Removable Singularities for Planar Quasiregular Mappings

Removable singularities for planar quasiregular mappings

Albert Clop

PhD Thesis

Memoria` presentada per aspirar al grau de doctor en Matematiques`

iii iv CERTIFIQUEM que aquesta memoria` ha estat realitzada per Albert Clop, sota la direccio´ dels Drs. Joan Mateu Bennassar i Joan Orobitg Huguet.

Bellaterra, octubre del 2006.

Joan Mateu Bennassar

Joan Orobitg Huguet

Introduction

A compact planar set E is said to be removable for bounded analytic functions if for any open set Ω ⊃ E, any bounded function f : Ω → C holomorphic on Ω \ E admits a holomorphic extension to Ω. It is enough to care about the case Ω = C. The Painleve´ problem consists on characterizing these sets in metric and geometric terms.

The analytic capacity of a compact set E, introduced by Ahlfors [3], is deﬁned by

0 γ(E) = sup | f (∞)|; f ∈ H(C \ E), k f k∞ ≤ 1, f (∞) = 0 . This set function vanishes only on removable sets. Some excellent introductions to γ are given by H. Pajot [43] and J. Verdera [55]. As a matter of fact, H1(E) = 0 implies γ(E) = 0, while if dim(E) > 1 then γ(E) > 0. Thus, 1 is the critical dimension for the Painleve´ problem.

Some easier variants of this problem are obtained when replacing L∞ by other classes < of functions, such as BMO or Lipα for 0 α ≤ 1. To specify:

• Kaufman [31] showed that E is removable for BMO analytic functions if and only if M1(E) = 0.

• Dolzenkoˇ [16] proved that E is removable for Lipα analytic functions if and only if M1+α(E) = 0.

• It follows from Verdera’s work [54] that E is removable for VMO analytic func- 1 tions if and only if M∗(E) = 0.

1 As it can be seen, the main difference between these variants and the classical Painleve´ problem is that now removable sets are characterized by a Hausdorff content or, in some cases, other set functions, that in any case are not so complicated as analytic capacity.

The main purpose of this thesis is the study of removable sets for quasiregular mappings of the plane. Given a domain Ω ⊂ C, a function f : Ω → C is said to be µ- Ω 1,2 Ω quasiregular on if it belongs to the Sobolev space Wloc ( ) and its derivatives satisfy at almost every point the so called Beltrami equation,

∂ f = µ ∂ f

K−1 < for some measurable function µ, the Beltrami coefﬁcient, such that kµk∞ = K+1 1. If we forget µ and just care about K, then the above PDE is equivalent to the distortion inequality, |D f (z)|2 ≤ K J(z, f ) a.e. z ∈ Ω, where |D f (z)| is the operator norm of the differential matrix of f at the point z, and J(z, f ) is its Jacobian determinant. Then we talk about K-quasiregular mappings. Bijec- tive quasiregular mappings are called quasiconformal. When K = 1 (i.e. if µ = 0) we recover, respectively, the classes of analytic functions and conformal mappings on Ω. For several reasons, this is considered one of the most convenient ways of extending the notion of analycity. As a matter of fact, notice that the distortion inequality has an analog in Rn. However, our interest comes not only from this point of view, but also from the fact that these functions are (at least in the plane) solution to a more or less general model of an elliptic PDE of ﬁrst order. As introductory references to this subject, we cite Ahlfors [2], Lehto and Virtanen [37], Reshetnyak [48] and Rickman [49].

In this dissertation we will consider Painleve’s´ problem, as well as some of its variants, in the quasiregular context. More precisely, a compact set E is said to be removable for bounded K-quasiregular mappings if for every open set Ω ⊃ E, any bounded function f : Ω → C, K-quasiregular on Ω \ E, admits a K-quasiregular extension to Ω. It also

2 makes sense to replace the boundedness by other properties, such as being in BMO or in Lipα. Moreover, we can even talk about removable sets for µ-quasiregular mappings, when our interest focusses on the Beltrami coefﬁcient, rather than on K. In any case, as in the holomorphic setting, we just need to understand in all situations the case Ω = C.

It follows from Uy’s work [53] that if E has positive Lebesgue measure, then E is not removable, neither for K = 1. Conversely, given a compact set E of zero area, and a function f quasiregular on C \ E, in order to say that f can be quasiregularly extended to the whole plane, one just must make sure that f extends as a W 1,2 function, in a neighbourhood of E. To explain it, such an f already satisﬁes a Beltrami equation (or a distortion inequality) almost everywhere. Hence, only the integrability of its distributional derivatives has to be checked. In other words, we are facing a simpliﬁcated version of a Sobolev removability problem.

