn

− (α − 1)

log rn
log ˜rn

while the overall rotation is

arg (f (˜rn) − f (0))
log |f (˜rn) − f (0)| =

αγ log ˜rn − αγ log rn + O(1)

α log ˜rn + log |βn,i| − (α − 1) log rn

.

(2.3.1)

(2.3.2)

Each βn,i has the same modulus βn; the only potential difference is the exact rotation. We

can easily compute this number from its definition, finding that

(cid:35)α−1

(cid:34)n−1(cid:89)

k=0

˜rk
rk

βn =

As a consequence, we have that

log |βn,i|
log ˜rn

= (α − 1)

n−1(cid:88)

k=0

log ˜rk − log rk

log ˜rn

Because ˜rk < rk < 1, we can estimate all the terms roughly by the final term (provided that

˜rk/rk is decreasing, which it will be), finding

(cid:12)(cid:12)(cid:12)(cid:12)log |βn,i|

log ˜rn

(cid:12)(cid:12)(cid:12)(cid:12) ≤ 2(1 − α)n

log ˜rn−1
log ˜rn

(2.3.3)

25

We have already defined rk+1 = 1

4 ˜rk, and now we make the selection that

˜rk = rk2

k

and the above error estimate (2.3.3) tends to zero. As an immediate consequence of this

selection, we have that the stretching tends towards α, while the rotation tends towards γ.

This completes the proof.

Now we will go in the other direction, finding that any countable set is a stretching

set with the worst possible exponent. As a nice application, this shows that an interesting

multifractal spectrum bound in the style of [10] is not possible for Minkowski dimension; see,

e.g. Chapter 5 of [45] for constructions of countable sets with large Minkowski dimension.

There are countable sets whose lower Minkowski dimension is arbitrarily close to 2, and

these can exhibit stretching of exponent 1/K at every point. The key idea here will be that

sums of radial stretches are quasiconformal maps; in general, it is quite rare for a sum of

quasiconformal maps to be quasiconformal (let alone injective). This idea will not work for

rotations.

Note, however, that this contrasts starkly with the possibilities in other dimensions. For

example, a one dimensional set containing a smooth curve or a segment can never be a

stretching set for an exponent other than 1. To see this, consider the fact that if f stretches

with exponent α > 1 at every point within a line segment, f is flat at every point within

that line. Explicitly, if f is viewed as a single-variable function on this line, it is (classically)
differentiable with derivative zero at every point, hence non-injective. Considering f−1 shows

why f cannot stretch with exponent α < 1.

Theorem 2.3.4. Given a countable set Λ ⊆ D, there is a K-quasiconformal mapping f such

26

that for each λ ∈ Λ there is a sequence of scales rm decreasing to zero for which

log |f (λ + rm) − f (λ)|

log rm

=

1
K

.

lim
m→∞

Recall that 1/K is the most extreme possible exponent due to [10].

Proof. Let us begin with the radial stretches

fλ(z) = (z − λ)|z − λ| 1

K −1 + λ

when |z − λ| ≤ 1, and the identity otherwise. These are K-quasiconformal mappings that

satisfy a Beltrami equation with coefficient µλn. Moreover, their derivatives ∂zfλ have

constant sign where they are defined. To wit,

(cid:18) 1

(cid:19)

|z − λ| 1

K −1

∂zfλ =

+

1
2

2K

within the disk λ + D, and 1 outside. It follows that if we sum such solutions, we can still

have a solution to a Beltrami equation; in particular, assuming that derivatives and sums

commute in this context, we have

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∂z

∞(cid:88)

n=1

1
2n fλn(z)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

1
2n ∂zfλn(z)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞(cid:88)
∞(cid:88)

n=1

n=1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

=

≤

=

n=1

K − 1
K + 1

27

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

1

1
2n µλn(z)∂zfλn(z)
2n(cid:107)µλn(cid:107)∞|∂zfλn(z)|
∞(cid:88)

1
2n ∂zfλn(z)

n=1

K − 1
K + 1

∂z

=

∞(cid:88)

n=1

1
2n fλn(z)

Now given a countable set, we can therefore define a function

∞(cid:88)

n=1

f (z) =

1
2n fλn(z).

(2.3.4)

Modulo swapping the derivatives and the sum, we have shown that f satisfies a Beltrami
equation with coefficient bounded by (K−1)/(K +1). This condition will follow very quickly
from the dominated convergence theorem. Fix a test function ϕ ∈ C∞
0 (C) and integrate by

parts:

(cid:90)

(cid:90)

f ∂xϕ =

N(cid:88)
(cid:90) 1
N(cid:88)
where we have used the fact that |f (z)| ≤(cid:80)
2n|fλn(z)| ≤(cid:80)

1
2n fλn∂xϕ

2n fλn∂xϕ

= lim
n→∞

lim
n→∞

n=1

n=1

2n (|λ| + 1 + |z|) ≤ 2 + |z|
from the estimate |fλ(z)| ≤ |z − λ|1/K + |λ| ≤ 2 within the disk λ + D, and |z| otherwise.

n

n

1

1

Thus f is bounded on the support of ϕ, and the above follows. Now integrate by parts in

each summand to get

(cid:90)

f ∂xϕ = − lim
n→∞

(cid:90) N(cid:88)

n=1

1
2n ∂xfλnϕ

Now ϕ is bounded on its support, and |∂xfλn| (cid:46)K |z − λn|1/K−1 is locally integrable (as
1/K − 1 > −1), and summing in n does not change this. Taking(cid:80)∞
2n|z − λn|1/K−1|ϕ|

n=1

1

28

as our dominating function, we again interchange the limits and find that

(cid:90) (cid:32) ∞(cid:88)

(cid:33)

(cid:90)

f ∂xϕ = −

∂xfλn

ϕ

n=1

as desired. Now we have that f has a weak derivative, which is a convergent sum of locally

L2 integrable functions. The same holds for ∂y, and hence both ∂z and ∂z. Now it follows
immediately that f ∈ W 1,2
loc (C) and satisfies a Beltrami equation; thus, the measurable
Riemann mapping theorem (see, for example, Theorem 5.3.2 of [8]) gives us the following

lemma:
Lemma 2.3.5. Given a countable set {λn}∞

n=1 ⊆ D, the function f defined in (2.3.4) is

K-quasiconformal.

We now claim that this function f has the correct stretching behavior at each point
in Λ. Fix λn ∈ Λ; we can assume that λn = 0. Morally, we proceed as follows: there are

contributions to the stretching from terms on two scales, the nearby and the far away. We can

arrange it so that nearby points λm only have very large indices, so that the exponentially

decaying weights will render this negligible; on the other hand, far away points have the

advantage of the smoothness of the radial stretches.

Let us make this precise. We will show that

|f (r) − f (0)| = cr1/K + o(r1/K )

(2.3.5)

with a non-zero constant c, from which the theorem will follow. First of all, it is clear that

the term n = m contributes exactly 1

2m r1/K ; we will estimate away the remaining terms.

29

To this end, we have for terms with m (cid:54)= n that the difference is

(cid:88)

m(cid:54)=n

1

2m (r − λm)|r − λm| 1

K −1 − 1

2m (−λm)| − λm| 1

K −1

After factoring a term −λm| − λm| 1

inequality, we need to estimate

K −1 from each summand and applying the triangle
(cid:18)

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)1 − r

λm

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .
(cid:12)(cid:12)(cid:12)(cid:12)1/K−1 − 1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)

m(cid:54)=n

1

2m|λm|1/K

1 − r
λm

To deal with the term within the absolute value, we need a simple estimate of a particular

function:

Lemma 2.3.6. If K > 1,

(cid:12)(cid:12)(cid:12)(1 + z)|1 + z|1/K−1 − 1

(cid:12)(cid:12)(cid:12) ≤ C0 min

(cid:110)|z|,|z|1/K(cid:111)

.

for a constant C0 depending only on K.

Proof. For large values of |z|, the triangle inequality implies that this is controlled by a
constant multiple of |z|1/K , which is smaller (up to a constant) than |z|. So let us assume
that |z| is small, e.g. |z| ≤ 1
If y = 0, |1 + z| = 1 and

2. Write |1 + z| = 1 + y with y real and |y| ≤ |z|.

(1 + z)|1 + z|1/K−1 − 1 = z.

30

Otherwise, select λ so that λy = z; then Taylor expansion gives

(1 + z)|1 + z|1/K−1 − 1 = (1 + λy)(1 + y)1/K−1 − 1

λ +

− 1

y + O(y2) − 1

(cid:19)

(cid:18)

1
K

(cid:18) 1

K

= 1 +

(cid:18)

=

λ +

= z +

(cid:19)
(cid:19)

1
K
− 1

− 1

y + O(y2)

y + O(y2)

= z + O(|z|) + O(|z|2)

from which the lemma follows.

Now we are ready to make the division into two scales. The cutoff point is to separate

in the following way: Since the sequence is fixed, we can choose r small enough that

(cid:18)

(cid:19) 1

1−1/K

r

|λm| ≥

1

2n+1C0

=⇒ m ≥ n + a + 10

where a is chosen so that 2a > C0; C0 here is the constant of Lemma 2.3.6. That is, when
|λm| is smaller than a very large constant multiple of r, the index must be very large.

The far scale is for terms when (r/|λm|)1−1/K < 1/2n+1C0. In this case we have the

lemma’s linear estimate available, and the sum over these indices m is at most

(cid:88)

m far

C

|λm| = C0r1/K (cid:88)

m far

1
2m

(cid:19)1−1/K

(cid:18) r

|λm|

<

r1/K
2n+1

1

2m|λm|1/K r

which is enough. Note that we have no control over the index m here.

Next is the nearby scale where we have the opposite inequality; now m must be large but

we have worse control on the summands. Using the non-linear estimate from the lemma, we

31

find that the contribution is at most

(cid:88)

(cid:18) r
2m|λm|1/K
having used the fact that(cid:80)

m near

C0

1

|λm|

(cid:19)1/K

(cid:88)

m near

= C0

1

2m r1/K ≤ C0

2n+a+9 r1/K <

r1/K
2n+9

m≥N

2m = 1
1

2N +1 .

Combining these two estimates, the contribution from all indices m (cid:54)= n is of the order
r1/K with constant significantly less than 2−n. This proves (2.3.5) and is the desired result.

2.4 Dimension Greater than Zero

To prepare for the main result, we will define a particular class of gauge functions. These

will be gauges which lead to minor perturbations of the pure Hausdorff meaures, without

changing the dimension. The perturbations should be chosen to tend to zero slowly enough

to guarantee this, and will contain some sort of embedded convexity condition.

Definition 2.4.1. We will say that a gauge function Λ(r) = rdh(r) is admissible if h(r) is
continuous, nonnegative, non-decreasing on [0,∞), and satisfies the following decay condition
at the origin: For every 
 > 0, there exists a constant C
 such that for any 0 < r ≤ s ≤ 1,

(cid:16) r

(cid:17)


s

.

h(r)
h(s)

≥ C


It will be proven later that functions of the form (log(1/r))−β for β > 0 are admissible,

giving a rich class of examples. We now come to the first theorem of the section.

Theorem 2.4.2. Let Λ be an admissible gauge function. Fix K and α ∈ (1/K, 1), setting

32

d = FK (α, 0). Then there is a set E with positive gauged Hausdorff measure HΛ(E) and a

K-quasiconformal map f so that f stretches like α at every point in E.

Proof. The main construction of the proof is taken from [63], although our choice of pa-

rameters will be different. We retain the notation from that paper, and for the sake of

self-containment give a brief description of the construction. At each stage of the con-

struction, we will pack a disk completely with disjoint disks, and then shrink these disks

appropriately to build a set of the desired Hausdorff dimension. The quasiconformal map

will stretch these shrunken disks appropriately.

Step 1. Select m1,1 disjoint disks Di

1,1 of radius R1,1 within the unit disk, followed by

m1,2 disjoint disks (and disjoint from the previously constructed disks as well) Di

1,2 of radius

R1,2, and so on. In this manner we pack the unit disk completely in area, leading to

∞(cid:88)

m1,jR2

1,j = 1.

j=1

It is important to note that we can assume that every R1,j is smaller than some fixed δ1 > 0,

which is as small as we desire. Also for each radius associate a parameter σ1,j > 0; these

will be chosen later, but are all quite small.

Next, we construct a first approximation of our quasiconformal map. Denote the center

of the disk Di

1,j as zi

1,j. Let ψi

1,j(z) = zi

1,j + (σ1,j)K R1,jz, and define disks

Di

j = D(zi
j)(cid:48) = D(zi

(Di

1,j, R1,j) =

1

(σ1,j)K

1,j(D)
ψi

1,j, (σ1,j)K R1,j) = ψi

1,j(D)

33

Then our first approximation is

(σ1,j)1−K (z − zi

1,j) + zi

1,j,

ϕ1(z) =

(z − zi

1,j) + zi

1,j,



(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z−zi

1,j
R1,j

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1

K −1

z,

j)(cid:48)
z ∈ (Di
j)(cid:48)
j \ (Di
z /∈(cid:83) Di

z ∈ Di

j.

This is K-quasiconformal, being a modification of a radial stretch, and is conformal except

for the annular regions between small disks (Di

j)(cid:48) and their dilates Di

j. In particular, it is

important to note that ϕ1 maps the disks of radius (σ1,j)K R1,j onto other disks of radius

σ1,jR1,j.

Step 2. We repeat the idea of the construction from the previous step. Choose m2,1

disjoint disks Di

2,1 with centers zi

2,1 of radius R2,1, and so on; again these will be subject to

the constraint

∞(cid:88)

j=1

m2,jR2

2,j = 1.

Again, we can choose R2,j to be bounded by some δ2 > 0, but as small as needed; this is

the difference from step 1, as we may wish to have δ2 < δ1. Next, we choose σ2,j > 0.

As before, we follow this with an approximation of the quasiconformal map. Set ψn

2,k(z) =

zn
2,k + (σ2,k)K R2,kz, a radius r{2,k},{1,j} = R2,kσ1,jR1,j and define disks

(cid:32)

34

j,k, r{2,k},{1,j}) = ϕ1

j,k = D(zi,n
Di,n
j,k)(cid:48) = D(zi,n

(Di,n

1

1,j ◦ ψn
ψi

2,k(D)

(σ2,k)K

(cid:16)

j,k, (σ2,k)K r{2,k},{1,j}) = ϕ1

1,j ◦ ψn
ψi

2,k(D)

(cid:33)
(cid:17)

Now we define

g2(z) =



(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

z,

j,k) + zi,n
j,k,

(σ2,k)1−K (z − zi,n
K −1

z−z

i,n
j,k

r{2,k},{1,j}

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1

(z − zi,n

j,k) + zi,n
j,k,

j,k)(cid:48)
z ∈ (Di,n
j,k)(cid:48)
j,k \ (Di,n
z ∈ Di,n

otherwise.

Finally, our second approximation is ϕ2 = g2◦ϕ1. As before, this is a K-quasiconformal map

equal to the identity outside the unit disk; the most important thing to note is that this map

behaves essentially as a radial stretch, sending certain disks of radius (σ1,jσ2,k)K R1,jR2,k

to certain other disks of radius (σ1,jσ2,k)R1,jR2,k.

Induction step. Assuming that N − 1 steps of the construction have been fulfilled, we

repeat the process, getting disks Di

N,j with centers zq

N,p, radii RN,p and satisfying

∞(cid:88)

mN,jR2

N,j = 1.

j=1

As before, we have a constraint RN,j < δN and parameters σN,j > 0.

We proceed with the next approximation of the quasiconformal map. Define radii

r{N,p},{N−1,h},...,{1,j} = RN,pσN−1,hr{N−1,h},...,{1,j}

and maps ψq

N,p(z) = zq

N,p + (σN,p)K RN,pz. For multiindices I = (i1, ..., iN ) and J =

(j1, ..., jN ), we define disks

DI

J = D(zI

J , r{N,p},...,{1,j}) = ϕN−1

(cid:32)

(cid:33)

1

(σN,p)K

ψ

i1
1,j1

◦ ··· ◦ ψ

iN
N,jN

(D)

35

(DI

J )(cid:48) = D(zI

J , (σN,p)K r{N,p},...,{1,j}) = ϕN−1

(cid:16)

ψ

i1
1,j1

◦ ··· ◦ ψ

iN
N,jN

(cid:17)

(D)

As usual, we set

gN (z) =



(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

z,

(σN,p)1−K (z − zI
J ) + zI
J ,
K −1

z−zI
J

r{N,p},...,{1,j}

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1

(z − zI

J ) + zI
J ,

J )(cid:48)
z ∈ (DI
J )(cid:48)
J \ (DI

z ∈ DI

otherwise.

This map is K-quasiconformal, conformal outside of the union of all the annuli and

preserves the disks DI

J . We finally set ϕN = gN ◦ ϕN−1, noting that this is the identity

outside the unit disk and maps disks of radius (σ1,j1

radius (σ1,j1

··· σN,jN

)R1,j1

··· RN,jN

.

··· σN,jN

)K R1,j1

··· RN,jN

to disks of

We now take the limits resulting from this construction. As ϕN is a K-quasiconformal
map which is the identity outside of D, compactness of quasiconformal maps allows us to

select a K-quasiconformal limit

f = lim

n→∞ ϕN

with convergence in the Sobolev space W 1,2
loc .

To recap, the result of the above construction is a Cantor type set E whose building

blocks at generation N are disks with radius

(cid:16)

(cid:16)

(cid:17)

. . .

(σN,jN

)K RN,jN

(cid:17)

sj1...jN

=

(σ1,j1

)K R1,j1

36

which are mapped to disks of radius

(cid:16)

(cid:16)

(cid:17)

. . .

σN,jN

RN,jN

(cid:17)

tj1...jN

=

σ1,j1

R1,j1

where we can choose σi,ji

more or less freely, subject to the constraint that they are all small.

Now we will select our parameters σk,jk

. We will choose them subject to the governing

equation

R2

1,j1

··· R2

N,jN

= (R1,j1

··· RN,jN

)Kd

)d(σ1,j1
σK
1,j1

··· σN,jN
··· σK

N,jN

· h(R1,j1

··· RN,jN

).

(2.4.1)

If we write σk,jk

= R

2−d
Kd
k,jk

ηk,jk

, the condition is equivalent to

(cid:16)

1 = ηKd
1,j1

··· ηKd
N,jN

h

R

2/d
1,j1

··· R

2/d
N,jN

ηK
1,j1

··· ηK

N,jN

(cid:17)

.

(2.4.2)

To see the relevance of the governing equation, note that if we sum over all the building

blocks of our construction at level N , our choice of parameters gives us

(cid:88)

j1,...,jN

m1,j1

··· mn,jnsd

j1,...,jnh(sj1,...,jn) =

(R1,j1

··· RN,jN

)2 = 1

(cid:88)

j1,...,jN

This is suggestive of the desired result, namely that the constructed set has positive measure

in the gauge rdh(r).

We now have three questions left to address: whether we can actually select our parame-

ters σ in this manner, whether the Cantor set will exhibit the correct stretching, and whether
the set has positive measure with respect to HΛ.

37

First, we consider the satisfiability of the governing equation for σk,jk

; the selection is

made inductively. Looking at the second form of our governing equation, and recalling that

h is continuous, it is immediately clear that we can select ηN,jN

to satisfy the equation -

the right hand side tends to zero as ηN,jN

does, and to infinity as ηN,jN

does. The only

concern is that ηN,jN

First, notice that RN,jN

might be so large as to defeat our requirement that σN,jN
σK
N,jN

< 1; if it were not, then we would have

is small.

R2

1,j1

··· R2

N,jN

(cid:16)
= (R1,j1
· h
(cid:16)
≥ (R1,j1
· h

= R2

1,j1

)Kd

(cid:17)

··· RN−1,jN−1
··· RN,jN
R1,j1
··· RN−1,jN−1
R1,j1
··· R2

)d(σ1,j1
σK
1,j1
)d(σ1,j1
··· RN−1,jN−1
σK
1,j1
N−1,jN−1

··· σN−1,jN−1
··· σK
(RN,jN
N,jN
··· σN−1,jN−1
··· σK
N−1,jN−1

)Kd

(cid:17)

σK
N,jN

)d

contradicting the fact that each Rk,jk

is much smaller than 1.

The above computation also suggests how to bound each σN,jN

, by playing the governing

equation off itself at different generations. In this manner, essentially just rearranging the

above, we find that

R2

N,jN

= Rd

N,jN

σKd
N,jN

(cid:16)

h

(cid:18)

h

R1,j1

··· RN,jN
R1,j1
··· RN−1,jN−1

σK
1,j1

σK
1,j1

(cid:17)

(cid:19)

··· σK
··· σ

N,jN

N−1,jK

N−1

Rearranging for σN,jN

and applying our growth condition with exponent 
, we find that

(cid:33)


σKd
N,jN

≤ R2−d
N,jN

1
σN,jN
RN,jN

1
C


.

(cid:32)

38

Consequently,

≤

σN,jN

1

1/K(d+
)



C

2−d−

Kd
N,jN

.

R

As long as 
 is chosen small enough that 2 − d − 
 > 0, we may choose all δN small enough

to result in σN,jN

< 1/100 as desired.

Next, we proceed to the stretching. Following the general approximation lemma 2.3.1, it

is sufficient to show that

log tj1,...,jN
log sj1,...,jN

→ α

as N → ∞. In this direction, observe that

log tj1,...,jN
log sj1,...,jN

=

=

(cid:80)N
(cid:80)N
i=1 log Ri,ji
(cid:16)
i=1 log Ri,ji
(cid:16)
1 + 2−d
1 + K 2−d

+(cid:80)N
+ K(cid:80)N
(cid:17)(cid:80)N
(cid:17)(cid:80)N

Kd

i=1 log Ri,ji
i=1 log Ri,ji

Kd

i=1 log σi,ji
i=1 log σi,ji

+(cid:80)N
+(cid:80)N

i=1 log ηi,ji
i=1 log ηi,ji

.

Now provided that the perturbation terms are negligible with comparison to the radii terms,

the stretching result follows. Indeed, in that case the quotient tends towards

1 + 2−d
1 + K 2−d

Kd

Kd

2 + (K − 1)d

2K

=

= α.

Thus, we need to prove that

tends to zero as N grows.