Chapter 2 is devoted to bounded and BMO K-quasiregular removability. In general, the Stoilow factorization Theorem (see for instance [37]) ensures that if f is K-quasiregular on C \ E, then f = h ◦ φ, where φ : C → C is K-quasiconformal and h is holomorphic on C \ φ(E). Moreover, both f and φ have the same Beltrami coefﬁcient. Notice also that

E φ(E) φ

f h

Figure 1 f is bounded if and only if h is, and Reimann [46] showed that a similar thing happens with BMO. Therefore, extending f K-quasiregularly is equivalent to extendig h holomorphically. Thus, in the bounded case we need to check under which conditions is it true that γ(φ(E)) = 0

3 whenever φ is K-quasiconformal. At the same time, for the BMO problem the condition H1(φ(E)) = 0 must be checked. In any case, is precisely at this point when the distortion problem appears. For instance, one is ﬁrst led to the dimension distortion formula of K. Astala [4], which says that if dim(E) = t, then

2K dim(E) dim(φ(E)) ≤ t0 = 2 + (K − 1) dim(E) for any K-quasiconformal mapping φ. Using this inequality, it is shown in [4] that a 1 < 2 sufﬁcient condition for H (φ(E)) = 0 (and therefore for γ(φ(E)) = 0) is t K+1 , or 0 < > 2 equivalently, t 1. In [4] it is also shown that for any real number t K+1 there exists a set E and a K-quasiconformal mapping φ such that dim(E) = t and dim(φ(E)) > 2 1. Hence, the number K+1 appears as the critical dimension for these K-quasiregular problems of removability. Afterwards, the natural following step to this dimensional setting consists on looking at the K-quasiconformal distortion of Hausdorff measures. For instance, it is well known (see for instance Ahlfors [2, p.33]) that

|E| = 0 ⇒ |φ(E)| = 0.

In this direction we prove the following Theorem.

Theorem 1 ([5]). Let E be a compact set, and φ a planar K-quasiconformal mapping. Then,

2 1 H K+1 (E) = 0 ⇒ H (φ(E)) = 0.

0 Therefore, there is absolute continuity of φ∗Ht with respect to Ht both for t0 = 1 and 0 2 t = 2, that is, t = K+1 and t = 2. For general t ∈ (0, 2) we cannot prove the absolute ∗ t0 t > 2 continuity of φ H with respect to H . However, for t K+1 we obtain a weaker result concerning Bessel capacities.

2 It follows from the above theorem and the BMO invariance that if H K+1 (E) = 0 then E is removable for BMO K-quasiregular mappings. Unexpectedly, for the L∞ problem

4 we can go further and obtain the following result.

Theorem 2 ([5]). 2 Let K > 1, and E be a compact set with σ-ﬁnite H K+1 (E). Then, E is removable for bounded K-quasiregular mappings.

In the proof of this Theorem we use several standard quasiconformal tools, together with the optimal integrability result for K-quasiregular mappings obtained by Astala and Nesi [10]. It also uses the fact that the lower 1-dimensional Hausdorff content behaves precisely as the analytic VMO capacity [54]. Moreover, there is a difﬁcult theorem appearing in [5] which asserts that if ψ is any K-quasiconformal mapping, and E is a subset of any rectiﬁable set, then

2 dim(ψ(E)) > . K + 1

In other words, these sets may not attain the maximal distortion. Furthermore, we can complete the proof by the results on analytic capacity from David [15] and Tolsa [51].

It can be seen that the construction of nonremovable sets for bounded (or BMO) K- quasiregular mappings is equivalent to some critical examples of extremal K-quasiconformal distortion. In that sense, we found sets E and K-quasiconformal mappings φ for which 2 > dim(E) = K+1 and γ(φ(E)) 0. As a consequence, one obtains the following result.

Theorem 3 ([5]). > 2 Let K 1. There exists compact sets E of dimension K+1 , non removable for bounded K- quasiregular mappings.

Our construction gives examples of compact sets E of dimension t and K-quasiconformal 0 mappings φ such that 0 < Ht (φ(E)) < ∞. However, we do not know if it is possible to 0 ﬁnd E and φ for which 0 < Ht(E) < ∞ and 0 < Ht (φ(E)) < ∞.

In Chapter 3, we consider the removability problem for K-quasiregular mappings, but ∞ with the Lipα condition instead of the L norm. Recall that a function f belongs to Lipα

5 if | f (z) − f (w)| ≤ C |z − w|α whenever |z − w| < 1. Although we again go back through Stoilow factorization, there are in this new setting some differences with respect to se situation in Chapter 2. As we have explained, the latter is strongly related to the way that general K-quasiconformal 2 mappings distort sets of dimension K+1 . A good understanding of this distortion is enough for the complete solution of the removability problem, since both BMO and L∞ are K-quasiconformally invariant spaces of functions. However, it comes from Mori’s

Theorem [41] that every K-quasiconformal mapping is of class Lip 1 , and the radial K 1 −1 stretching φ(z) = z |z| K shows that this Holder¨ exponent cannot be improved. It then follows that, unfortunately, the space Lipα is not K-quasiconformally invariant. Because of this non invariance, techniques of Chapter 2 will need to be modiﬁed.