(cid:80)N
(cid:80)N

i=1 log ηi,ji
i=1 log Ri,ji

SN :=

To get this result, first notice that SN is negative: the product of all ηi,ji

is greater than

39

1 (as h is small), while each Ri,ji

is less than 1; see (2.4.2). From this, it follows that

0 ≥ KdSN =

=

≥

=

Kd(cid:80)N
(cid:80)N
(cid:16)

− log h

(cid:16)

i=1 log ηi,ji
i=1 log Ri,ji

R

2/d
1,j1

··· R

2/d
N,jN

(cid:80)N
(cid:80)N

2
/d
1,j1

− log

C
R

(cid:80)N

− log C

i=1 log Ri,ji

i=1 log Ri,ji
··· R

2
/d
N,jN

k=1 log Ri,ji
− 2

− K
SN
d

(cid:17)

ηK
1,j1

··· ηK

N,jn

ηK

1,j1

··· ηK

N,jN

(cid:17)

where in the inequality we have used that h(r) ≥ C
r
 provided that r is sufficiently small,

e.g. that N is sufficiently large; this is the admisibility condition (2.4.1) applied with s = 1.

To be precise, we require that N is large enough that R

2
/d
1,j1

··· ηK

N,jn

< 1. Now rearranging

the result, we get

(cid:18)

1

K(d + 
)

(cid:19)(cid:32)

−

0 ≥ SN ≥

(cid:33)

(cid:80)N

log C

i=1 log Ri,ji

− 2

d

It follows that we have

|SN| ≤ log C

N log 2

+ O(
) = O(
)

provided that N is chosen large enough given 
. Taking 
 to zero gives the required stretching.

Now all that remains is to show positivity of the measure of the Cantor set. Our starting

point is an estimate analogous to equation (3.17) in [63]; if D is a building block at generation
N − 1,

(cid:88)

r(Bn)dh(r(Bn)) =

Bn children of D

jN

R1,j1

··· RN,jN

σK
1,j1

··· σK

N,jN

(cid:17)

(cid:88)

(cid:16)

mN,jN

Λ

40

R1,j1

··· RN−1,jN−1
σK
mN,jN
N,jN

(RN,jN

σK
1,j1

)dh

R1,j1

(cid:17)

σK
1,j1

··· σK

N,jN

(cid:105)d
··· RN,jN

··· σK

N−1,jN−1

(cid:16)

(cid:104)
·(cid:88)
(cid:88)

jN

jN

=

=

mN,jN

R2

1,j1

··· R2

N,jN

= R2

1,j1

= (R1,j1

··· R2
N−1,jN−1
··· RN−1,jN−1
R1,j1

··· RN−1,jN−1

)d(σ1,j1
σK
1,j1

(cid:16)

· h

··· σN−1,jN−1
··· σK
N−1,jN−1

)Kd

(cid:17)

= Λ(r(D))

(2.4.3)

where we have used the governing equation (2.4.1) at generations N and N − 1. As a conse-
quence, we can iterate this result to find that if {Bn} is a finite collection of building blocks

all contained in D (not necessarily of the same generation), and BN,k are the generation N

descendents of Bn,

(cid:88)

Bn

Λ(r(Bn)) =

(cid:88)

BN,k

Λ(r(BN,k)).

We now wish to prove a Carleson style packing condition, from which positivity of measure

will follow. We will state this as a separate lemma, similar to Lemma 3.2 of [63].

Lemma 2.4.3. Let B be an arbitrary disk and Bn disjoint building blocks of E. There is an
absolute constant C1 independent of the family C = {Bn} such that

(cid:88)

Bn∈C
Bn⊂B

Λ(r(Bn)) ≤ C1Λ(r(B)).

Once the lemma has been proven, the positivity of the gauged Hausdorff measure follows

41

immediately. So let us fix such a family C; we may assume that r(B) ≤ 1, since the above
computation (2.4.3) shows that the lemma holds when B = D. Choose the integer H
such that all the Bn are contained in some building block at generation H − 1, but not at
generation H; then let {BH
kp
family. Note that the lemma holds with B = BH−1
B = D) and so we will assume that

}m
p=0 be the complete list of ancestors at generation H of our

(by the same reasoning that it holds for

i0

r(B) ≤ r(BH−1

i0

For each of these generation H disks, let (cid:103)BH
r((cid:103)BH

) =

kp

kp

) = sj1,...,jH−1

.

be the concentric dilate with radius

sj1,...,jH
σK
H,jH

.

Provided that the multiindices I = (i1, ..., iH ) and J = (j1, ..., jH ) are chosen appropriately,
J )(cid:48) from the construction of the Cantor set; now as each σN,p is

these disks are the disks (DI

small (e.g. less than 1/100) and since B meets each BH
kp

, we find that

r((cid:103)BH

kp

).

2r(B) ≥ 99
100

Moreover, we have the containment (cid:103)BH

kp

⊆ 4B. We now can compute:

Λ(r(BH
kp

))

(cid:88)

Bn∈C

Λ(r(Bn)) ≤ m(cid:88)
(cid:104)
· m(cid:88)

p=0

=

(cid:32)

p=0

(cid:105)d

σK
1,j1

R1,j1

··· σK

H−1,jH−1

RH−1,jH−1

(cid:33)d

(cid:32)

(cid:33)

σK
H,jHkp

RH,jHkp

h

R1,j1

42

··· σK

H,jHkp

= sd

j1,...jH−1

h(sj1,...jH−1

)

= sd

j1,...jH−1

h(sj1,...jH−1

)

m(cid:88)
m(cid:88)

p=0

1
π

p=0

R2
H,jHkp

Area(Dp)

where we have defined Dp = D(z

kp
H,jHkp

, RH,jHkp

), recalling that these are disks chosen

during the induction step of the Cantor set’s construction, called Di
N,j. The second to last
equality follows from applications of the governing equation at generations H and H − 1.

Now since

and

it follows that

r((cid:103)BH

kp

) =

sj1,...,jHkp
σK
H,jHkp

r((cid:103)BH

kp

)

r(Dp) = RH,jHkp

=

sj1,...,jH−1

(cid:88)

Bn∈C

Λ(r(Bn)) ≤ sd

j1,...jH−1

h(sj1,...jH−1

(cid:34)

)

(cid:34)

(cid:35)2
(cid:35)2−d

r(4B)

sj1,...,jH−1

(cid:46) r(B)dh(sj1,...,jH−1

)

r(B)

sj1,...,jH−1

Finally, recall the condition (2.4.1) that for any 0 < x < y ≤ 1, we have

(cid:18)x

(cid:19)2−d

h(x)
h(y)

≥ C

y

43

Applying this to the above, it follows that

(cid:88)

Λ(r(Bn)) (cid:46) r(B)dh(r(B)) = Λ(r(B))

Bn∈C

as desired.

We now move to the rotation results.

Theorem 2.4.4. Let Λ be an admissible gauge function. Fix K and parameters α, γ so that
α(1 + iγ) ∈ BK and α < 1, setting d = FK (α, γ). Then there is a set E with positive gauged
Hausdorff measure HΛ(E) and a K-quasiconformal map f so that f stretches like α and

rotates like γ at every point in E.

Proof. This proof will very closely follow Hitruhin’s modifications in [33] to add rotation to

the previous theorem. We select K < 1/α and let f be the K-quasiconformal map previously
constructed; the corresponding Cantor set has positive Hrdh(r) measure, where

d = 1 + α − K + 1
K − 1

(1 − α).

Now all we need to do is modify the construction of ϕn for each n by replacing the old gn

by

gn(z) =



(σn,jn)1−K (z − zI

J )eiθI
−1+iαγ K−1
K(1−α)

J + zI
J ,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z−zI

J
r(DI
J )

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1

K

J )(cid:48)
z ∈ (DI

(z − zI

J ) + zI
J ,

z ∈ DI

J )(cid:48)
J \ (DI

z,

otherwise.

where the change in argument over the annulus DI

J \ (DI

J )(cid:48) is θI

J , and makes the map

continuous across the boundary crossings. Let f denote the resulting map using ϕn and gn,

44

rather than the old versions ϕn and gn.

Since the paper [33] has already shown that d = FK (α, γ) is the desired dimension,

and the previous theorem improves this to the perturbed Hausdorff gauge function, all that

remains is to check that the rotational behavior is correct. That is, we need to show that

arg(f (z0 + rn) − f (z0))
log |f (z0 + rn) − f (z0)| = γ

lim
n→∞

for a suitable choice of scales rn → 0, and z0 in a large subset of the Cantor set. Following
the argument in [33], we end up with the result that the total rotation as we move from ∞

to a disk at scale n is

arg (f (z0 + rn) − f (z0)) = αγ

n−1(cid:88)

k=1

K − 1
(1 − α)

log σk,jk

+ O(n)

Now we select our parameters σk,jk

as before, but with d and K replacing d and K respec-

tively. With our usual notation

1−α
αK−1
k,jk

= R

ηk,jk

σk,jk

we can compute that

(cid:80)n−1

k=1 log Rk,jk

+(cid:80)n−1

+(cid:80)n−1

(cid:105)
+(cid:80)n−1

k=1 ηk,jk

(cid:17)

k=1 log Rk,jk

k=1 log Rk,jk

k=1 ηk,jk

arg(f (z0 + rn) − f (z0))
(cid:104) 1−α
log |f (z0 + rn) − f (z0)|
(cid:80)n−1
αγ K−1
1−α
αK−1

(cid:16)

=

α

K 1−α
αK−1
αγ K−1
αK−1
α(K 1−α
αK−1

≈

+ 1)

45

αγ(K − 1)

αK(1 − α) + αK − 1

=

= γ

as desired, where we have used the previous result that(cid:80)N
to(cid:80)N

k=1 Rk,jk

. In particular, letting n → ∞, the infinitesimal rotation is exactly γ.

k=1 ηk,jk

is negligible in comaprison

Now we would like to generalize this theorem to include stretching exponents greater
than 1; this can be done by considering the inverse function f−1, which inverts the stretch-

ing exponent and changes the sign of the rotation exponent. However, without assuming

additional constraints on the perturbation h, it does not seem (to the best of the author’s
knowledge) possible to identify a gauge function h(cid:48) for which

Hrd(cid:48)

h(cid:48)(r) (f (E)) > 0.

It turns out that the key obstacle is a lack of decay in h; taking Section 4 of [63] as inspiration,
we will impose the additional condition that for all t > 0, h(t) (cid:46) h(tK ). Powers of logarithms
such as (log 1/r)−β clearly satisfy this condition, so we still have a useful class of examples.

Theorem 2.4.5. Let Λ(r) = rdh(r) be an admisible gauge function, with the additional
constraint that h(rK ) (cid:46) h(r) for all r > 0. Let E and f be the stretching and rotation set
and quasiconformal map constructed in Theorem 2.4.4, with exponents α and γ. Then f−1
stretches with exponent 1/α and rotates with exponent −γ at every point in f (E) and f (E)

has positive measure with respect to the gauge function

Λ(cid:48)(r) = rd(cid:48)

hd(cid:48)/Kd(r).

46

Since we have the additional decay constraint on h, the proof of this is a minor modifi-

cation of that of Theorem 4.2 of [63], again proceeding through a Carleson type estimate.

Rather than repeat a sketch of the argument, we will compare our conditions on h to those

of Uriarte-Tuero. First of all, for technical reasons, it is important to use only a finite family

of disks (as in [63]) at each generation of the Cantor set (rather, what is important is that

there is a minimum choice of Rn,j at each generation n, so that the construction of the next
scale takes place on stricly smaller scales). In particular, choosing a sequence 
n → 0 very
quickly and packing an (1 − 
n) portion of the unit disk at each generation will only change

the measure by a factor(cid:81)∞

n=1(1 − 
n) ≈ 1, so the finiteness condition is not an obstacle.

Secondly, it is required in Uriarte-Tuero’s construction that h(t) is a (strictly) increasing
function for which tα/h(t) → 0 as t → 0 for each α > 0, that h1/(2−d)(t)/t is decreasing in t,
and the logarithmic-type condition h(t) (cid:46) h(tK ). The first and fourth conditions hold here,

as does the second by the definition of admissibility. Furthermore, admissibility applied with
exponent 
 = 2 − d gives us that if r < s,

(cid:17) 

2−d = C


1
2−d (r)/r
1
2−d (s)/s

h

h

≥ s
r

C


(cid:16) r

s

Although this does not show that h1/(2−d)(t)/t is decreasing, it is almost decreasing (and in

fact, if s is small enough we may assume that C
 > 1, in which case we do have a decreasing

function); it turns out that this is enough for the proof of the theorem to go through with

minor modifications of the constants. It is worth pointing out, however, that the logarithmic-

type decay condition is independent of the admissibility condition in the sense that neither

is strong enough to imply the other.

Now to show the usefulness of the theorems, it would be nice to give an explicit and

47

interesting gauge function. Fortunately, logarithmic perturbations of rd are admissible, so

we have a variety of gauges for which the theorem holds.

Corollary 2.4.6. There are positive measure stretching and rotation sets associated to the

gauges Λ(r) = rd(cid:16)

(cid:17)−β

log 1
r

for every β > 0.

To be precise, this is not a well-defined gauge function for r ≥ 1; we ought to cut it off at

some point between 0 and 1 so it does not blow up. However, as we only really care about
the behavior as r tends to zero, this is a point we will ignore; we will assume that s ≤ 1
100.

Proof. We will show that the gauge functions Λ(r) = rd(cid:16)

are admissible for all
β > 0; all that we need to prove is the growth condition. Fix 0 < r ≤ s with s small, and

(cid:17)−β

log 1
r


 > 0. We need to show that there exists a constant C
,β for which

(cid:16)r

(cid:17)


s

h(r)
h(s)

≥ C
,β

or alternatively, that

s
(log(1/s))β
r
(log(1/r))β

is bounded below independent of r and s. We may just as well consider the functions

g(r, s) =

s
/β log s
r
/β log r

.

on the triangular domain {(r, s) : 0 < r ≤ s ≤ 1

100}.

First, let us fix s; we minimize the function over r. The r-derivative is

(cid:21)

log r + 1

,

(cid:20) 


β

∂g
∂r

=

s
/βr
/β−1
(r
/β log r)2

(− log s)

48

which changes sign from negative to positive at r = e−β/
. We now split into two cases,

depending on the size of s.

The first case is that s ≥ e−β/
, so that g(r, s) is in fact minimized at r = e−β/
. In this

case, we have

g(r, s) ≥ g(e−β/
, s) = −
e
β

s
/β log s.

If we again differentiate, but in s, we get

s
/β−1

−
e
β

(cid:21)

log s + 1

(cid:20) 


β

which is negative due to the fact that s ≥ e−β/
. Hence, this quantity is minimized when

s = 1

100; this gives a lower bound of

(cid:18)
e−β/
,

(cid:19)

1
100

∀ 0 < r ≤ s, s ≥ e−β/
.

g(r, s) ≥ g

This of course only depends on β and 
, which is good enough.

The second case is that s < e−β/
. Here, we may compute the s-derivative, finding that

(cid:20) 


(cid:21)
s
/β−1 log s + s
/β+

(cid:20) 


(cid:21)

s
/β−1
r
/β

=

1

log r

β

log s + 1

.

∂g
∂s

=

1

r
/β log r

β

Since r < 1, this is positive; therefore, g increases from its minimum value of 1 at the bottom

of the domain where r = s, and is again bounded below.

We can also apply this technique to get positive results for Riesz capacities. Recall that

49

for a set E, the (β, p)−Riesz capacity ˙Cβ,p is defined by

˙Cβ,p(E) = inf(cid:8)(cid:107)g(cid:107)p : g ∗ Iβ ≥ χE

(cid:9)

where up to a normalization, Iβ(z) = |z|−(2−β) is the Riesz kernel; see, e.g.

[1] for more

details. There is also a dual characterization by Wolff’s theorem that

(cid:110)

˙Cβ,p(E) (cid:39) sup

µ(E) : supp(µ) ⊆ E,

˙W µ

β,p(z) ≤ 1∀z ∈ C(cid:111)

where the homogeneous Wolff potential

˙W µ

β,p is

(cid:90) ∞

0

˙W µ

β,p(z) =

(cid:32)

µ(cid:0)B(z, r)(cid:1)

r2−βp

(cid:33)p(cid:48)−1

dr
r

.

Furthermore, it is important to note that the Riesz capacity is homogeneous of degree 2−βp,

which will correspond with the Hausdorff dimension of the set under consideration. Our main

result here is the following theorem:

Theorem 2.4.7. Fix any parameter τ = α(1 + iγ) ∈ BK with α ∈ (1/K, 1), and a pair
(β, p) with 1 < p < ∞ and 2 − βp = FK (α, γ). There is a K-quasiconformal map f and a

set E such that f stretches with exponent α and rotates with exponent γ at every point in
E, and E has positive (β, p)−Riesz capacity.

In particular, this shows that there cannot be a theorem improving the results of [10] to

the level of Riesz capacity zero for any choice of parameters with the correct homogeneity.

This stands in sharp contrast with the results of [5], in which Riesz capacities were used to

give sharper results than gauge functions alone can give. In [5], at the critical homogeneity,

50

there was a range of parameters (β, p) in which extremal examples could exist, beyond

which there was a negative result showing the sharpness of Riesz capacities. However, in our

case, all possible indices have associated examples; thus an analogue of their theorem is not

possible.

Proof. This theorem is actually much easier to prove than the last one, as the Riesz ca-

pacities of these Cantor type sets have already been estimated in [5]. We will first make

the construction for a fixed (β, p), and then extend it in such a way that the set will have

positive Riesz capacities for all parameter choices simultaneously. Let E be a Cantor type

set as constructed in Theorem 2.4.2; our choice of parameters will be

(cid:18) k + 1

(cid:19)δ

k

2−d
dK
k,jk

= R

σk,jk

with δ to be chosen soon. Following the techniques of the previous proofs, we can compute

that the stretching exponent is α at every point of E, while the rotation exponent is γ on a

large subset of E.

Now it remains to understand the Riesz capacity of this set. Per Lemma 8.1 of [5], if ν

is the naturally distributed measure on E, its Wolff potential is

at each x ∈ E. If we select, e.g.

n=2

∞(cid:88)

β,p (cid:39)
˙W ν

1

ndK(p(cid:48)−1)δ

δ = 1 +

1

dK(p(cid:48) − 1)

51

then this series is convergent, the Wolff potential is uniformly bounded, and therefore the
set has positive (β, p)−Riesz capacity.

Now we need to extend this from a particular parameter choice to all simultaneously.

Carry out the above construction, but localized to a disk of radius 1/2. Fix a new choice

(β1, p1) with p1 > p, and carry out the construction with this parameter choice (meaning,

with the updated value of δ) in a disjoint disk of radius 1/4. Continue in this manner with

(β2, p2) with p2 > p1, and so on; this gives a set with positive Riesz capacity for a sequence
(βn, pn) with 2 − βnpn = d for every n. If pn → ∞, a comparison theorem (e.g. Theorem

5.5.1(b) of [1]) shows that E has positive capacity for all parameter choices.

It is worth remarking that this theorem actually follows from the previous one, with

the correct choice of h (at least for these Cantor type sets). If we choose the gauge to be
h(r) = (log 1/r)−1 for small enough r, then the resulting Cantor set must be larger, in a

sense, than one only with positive Riesz capacity. This follows from an estimate of the ηk,jk

.

Recall the generating relationship (2.4.2):

1 =

η1,j1

··· ηN,jN

(cid:17)Kd

(cid:16)

R

2/d
1,j1

··· R

2/d
N,jN

ηK
1,j1

··· ηK

N,jN

(cid:17)

.

h

We will show that the choice of ηk,jk

to satisfy this equation with this choice of gauge is

typically larger than (1 + 1/k)δ; in particular, that means that the Cantor set naturally asso-

ciated to this gauge is significantly larger than that constructed for positive Riesz capacity.

To this end, suppose that ηk,jk

≤ (1 + 1/k)δ for all k. Then we have

(cid:16)

(cid:16)

1 =

η1,j1

(cid:16)
··· ηN,jN
= (N + 1)Kdδh

R

2/d
1,j1

h
R
··· R

2/d
1,j1

2/d
N,jN

··· R

ηK
1,j1

2/d
N,jN

(N + 1)Kδ(cid:17)

(cid:17)Kd

(cid:16)

52

(cid:17)

··· ηK

N,jN

(cid:80)N

k=1 log

=

2
d

(N + 1)Kdδ

1

Rk,jk

− Kδ log(N + 1)

.

However, we can choose the radii Rk,jk

as small as we desire, making the right hand side

of this equation arbitrarily small. This leads to a contradiction, showing that this uniformly

bounded selection of ηk,jk
was in fact too small. Hence at least some of the selections ηk,jk
must have been larger than (1+1/k)δ, a contradiction. Moreover, asymptotically, the choices

of ηk,jk

must be much larger than (1 + 1/k)δ, and larger choices of ηk,jk

lead to a larger

Cantor set.

53

Chapter 3

Improved H¨older Continuity of

Quasiconformal Maps

This work first appeared in [22].

3.1

Introduction

A K-quasiconformal map f is an orientation-preserving homeomorphism between two do-

mains in the complex plane, lying in the Sobolev space W 1,2

loc and satisfying the distortion

inequality

|∂βf| ≤ K min

β

|∂βf|

max

β

for almost all z, where ∂β is the directional derivative in the direction β. These maps can

be realized as solutions to the Beltrami equation

∂zf = µ(z)∂zf

(3.1.1)

where the Beltrami coefficient µ(z) has the bound (cid:107)µ(cid:107)∞ ≤ K−1

K+1 < 1 and represents the com-

plex distortion of the function f . Such maps have useful geometric and regularity properties,

and provide a natural framework for generalizing conformal maps. They arise naturally in

54

a number of applications, and are closely related with the solutions to elliptic PDEs in the

plane.

In this paper, we will be concerned with the precise degree of regularity and smoothness

properties of quasiconformal maps; we will be most interested in determining what H¨older

continuity such maps have (that is, which Lipschitz class the functions lie in). A function f

defined on an open set Ω is said to be locally α-H¨older continuous if for each compact set
E ⊆ Ω there exists a constant C = C(f, E) with

|f (z1) − f (z2)| ≤ C|z1 − z2|α

for all z1, z2 ∈ E; equivalently, f lies in the Lipschitz class Cα(Ω). It is well known that

K-quasiconformal maps are H¨older continuous with exponent 1/K, due to an old theorem

of Mori [46]. More recently, quantitative upper and lower bounds on the size (in the sense

of Hausdorff measure and dimension) of the set where f can attain the worst-case H¨older

regularity were given by Astala, Iwaniec, Prause, and Saksman [10] and the author [21].

However, the exponent 1/K is not always optimal for a K-quasiconformal map. For
example, there are bilipschitz K-quasiconformal maps defined on C which are not (K − 
)-

quasiconformal for any 
 > 0. A particular case of this is a map exhibiting rotation, such
as z|z|iγ for an appropriately chosen exponent γ ∈ R. Therefore, it is apparent that the

exact H¨older regularity of a quasiconformal map depends on more than just the magnitude

of the complex distortion and should instead encode something about the structure of the

distortion. A result of Ricciardi [56] gave a great deal of information; in that paper, it

was shown that if f is a solution to the Beltrami equation ∂zf = µ∂zf then f is α-H¨older

55

continuous with exponent

α ≥

(cid:32)

sup
Sρ,x⊂Ω

1
|Sρ,x|

(cid:90)

Sρ,x

|1 − η2µ|2
1 − |µ|2 dσ

(cid:33)−1

(3.1.2)

where Sρ,x is a circle with radius ρ centered at x ∈ Ω, η is an outward unit normal, and

dσ is the arclength measure. This was proven through a sharp Wirtinger inequality. The

integrand here also appeared in [55] in the context of ring modules; for more information, as

well as some related estimates and theorems about extremizers, see the book [18] of Bojarski,

Gutlyanskii, Martio, and Ryazanov.

An important application of the regularity results for quasiconformal maps is their con-

nection with solutions to elliptic partial differential equations of the form

div(A∇u) = 0

(3.1.3)

where z (cid:55)→ A(z) is an essentially bounded, symmetric, measurable, matrix-valued function
satisfying λ(cid:104)ξ, ξ(cid:105) ≤ (cid:104)ξ, A(z)ξ(cid:105) ≤ Λ(cid:104)ξ, ξ(cid:105) for some 0 < λ ≤ Λ < ∞ at almost every z. Just as

there is a correspondence between the Cauchy-Riemann equation ∂zf = 0 and the Laplacian

∆u = 0 (where f = u + iv and v is the harmonic conjugate of u), there is a correspondence
between the C-linear Beltrami equation (3.1.1) and the divergence form elliptic equation

(3.1.3), where f = u + iv and v is the A-harmonic conjugate of u. The exact details of this

correspondence can be found in, e.g. Chapter 16 of [8].

Solutions to these equations are known to be H¨older continuous, and the study of their

regularity has a long history. H¨older continuity of solutions to (3.1.3) was shown by De

Giorgi [28]; later, Piccinini and Spagnolo [52] gave a quantitative estimate that the H¨older

56

continuity exponent of u is at least(cid:112)λ/Λ (as well as further improved bounds for the case

of an isotropic matrix A). More recently, Ricciardi [57] showed that the H¨older exponent is

at least

α ≥

(cid:32)

(cid:90)

sup
Sρ,x⊂Ω

1
|Sρ,x|

Sρ,x

(cid:33)−1

(cid:104)η, Aη(cid:105)dσ

.

The main result of this paper will be an improvement of the H¨older continuity exponent

given in (3.1.2) to incorporate an extra term involving the geometry of the underlying map.

In particular, we will show that

Theorem 3.1.1. Let f : Ω → Ω(cid:48) be a continuous and W 1,2
∂zf = µ(z)∂zf with |µ(z)| ≤ K−1

K+1. Then f is α-H¨older continuous for some exponent α

loc solution to the Beltrami equation

satisfying

(cid:34)

α ≥

|f (Dρ,x)|

H1(cid:0)f (Sρ,x)(cid:1)2

sup
Sρ,x⊂Ω

1
|Sρ,x|

4π sup
Sρ,x⊂Ω

(cid:90)

Sρ,x

|1 − ¯η2µ|2
1 − |µ|2 dσ

(cid:35)−1

where Sρ,x is the circle centered at x with radius ρ, η is the outward unit normal, and σ is

arclength measure.

Here, it is important to note that the isoperimetric inequality guarantees that

|f (Dt)|
H1(f (St))2 ≤ 1

4π

for all t (and is frequently strictly less than 1); here, Dt is the disk centered at the origin with

radius t. Our result therefore gives an improvement over the previously known regularity

whenever we can impose an upper bound on 4π

|f (Dt)|
H1(f (St))2 ; for example, any affine map which
stretches differently in two orthogonal directions will exhibit this. Furthermore, we can use

this information to determine the structure of the extremizers for H¨older continuity. We

57

have the following definition:

Definition 3.1.2. Let f be a K-quasiconformal map, a K-quasiregular map, or a solution

to (3.1.3) with ellipticity constants satisfying (cid:112)Λ/λ = K. We say that f is an extremizer

for H¨older continuity at the origin if f is not more than 1/K-H¨older continuous there. In
particular, for each 
 > 0, there is a sequence rn → 0 such that |f (rn) − f (0)| ≥ r

1/K+

n

.

We can now give the form of the complex distortion of a K-quasiconformal map that

exhibits the worst-case regularity. Motivated by the fact that the Beltrami coefficient of the
radial stretch z|z|1/K−1 is −kz/z with k = K+1

K−1, we have the following result:

Theorem 3.1.3. Suppose that f is K-quasiconformal and an extremizer for H¨older conti-

nuity at the origin. Write the Beltrami coefficient in the form µ(z) = z
there is a sequence of scales tn → 0 for which

z (−k + 
(z)). Then

(cid:90) 1

tn

1
r

(cid:90)

(cid:18) 1

|Sr|

Sr

(cid:19)

(cid:18)

(cid:19)

1
tn

Re 
(z) dσ

dr = o

log

.

There is an analogous theorem for the geometric distortion properties of such extremizers,

and how far they can be from a map which preserves circularity:

Theorem 3.1.4. Suppose that f is K-quasiconformal and an extremizer for H¨older conti-

nuity at the origin. Define a function δ by
tn → 0+ such that

|f (Dt)|
H1(f (St))2 =

(cid:18)

dr = o

log

.

1
tn

(cid:90) 1

δ(r)

r

tn

1

4π(1+δ(t)) . Then there is a sequence

(cid:19)

Our last main results are on the regularity and H¨older continuity extremizers of solutions

to the elliptic equation (3.1.3). Suppose u ∈ W 1,2

loc (Ω) is a continuous solution to div(A∇u) =

0 on a simply connected domain Ω, that A is symmetric, measurable, and satisfies the

58

ellipticity bound 1

K|ξ|2 ≤ (cid:104)ξ, A(z)ξ(cid:105) ≤ K|ξ|2 almost everywhere. Let v be the A-harmonic

conjugate of u and f = u + iv. We will show that

Theorem 3.1.5. If u is an extremizer for H¨older continuity at the origin, u must be of the

form

u = Φ ◦ g

where Φ is harmonic with non-vanishing gradient at g(0), g is K-quasiconformal, and g

exhibits the worst-case regularity for a K-quasiconformal map. In particular, the bounds of

the previous two theorems apply to the Beltrami coefficient and circular distortion of g.

The outline of this paper is as follows.

In Section 2, we prove the main theorem on

H¨older continuity of quasiconformal maps. In Section 3, we use this information to classify

the extremizers and study geometric distortion, and extend the results to quasiregular maps.

In Section 4, we apply the results of the previous sections to solutions to elliptic PDEs.

3.2 Estimate of H¨older Exponent

The main result of this section is estimate the exact degree of H¨older continuity of a K-

quasiconformal map through the behavior of the associated Beltrami coefficient.

Theorem 3.2.1. Let f : Ω → Ω(cid:48) be a continuous and W 1,2
equation ∂zf = µ(z)∂zf with |µ(z)| ≤ K−1

loc (Ω) solution to the Beltrami

K+1. Then f is α-H¨older continuous for some

exponent α satisfying

(cid:34)

α ≥

|f (Dρ,x)|

H1(cid:0)f (Sρ,x)(cid:1)2

sup
Sρ,x⊂Ω

1
|Sρ,x|

4π sup
Sρ,x⊂Ω

59

(cid:90)

Sρ,x

|1 − ¯η2µ|2
1 − |µ|2 dσ

(cid:35)−1

where Sρ,x is the circle centered at x with radius ρ, η is the outward unit normal, and σ is

arclength measure.

Here, the suprema can be regarded as the essential supremum over the radii ρ for each

point x. Note that the isoperimetric inequality guarantees that

|f (Dρ,x)|

H1(cid:0)f (Sρ,x)(cid:1)2 ≤ 1

4π sup
Sρ,x⊂Ω

and so we recover the result (3.1.2) of Ricciardi for the case of the homogeneous Beltrami

equation; moreover,

(cid:90)

|1 − ¯η2µ|2
1 − |µ|2 dσ ≤

Sρ,x

(cid:90)

Sρ,x

(1 + |µ|)2
1 − |µ|2 dσ =

(cid:90)

Sρ,x

1 + |µ|
1 − |µ|dσ ≤ K|Sρ,x|

for almost every circle (that is, for almost every positive radius). This recovers the classic

exponent of 1/K.

The theorem also shows that H¨older continuity of a quasiconformal map is locally deter-

mined by the structure of the Beltrami coefficient. For example, if there is an open set E
where (cid:107)µχE(cid:107)∞ < (cid:107)µ(cid:107)∞, then the quasiconformal map displays better-than-expected H¨older

continuity on the entirety of the open set. As will be proved later on, this idea also gives

powerful constraints on the complex distortion of a map which has the worst-case H¨older

continuity (even at a single point). It is worth mentioning that these results also follow from

Stoilow factorization. We now turn to the proof of Theorem 3.2.1.

Proof. Without loss of generality, we will look at circles centered at the origin. Our starting

ϕ(t) = (cid:82)Dt

point will be an adaptation of a classical argument of Morrey [47]. To this end, define
Jf = |f (Dt)|. If we can show that ϕ(t) ≤ ϕ(1)t2c, then quasisymmetry shows

60

that the worst case length distortion is controlled by

|f (teiθ) − f (0)|2 ∼K |f (Dt)| ≤ ϕ(1)t2c

which implies a H¨older exponent no worse than c. Our task is therefore to estimate ϕ, which

we will do by controlling ϕ by its derivative.

In order to do this, we will compute the circumference of the quasicircle f (St) explicitly,

where St has radius t and is centered at the origin. Parameterize the quasicircle by γ(θ) =
f (teiθ) for θ ∈ [0, 2π]; note that for almost every t, the quasicircle has positive and finite
length. Indeed, since f ∈ W 1,2
loc , f is absolutely continuous on the circle St for almost every
t ∈ [0,∞). Likewise, since the Beltrami coefficient µ is defined almost everywhere with

respect to the area measure, Fubini’s theorem guarantees that µ is defined at almost every

point (with respect to arclength) on almost every circle. We also have that fz = µfz almost

everywhere in the plane, so almost everywhere on almost every circle. Thus, we can compute

the length by

Length =

(cid:90) 2π

0

(cid:12)(cid:12)(cid:12)(cid:12) d

dθ

(cid:12)(cid:12)(cid:12)(cid:12) dθ.

γ(θ)

Writing f = u + iv, we have that

(cid:12)(cid:12)(cid:12)(cid:12) d

dθ

(cid:12)(cid:12)(cid:12)(cid:12)2

γ

1
t2

(cid:12)(cid:12)(cid:12)(cid:12) d
(cid:12)(cid:12)−tux sin θ + tuy cos θ + i(cid:2) − tvx sin θ + tvy cos θ(cid:3)(cid:12)(cid:12)2

[u(t cos θ, t sin θ) + iv(t cos θ, t sin θ)]

(cid:12)(cid:12)(cid:12)(cid:12)2

=

=

dθ

1
t2
1
t2
x sin2 θ + u2

= u2

y cos2 θ − 2uxuy sin θ cos θ

+ v2

x sin2 θ + v2

= |fx|2 sin2 θ + |fy|2 cos2 θ − 2 sin θ cos θ(cid:0)uxuy + vxvy

y cos2 θ − 2vxvy sin θ cos θ

61

(3.2.1)

(3.2.2)

(cid:1)

If we write the Beltrami equation in terms of x− and y−derivatives rather than the Wirtinger

derivatives, we find that

fx + ify = µ(fx − ify) =⇒ fx(1 − µ) = −ify(1 + µ) =⇒ fy = i

1 − µ
1 + µ

fx.

(3.2.3)

Furthermore,

Re(fxfy) = Re(cid:0)(ux − ivx)(uy + ivy)(cid:1) = uxuy + vxvy.

Thus, the final term in (3.2.2) can be replaced with

uxuy + vxvy = Re(fxfy)

(cid:18)

(cid:19)

(cid:19)

= Re

fxi

1 − µ
1 + µ
− Im

fx
1 − µ
1 + µ

(cid:18)
= |fx|2
|1 + µ|2 Im(cid:0)(1 − µ)(1 + µ)(cid:1)
= − |fx|2
= − |fx|2
|1 + µ|2 Im(µ − µ)
2|fx|2 Im µ
|1 + µ|2

=

(3.2.4)

We also wish to rewrite fx in terms of the Jacobian, so as to relate ϕ to ϕ(cid:48). We have

Jf = |fz|2 − |fz|2 = (1 − |µ|2)|fz|2.

On the other hand, fx = fz + fz = (1 + µ)fz, so that

1 − |µ|2
|1 + µ|2|fx|2

Jf =

62

(3.2.5)

Combining the equations (3.2.2)-(3.2.5), we find that the length H1(cid:0)f (St)(cid:1) of the quasicircle

f (St) can be written as

|1 + µ|

(cid:90) 2π
(cid:112)1 − |µ|2
(cid:90) 2π
f(cid:112)1 − |µ|2

1/2

=

J

0

0

1/2
J
f

(cid:12)(cid:12)(cid:12)(cid:12)1 − µ

(cid:115)
(cid:113)|1 + µ|2 sin2 θ + |1 − µ|2 cos2 θ − 4 sin θ cos θ Im µ tdθ.

cos2 θ − 4 sin θ cos θ

Im µ
|1 + µ|2 tdθ

(cid:12)(cid:12)(cid:12)(cid:12)2

sin2 θ +

1 + µ

(3.2.6)

It remains to simplify the term within the square root. We can expand it to find that

|1 + µ|2 sin2 θ+|1 − µ|2 cos2 θ − 4 sin θ cos θ Im µ

= 1 + |µ|2 + 2 Re µ sin2 θ − 2 Re µ cos2 θ − 4 sin θ cos θ Im µ
= 1 + |µ|2 − 2 Re µ cos 2θ − 2 Im µ sin 2θ

= 1 + |µ|2 − 2 Re(cid:0)(cos 2θ − i sin 2θ)(Re µ + i Im µ)(cid:1)

= 1 + |µ|2 − 2 Re(e−2iθµ)
= |1 − e−2iθµ|2

(3.2.7)

Noting that eiθ = η is the outer normal from the circle, combining this with (3.2.6) we arrive

at

H1(cid:0)f (St)(cid:1) =

(cid:90) 2π

0

(cid:18)|1 − η2µ|2

(cid:19)1/2

1 − |µ|2

1/2
J
f

t dθ.

(3.2.8)

We are now ready to make the estimate of ϕ. Denote

|f (Dt)|

H1(cid:0)f (St)(cid:1)2

A = 4π sup

t

and C = sup

t

1
|St|

(cid:90)

St

|1 − η2µ|2
1 − |µ|2 dσ.

63

recalling that dσ = tdθ is arclength. We then have for almost every t that

ϕ(t) =

|f (Dt)|

H1(cid:0)f (St)(cid:1)2H1(cid:0)f (St)(cid:1)2
(cid:32)(cid:90) 2π
(cid:18)|1 − η2µ|2
(cid:90) 2π
(cid:32)

1 − |µ|2

(cid:90)

|1 − η2µ|2
1 − |µ|2 t dθ
|1 − η2µ|2
1 − |µ|2 dσ

A

A

0

0

St

≤ 1
4π
≤ 1
4π

At

1
2πt
ACtϕ(cid:48)(t)

=

1
2
≤ 1
2

(cid:33)2

t dθ

1/2
J
f

(cid:19)1/2
(cid:90) 2π
(cid:33)(cid:18)(cid:90) 2π

Jf t dθ

0

(cid:19)

Jf t dθ

(3.2.9)

0

for almost every t. We then find that

(cid:104)

(cid:105)
t−2/AC ϕ(t)

d
dt

= t−2/AC−1

(cid:20)

ϕ(t) + tϕ(cid:48)(t)

− 2
AC

(cid:21)

≥ 0.

almost everywhere. Integrating this inequality over [t, 1] leads to ϕ(t) ≤ ϕ(1)t2/AC , which

is the desired result.

3.3 Extremizers for H¨older Continuity

Here we will study the structure of the Beltrami equation for the extremizers of H¨older

continuity. Recall that by Mori’s Theorem, a K-quasiconformal map is at least 1

K -H¨older

continuous. We will show that, in some sense, the extremizers must have Beltrami coefficients
which are very close to the coefficient for a pure radial stretch. Denote k = K−1

K+1; since the

64

Beltrami coefficient has |µ| ≤ k, we can always write

µ(z) = e2iθ (−k + 
(z))

(3.3.1)

with θ being the argument of z, and 
 some function such that has nonnegative real part.
Note that −ke2iθ is precisely the Beltrami coefficient of the radial stretch z|z|1/K−1. We

now have our result:

Theorem 3.3.1. Suppose that f is K-quasiconformal and an extremizer for H¨older continu-

ity at the origin. Write the Beltrami coefficient in the form (3.3.1). Then there is a sequence
of scales tn → 0 for which

(cid:90) 1

(cid:90)

1
r

tn

Sr

(cid:18)

(cid:19)

1
tn

Re 
(z) dτ dr = o

log

where dτ = dσ

2πr is normalized arclength.

titative sense - otherwise, the integral(cid:82) 1

Morally, this says that the circular averages of Re 
(z) are tending to zero in some quan-

dr
r would give some non-zero fraction of log 1/tn.

tn

(cid:90)

Proof. We proceed through two steps; the first is to estimate the impact that our perturba-

tion by 
 has on

g(t) :=

|1 − e−2iθµ|2

1 − |µ|2

dτ

Sr

and the second is to sharpen the estimate of |f (Dt)| accordingly. Recall that Theorem 3.2.1
(and in particular equation (3.2.9) with the estimate A ≤ 1) tells us that

ϕ(r) ≤ rg(r)
2

ϕ(cid:48)(r)

65

for almost every r ∈ [0, 1]. Integrating this differential inequality, we find that

(cid:90) 1

t

ln

ϕ(1)
ϕ(t)

=

ϕ(cid:48)(r)
ϕ(r)

dr ≥

2

rg(r)

t

(cid:90) 1

dr

(3.3.2)

Later on, we will prove that there is a constant c1 > 0 such that

(cid:90)

Sr

g(r) ≤ K − c1

Re 
 dτ

(3.3.3)

so that

1

g(r)

≥ 1
K

·

(cid:20)

1 + c

(cid:90)

Sr

(cid:21)

Re 
 dτ

≥ 1
K

Re 
 dτ

Sr

with c = c1/K > 0. With this estimate in mind, we can continue (3.3.2) to find that

1

1 − (c1/K)(cid:82)
(cid:90) 1

ln

ϕ(1)
ϕ(t)

2
r

≥

· 1
K
= ln t−2/K +

t

(cid:21)

(cid:20)

(cid:90)
(cid:90) 1

t

(cid:90)

Sr
1
r

2c
K

1 + c

Re 
 dτ

dr

Rearranging this gives us

ϕ(t) ≤ ϕ(1)t2/K exp

(cid:18)

−2

c
K

(cid:90) 1

t

1
r

Re 
 dτ dr

(3.3.4)

Sr

(cid:90)

Sr

(cid:19)

Re 
 dτ dr

(3.3.5)

Now if f is no more regular than 1/K-H¨older continuous and γ > 0, there is a sequence of
scales rn → 0 (depending on γ) for which ϕ(rn) ≥ r

ϕ(1) for all n. We therefore have

2/K+γ
n

2/K+γ
r
n

≤ r

2/K
n

exp

−2

c
K

(cid:18)

(cid:90) 1

(cid:90)

1
r

rn

Sr

(cid:19)

Re 
 dτ dr

(3.3.6)

66

which is the key estimate behind our constraint on the structure of Re 
. Equivalently,

(cid:90)

1
r

(cid:90) 1
(cid:90)

rn

c
K

(cid:90) 1

1
r

rn

Sr

Re 
 dτ dr

Sr

γ log rn ≤ −2

=⇒ γK
2c

log

≥

1
rn

Re 
 dτ dr

(3.3.7)

Taking a sequence of γm → 0 and choosing scales tm appropriately gives the desired estimate.
All that remains now is to show (3.3.3). Writing µ = e2iθ(−k + Re 
) and choosing

w = −k + Re 
 we have that

|1 − e−2iθµ|2

1 − |µ|2

|1 − w|2
1 − |w|2 .

=

We will show that

|1 − w|2
1 − |w|2 − 1 + k
1 − k

(cid:46) − Re 


(3.3.8)

where the implied constant only depends on k; the estimate (3.3.3) follows immediately from

integrating this over Sr, recalling that 1+k
1−k = K and that dτ is a probability measure. It is
worth mentioning that the estimate |1 − w|2/(1 − |w|2) − (1 + k)/(1 − k) ≤ 0 is immediate

from the triangle inequality, but that we need to sharpen it a little bit. To carry out this

estimate, observe that since w is real we have

|1 − w|2
1 − |w|2 − 1 + k
1 − k

=

=

=

1 − w
1 + w
(1 − w)(1 − k) − (1 + k)(1 + w)

− 1 + k
1 − k

(1 + w)(1 − k)

−2(w + k)
(1 + w)(1 − k)

67

−2 Re 


(1 + w)(1 − k)

=

(3.3.9)

Now w ≤ k since 
(z) lies within the disk centered at k with radius k, and so we have

|1 − w|2
1 − |w|2 − 1 + k
1 − k

≤

−2 Re 


(1 + k)(1 − k)

which proves (3.3.3).

In a similar manner, we can study the local distortion properties of an extremizer. In-

finitesimally, a quasiconformal map must take circles to ellipses with bounded eccentricity;

we will show that (in an appropriate sense), the extremizers for H¨older continuity must al-

most take circles to circles. Our approach to this is similar to the previous theorem: we will
write |f (Dt)|/H1(f (St))2 as a perturbation of 1/4π and control the perturbation.

Theorem 3.3.2. Suppose that f is K-quasiconformal and an extremizer for H¨older conti-

nuity at the origin. Define functions h and δ by

h(t) :=

|f (Dt)|
H1(f (St))2 =

1

4π(1 + δ(t))

.

Then there is a sequence tn → 0+ such that

(cid:90) 1

δ(r)

r

tn

(cid:18)

(cid:19)

.

1
tn

dr = o

log

Proof. As a first remark, the isoperimetric inequality shows that h(t) ≤ 1
δ(t) ≥ 0 everywhere; also, δ(t) < ∞ almost everywhere.

In analogy with the previous

4π for all t, so

68

theorem, we can conclude from (3.2.9) that

ϕ(t) ≤ 2πKth(t)ϕ(cid:48)(t).

Rearranging and integrating leads to

ϕ(t) ≤ ϕ(1) exp

(cid:18)

−

(cid:90) 1

(cid:19)

dr

2πKrh(r)

t

(3.3.10)

Now since f is an extremizer for H¨older continuity, for any sequence γn → 0+ there is a
sequence of scales tn → 0+ for which ϕ(tn) ≥ ϕ(1)t

; combining this with (3.3.10)

2/K(1+γn)
n

leads to

2
K (1+γn)
t
n

≤ exp

(cid:90) 1

(cid:18)

−

dr

2πKrh(r)

tn

(cid:19)

.

Taking a logarithm and rearranging, we find that

(1 + γn) log tn ≤ −

2
K

= −

dr

2πKrh(r)

(cid:90) 1
(cid:90) 1
(cid:90) 1

tn

1 + δ(r)

2πKr

tn
= − 2
K
r
log tn − 2
K

2
K

tn

=

dr

1

4π(1+δ(r))

dr

(cid:90) 1

δ(r)

r

tn

dr.

(3.3.11)

Rearranging this leads to

as desired.

(cid:90) 1

δ(r)

r

tn

dr ≤ γn log

1
tn

Corollary 3.3.3. Suppose f is an extremizer for H¨older continuity at the origin and δ is

69

defined as in Theorem 3.3.2. Then for any δ0 > 0, the set {r : δ(r) > δ0} has zero lower

density at 0.

Proof. Fix δ0 > 0 and suppose, intending a contradiction, that the lower density of {r :
δ(r) > δ0} had lower density at least η > 0 at zero. Then there exists a scale 
 > 0 such

that for all γ < 
,

|{r : δ(r) > δ0} ∩ [0, γ]}|

γ

>

η
2

.

Consequently, there exists an M depending only on η such that for all γ < 
,

|{r : δ(r) > δ0} ∩ [γ/M, γ]}|

γ

>

η
100

.

Therefore, we can estimate that

(cid:90) γ

δ(r)

γ/M

r

dr ≥ δ0
γ

|[γ/M, γ] ∩ {r : δ(r) > δ0}|

>

=

· γη
100

δ0
γ
δ0η
100

(3.3.12)

Summing (3.3.12) over intervals [γ/M, γ], [γ/M 2, γ/M ] and so on, it is immediate that

(cid:90) 1

t

dr (cid:38) log

δ(r)

r

1
t

for all sufficiently small t. This contradicts the result of Theorem 3.3.2 and the corollary

follows.

Finally, as usual, there is a natural generalization of the quasiconformal result to the

70

quasiregular result.

Theorem 3.3.4. A K-quasiregular map g is an extremizer for H¨older continuity at the origin
if and only if it is of the form g = Φ ◦ f , where f is K-quasiconformal and an extremizer

for H¨older continuity at the origin, and Φ is conformal in a neighborhood of f (0).

Proof. Suppose g is a K-quasiregular extremizer. By the Stoilow factorization theorem,

there exists a holomorphic Φ and K-quasiconformal f such that

g = Φ ◦ f.

Since Φ is a smooth function, Φ◦ f is at least as regular as f is (in particular, if f is α-H¨older

continuous, then so is g); but since g is not more than 1/K-H¨older continuous, neither
is f . Furthermore, we must have Φ(cid:48)(f (0)) (cid:54)= 0 (which implies conformality); otherwise,
Φ(f (z)) − Φ(f (0)) vanishes to at least second order at 0, and Φ ◦ f is H¨older continuous at
0 with exponent at least min{1, 2/K} > 1/K.

On the other hand, if g = Φ ◦ f with f an extremizer and Φ conformal at f (0), we can
locally invert Φ as a smooth function, so that f = Φ−1 ◦ g. Hence f is at least as regular as

g is; but since f has the worst-case regularity, so must g.

3.4 Applications to Elliptic PDEs

Next, we will use the relationship between quasiconformal maps and solutions to elliptic

partial differential equations in order to deduce regularity results and classify the extremizers

for H¨older continuity. The starting point for this work is the correspondence laid down in

[8], Chapter 16. Throughout, we will assume that A(z) is a matrix valued function which is

71

measurable, symmetric, and satisfies the ellipticity bound

|ξ|2 ≤ (cid:104)A(z)ξ, ξ(cid:105) ≤ K|ξ|2

1
K

at almost every z ∈ Ω. Note that this implies that A(z) is positive definite, and an equivalent

formulation is the unified inequality

|ξ|2 + |A(z)ξ|2 ≤

(cid:19)

(cid:18)

K +

1
K

(cid:104)A(z)ξ, ξ(cid:105).

Consider the divergence form equation

div A(z)∇u = 0.

(3.4.1)

If Ω is a simply connected domain and u ∈ W 1,2
lemma guarantees that there exists an A-harmonic conjugate v; that is, v ∈ W 1,2

loc (Ω) is a solution to (3.4.1), the Poincar´e

loc (Ω) solves

∇v = ∗A(z)∇u

where ∗ is the Hodge star operator, viewed as the matrix

 0 −1

 .

∗ =

1

0

Define f = u+iv; we now claim that the ellipticity condition implies that f is K-quasiregular.

Following Theorem 3.3.4, we will be able to use this to determine the regularity and extrem-

72

izers for H¨older continuity. To see that f is actually quasiregular, we may compute that

(cid:107)Df(cid:107)2 = |∇u|2 + |∇v|2 = |∇u|2 + | ∗ A(z)∇u|2 ≤

K +

1
K

(cid:104)A(z)∇u,∇u(cid:105).

(cid:18)

(cid:19)

It is immediate to check that (cid:104)A(z)∇u,∇u(cid:105) = J(z, f ) is the Jacobian of f , and we therefore
have (cid:107)Df(cid:107)2 ≤ (K + 1
derivatives, this implies that |∂zf| ≤ K−1

K )J(z, f ). Rewriting the Hilbert-Schmidt norm in terms of Wirtinger
K+1|∂zf| as desired. This brings us to the first theorem

on regularity, which is a direct application of Theorem 3.2.1:

Theorem 3.4.1. Let u ∈ W 1,2

loc (Ω) be a continuous W 1,2

loc (Ω) solution to (3.4.1) on a simply

connected domain Ω, where v is its A-harmonic conjugate, and f = u + iv. Let µ(z) denote

the complex distortion of f . Then u is α-H¨older continuous with some exponent α, where

(cid:34)

α ≥

|f (Dρ,x)|

H1(cid:0)f (Sρ,x)(cid:1)2

sup
Sρ,x⊂Ω

1
|Sρ,x|

4π sup
Sρ,x⊂Ω

(cid:90) |1 − ¯η2µ|2

(cid:35)−1

1 − |µ|2 dσ

.

In particular, u = Re f is H¨older continuous with exponent at least 1/K.

In a similar manner, we can find the extremizers for H¨older continuity.

Theorem 3.4.2. With the assumptions and notation of Theorem 3.4.1, suppose that u is

an extremizer for H¨older continuity at the origin. Then there is a harmonic map Φ and a

K-quasiconformal map g such that g is an extremizer for H¨older continuity at the origin,
Φ has non-vanishing gradient in a neighborhood of g(0), and f = Φ ◦ g. In particular, the

Beltrami coefficient and circular distortion of g satisfy the bounds of Theorems 3.3.1 and

3.3.2 respectively.

Proof. This is essentially the same idea as the proof of Theorem 3.3.4. Since u is the real
part of a K-quasiregular map f , we can use Stoilow factorization to write u = (Re Ψ) ◦ g

73

with Ψ holomorphic and g being K-quasiconformal. As before, Ψ must have non-vanishing

gradient at g(0) (so as not to improve the regularity), and g must be an extremizer for H¨older

continuity at the origin; the result follows.

Finally, we also have a generalization of the result on extremizers to nonlinear elliptic

PDEs.

Theorem 3.4.3. Let Ω ⊆ C be a simply connected domain and suppose A : Ω × C → C is
measurable in z ∈ Ω and continuous in ξ ∈ C and satisfies the ellipticity condition

(cid:18)

(cid:19)

|ξ|2 + |A(z, ξ)|2 ≤

Then if u ∈ W 1,2

loc (Ω) is a solution to

K +

1
K

(cid:104)ξ,A(z, ξ)(cid:105).

div A(z,∇u) = 0

(3.4.2)

Then if u is an extremizer for H¨older continuity at the origin, u is the form of Theorem

3.4.2.

Proof. By Theorem 16.1.8 of [8], every solution u ∈ W 1,2

loc (Ω) of (3.4.2) solves a linear elliptic

equation

div A(z)∇u = 0

with A(z) a positive definite, symmetric measurable matrix field of determinant 1. Further-
K|ξ|2 ≤ (cid:104)A(z)ξ, ξ(cid:105) ≤ K|ξ|2 and the result follows from

more, we have the ellipticity bound 1

Theorem 3.4.2.

74

Chapter 4

Geometric Bounds for Favard Length

This work first appeared in [20].

4.1

Introduction

Given a set E in the plane, its Favard length is the average

(cid:90) 2π

0

Fav(E) =

|πθE| dθ

where πθ is orthogonal projection onto a line Lθ through the origin at angle θ to the positive
x-axis, and |.| is the length measure within the line Lθ. This quantity is comparable to the

Buffon needle probability of the set E; this is the probability that a needle dropped near the

set E passes through it.

The Favard length of a set carries a great deal of metric and geometric information about

the set. It is deeply related to rectifiability; Bescovitch proved in [17] that a set with positive

and finite length is purely unrectifiable if and only if it has Favard length zero. In such a case,

the dominated convergence theorem implies that the Favard lengths of the r-neighborhoods

of a bounded set E (that is, the set E(r) of points of distance less than r from E) must

tend to zero as r does. The exact rate of decay is another measure of the size of a set, and

is related to Minkowski dimension. It is also conjectured that Favard length is controlled by

75

analytic capacity in many circumstances.

In this paper, we will give geometrically motivated proofs for various properties of Favard

length. First, we will reprove a result of Mattila from [44] that connects the decay rate of the

Favard lengths of the neighborhoods of a set with the Hausdorff dimension of the underlying

set:

Theorem 4.1.1. Fix s ∈ (0, 1) and suppose that E ⊆ R2 is measurable, and A ⊆ S1

is measurable with positive (arc-length) measure. Suppose there exists a sequence of scales
rn → 0 such that

(cid:90)

A

|πθ (E(rn))| dθ ≤ Crs

n

for some C < ∞. Then dimH E ≤ 1 − s.

The original proof of this theorem relies on potentials and estimates of the energy of a

measure; we will prove this result here with a direct geometric argument.

In the next section, we will show how self-similarity leads to new and useful properties

of the sequence of Favard lengths. In particular, we will show that:

Theorem 4.1.2. Suppose that {An}n∈N is a sequence of sets such that An+1 ⊆ An for all

n, that each generation can be written as a union

N(cid:91)

An+1 =

riAn + βi

for some fixed set of contraction ratios ri > 0, and that (cid:80)

i=1

i ri = 1. Then for each θ, the

sequence {|πθAn|}n∈N is convex.

Note that since the sum of contraction ratios is 1, the sequence of sets converges to an

attractor which is a self-similar set of Hausdorff dimension at most 1 (and if the similitudes

76

satisfy the open set condition, it has Hausdorff dimension equal to 1). See, e.g., Chapter 4

of [45] for more details.

Convexity gives a powerful constraint on the decay rate of Favard lengths: the decay

within the first few generations controls the decay until much later stages. In particular, it

is very easy to recover the result that:

Corollary 4.1.3. If Kn is the n-th generation of the four-corner Cantor set, then Fav(Kn) (cid:38)

1/n.

Before we begin the proofs, we first define some notation. Given a set A, we will denote its
Lebesgue measure by |A|; depending on the context, this could mean the Lebesgue measure

within a line, or area measure in the plane, or arc-length measure in the circle. If it is clear

from context which one of these is meant, we will not specify.

4.2 Dimension and Favard Length

In this section, we will prove that sufficiently quick decay of Favard length of neighborhoods

of a set controls the Hausdorff dimension of the set. For a set E in some Euclidean space

and r > 0, we denote the r-neighborhood of E by

E(r) = {x : dist(x, E) < r}.

The decay rate of the Lebesgue measure of E(r) as r → 0 is connected with the Minkowski

dimension of the underlying set, as well as other notions of size. Our main result is a new

proof of the following theorem:

Theorem 4.2.1. Fix s ∈ (0, 1) and suppose that E ⊆ R2 is measurable, and A ⊆ S1

77

is measurable with positive (arc-length) measure. Suppose there exists a sequence of scales
rn → 0 such that

(cid:90)

A

|πθ (E(rn))| dθ ≤ Crs

n

for some C < ∞. Then dimH E ≤ 1 − s.

The contrapositive of this theorem appeared in [44] with sharper bounds involving both

the measure of E with respect to an appropriate measure and the measure of the angle set A;

here, we only have a result at the level of dimension. Mattila’s argument relies on studying

the energy of a measure; here, we use a direct geometric argument. The previous proof relies

on being able to find a measure supported on the set that satisfies certain decay conditions,

which is guaranteed for compact (or Suslin) sets by Frostman’s lemma. Our technique has

the advantage of avoiding questions of the existence of such a measure, so we do not need

any additional topological assumptions about the set.

Proof. We proceed in three steps. First, we need to find a particular direction where the

projection πθE has full dimension while simultaneously having almost sufficiently quick
decay of |πθE(rn)|. Secondly, we will use a H¨older inequality to control the sum of lengths

over a natural cover on the projection side; this gives control on the Hausdorff measure of
the projection. Finally, we tighten the bounds by adjusting exactly how quickly |πθE(rn)|

decays. Note that we do not need to differentiate between the sets πθ(E(r)) and (πθE)(r)

(that is, the neighborhood of a projection within a line); they are equal.

First, note that E has Hausdorff dimension at most 1; otherwise, a result of Marstrand in

direction, which would contradict that(cid:82)

[41], Chapter II, would imply that E has strictly positive projection length in almost every
n → 0. (Of

A |πθ(E)| dθ ≤(cid:82)

A |πθ(E(rn))| dθ ≤ Crs

course, this follows from Mattila’s work in [44] or Chapter 9 of [45]; however, we are trying

78

to avoid the use of potentials). For each n, we can consider a set of angles

An =(cid:8)θ ∈ A : dimH(πθE) = dimH E and |πθE(rn)| ≤ rs−


n

(cid:9) .

since(cid:82)

The first condition holds for almost all θ; this also follows from Chapter II of [41]. Secondly,

A |πθE(rn)| ≤ Crs

n, we can estimate the size of the exceptional set Ac

n by

|A ∩ Ac

n| ≤ Cr

n.

Passing to a subsequence of scales (which we also denote as rn) if necessary, we can assume

that (cid:80)

n |A \ An| < |A|; thus, there exists an angle ϕ ∈ (cid:84)

n An. In particular, πϕE has

Hausdorff dimension equal to that of E itself.

Next, we will control the Hausdorff measure of πϕE at dimensions a little above s. Note

that πϕE(rn) consists of a union of disjoint intervals In,k, each having length at least 2rn.

This forms a natural cover of πϕ(E). We can estimate the number of intervals in the cover

via

rs−

n ≥ |πϕE(rn)| =

(cid:88)

k

|In,k|

Using that |In,k| (cid:38) rn, we can rearrange this to find that there are at most rs−
−1
intervals. We are now in a position to estimate sums of the form(cid:80)

such
k |In,k|p for p ∈ (0, 1).
A direct application of H¨older’s inequality shows that if p ∈ (0, 1), q satisfies 1/p − 1/q = 1,

n

and µ is any measure,

(cid:90)

(cid:18)(cid:90)

(cid:19)1/p(cid:18)(cid:90)

f p dµ

(cid:19)−1/q

g−q dµ

f g dµ ≥

79

holds for measurable functions. Taking this with counting measure, we find that

rs−


n ≥(cid:88)
(cid:32)(cid:88)
(cid:32)(cid:88)

≥

k

k

(cid:38)

|In,k|

(4.2.1)

(cid:33)1/p(cid:32)(cid:88)
(cid:33)1/p(cid:16)

k

(cid:33)−1/q
1−q
(cid:17)−1/q

rs−
−1

n

|In,k|p

|In,k|p

k

Rearranging this leads to (cid:32)(cid:88)

(cid:33)1/p

|In,k|p

k

s−
+ 1
(cid:46) r
n

q (s−
−1)

.

The exponent can be simplified as

(cid:19)

(cid:18)

1 +

1
q

(s − 
) − 1
q

=

1
p

(s − 
 − (1 − p)) .

As long as s − 
 − (1 − p) ≥ 0, we can give a uniform upper bound on(cid:80)

k |In,k|p; this works
provided that p ≥ 1− s + 
. Furthermore, one can see from (4.2.1) that each In,k has radius
no larger than rs−


n , which tends to zero as n grows. Combining these observations leads to

H1−s+
(cid:0)πϕE(cid:1) < ∞.

so that dimH πϕ(E) ≤ 1 − s + 
.

Finally, take a smaller 
 and rerun the argument with a (potentially) new choice of ϕ.

80

Taking a sequence 
m → 0, we then get a sequence ϕm of angles and

dimH E = dimH πϕ(E) ≤ 1 − s + 
m → 1 − s.

This is the desired bound on dimension.

4.3 Self-similar Sets

In this section, we will show how self-similarity can be used to give lower bounds on the

Favard length, as well as control the behavior of the sequence of projection lengths. Our

result is

Theorem 4.3.1. Suppose that {An}n∈N is a sequence of sets such that An+1 ⊆ An for all

n, that each generation can be written as a union

N(cid:91)

An+1 =

riAn + βi

for some fixed set of contraction ratios ri > 0, and that (cid:80)

i=1

i ri = 1. Then for each θ, the

sequence {|πθAn|}n∈N is convex.

Proof. Let us define En,θ = πθAn and αn(θ) = |En,θ|; note that on the projection side, En,θ
is also self-similar and is generated by similitudes of the form Ti : x (cid:55)→ rix + πθβi, where x

is measured within the line Lθ. We then have

αn(θ) − αn+1(θ) = |En,θ| − |En+1,θ|
= |En,θ \ En+1,θ|

81

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

Tj(En,θ)

j=1

Ti(En−1,θ) \ N(cid:91)
(cid:0)Ti(En−1,θ) \ Ti(En,θ)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

Ti(En−1,θ \ En,θ)

i=1

i=1

i=1

=

≤

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N(cid:91)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N(cid:91)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N(cid:91)
≤ N(cid:88)

=

ri|En−1,θ \ En,θ|

i=1

= αn−1(θ) − αn(θ).

where we have used that {En,θ}n∈N is a decreasing sequence of sets for each θ, that each Ti

is a contraction by ri along with a translation, and that (cid:80)

i ri = 1. Rearranging this, we

find that

which is the desired result.

αn(θ) ≤ αn−1(θ) + αn+1(θ)

2

As a corollary, we can easily deduce lower bounds on the Favard length of the four-corner
Cantor set. Recall that the generations of this set are constructed by taking K0 = [0, 1]2;
then Kn is constructed by taking each square in Kn−1, dividing it into four sections of equal

width horizontally and vertically, and taking the four subsquares at the corners. The result
is a family of 4n squares of sidelength 4−n each. Alternatively, the set is generated by the
4 z + βi, with {βi : 1 ≤ i ≤ 4} = {(0, 0), (0, 3/4), (3/4, 0), (3/4, 3/4)}. We

similitudes fi(z) = 1

have the following result:

Corollary 4.3.2. The Favard lengths of the generations of the four-corner Cantor set satisfy
Fav(Kn) (cid:38) 1
n.

82

Proof. Fix n ∈ N. Note that if we take θ∗ = arctan 1/2, the projection πθ∗ maps the four
components of K1 to four intervals that only overlap on the boundaries (and therefore, πθKn
is the same interval for all n, as an application of self-similarity). Therefore, α0(θ∗)−α1(θ∗) =
0. Furthermore, θ (cid:55)→ α0(θ)−α1(θ) is piecewise C1 and the derivative is bounded by 10 (which
follows from a direct computation of the function α0 − α1); moreover, α0(θ) ≥ 1 for all θ.
Hence, there is an interval In of length 1

20n centered at θ∗ such that

0 ≤ α0(θ) − α1(θ) ≤ 1
2n

for all θ in the interval. Applying convexity iteratively leads to αn(θ) ≥ 1

2 for all θ ∈ In, and

so

as desired.

Fav(Kn) ≥

(cid:90)

αn(θ) dθ ≥ 1
40n

In

Note that the key idea here is that there is a special angle at which the projection acts

(more or less) bijectively on components. It follows that this technique is applicable to a

broad class of self-similar sets with such an angle - the Sierpinski gasket is another important

example. One hopes that tightening the losses of this technique would be sufficient to improve
the estimate past 1/n; it was proved in [15] that the Favard length of Kn is actually at least

c ln n/n.

It is worth mentioning that the function n (cid:55)→ |E(4−n)| is not generally convex without

the self-similarity assumption. For example, the set

{0, 1/4, 1/2, 3/4, ..., 100}

83

(or a small neighborhood of it) serves as a counterexample. A modification of this example

(using ever finer lattices around carefully selected points in the set), one can find examples
where n (cid:55)→ |E(4−n)| is neither eventually convex nor eventually concave. Rather, the

sequence exhibits “see-saw” behavior as it decays to zero.

84

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