Our ﬁrst result in this Lipα setting is the following.

Theorem 4. Let K > 1 and α ∈ (0, 1). Let E be compact, and assume that

2 (1+αK) H K+1 (E) = 0.

Then E is removable for α-Holder¨ continuous K-quasiregular mappings.

The proof of this result also uses strongly planar techniques. Roughly speaking, if C C C C f : → is a Lipα function on , K-quasiregular on \ E, then we show that ∂( f ◦ φ−1) = 0 in the sense of distributions (here φ is K-quasiconformal with the same Beltrami coefﬁcient of f ). As our main tools, we use the integrability properties of K- quasiconformal mappings, as well as a classical result on partitions of unity.

2 To show that the index d = K+1 (1 + αK) is sharp, extremal K-quasiconformal distortion is not enough, as it was in the L∞ problem. Again the principal reason is the non-invariance of the class Lipα under K-quasiconformal changes of variables. More- over, a compact set of dimension d may be K-quasiconformally sent to another set of

6 dimension bigger than 1 + α, which is the the critical dimension in the analytic case. The way to overcome these obstacles comes from an explicit example of extremal K- quasiconformal distortion.

Lemma 5. > 0 2Kt Let t ∈ (0, 2) and K 1. Denote t = 2+(K−1)t . There exists a compact set E and a K- quasiconformal mapping φ satisfying:

(a) Ht(E) is positive and σ-ﬁnite.

(b) dim(φ(E)) = t0.

t 1 K−1 (c) φ is Holder¨ continuous with exponent t0 = K + 2K t.

We must remark that in this Lemma the extremal distortion of dimension is nothing new. The interesting fact here is that the Holder¨ exponent of this mapping φ is strictly 1 above the usual K coming from Mori’s Theorem. This higher regularity ensures that we do not loose so much regularity when composing with φ than with any other K- quasiconformal mapping. Therefore, a similar procedure to that in [4] for the L∞ problem gives the following result.

Theorem 6. > 2 Let K ≥ 1 and α ∈ (0, 1). For any t K+1 (1 + αK), there exists a compact set E of dimension t, non removable for α-Holder¨ continuous K-quasiregular mappings.

2 We do not know what is happening for sets of dimension K+1 (1 + αK) and positive Hausdorff measure. Therefore, this remains as an open problem.

Among the consequences of the Main Lemma, we get not only the sharpness of the di- 2 mension K+1 (1 + αK) in the Lipα problem for K-quasiregular mappings, but also other applications. Recently, L. Kovalev [35] has found some of them in the setting of K- quasiregular gradient mappings. Furthermore, the techniques we used in the Lipα setting have been shown to apply in the more general ﬁeld of ﬁnite distortion mappings (see Sec- tion 3.4). In that case, 2α is the corresponding critical dimension.

7 When dealing with any removability problem, it is not strange to look for some sort of selfimprovement results. Phylosophically, these results are used to ensure that weak solutions to some equation are actually strong solutions. For instance, in Chapters 2 and 3, we will use repeatedly the fact that K-quasiconformal mappings exhibit better 1,2 Sobolev regularity than the usual Wloc appearing in its definition. In this setting, Weyl’s Lemma is an interesting tool, but has an unavoidable limitation, which is that it only works for constant coefficient operators. Therefore, the Beltrami equation seems to be a bad model for that kind of distributional results. The reason is that if f is a function then the identity hµ ∂ f , ϕi = −h f , ∂(µ ϕ)i (0.1) makes no sense, unless both µ and ϕ are functions in the class D (compactly supported C∞ functions). If one wants to work with more general Beltrami coefficients µ, then it is sufficient to ask that ∂ f acts on a bigger class of testing functions. As a matter of fact, 2 notice that equality (0.1) turns out to make sense if f ∈ Lloc, µ is a Beltrami coefficient of class W1,2, and our testing functions ϕ have also W1,2 regularity. This question is treated in Chapter 4. We are able to prove the following result, which may be interpreted as a Weyl’s Lemma in the quasiconformal setting.

Theorem 7. p C > Let f ∈ Lloc( ) for some p 2. Let µ be a compactly supported Beltrami coefﬁcient, such that µ ∈ W1,2(C). If h∂ f , ϕi = hµ ∂ f , ϕi

2,q C < for each ϕ ∈ D, then f ∈ Wloc ( ) for any q 2. In particular, f is µ-quasiregular.

One of the tools used in the proof of this result is that such good Beltrami coefﬁcients 1,p C < 2K give µ-quasiconformal mappings not only in Wloc ( ) for any p K−1 (this holds for K−1 2,q C any µ with kµk∞ ≤ K+1 ) but also in the second order Sobolev spaces Wloc ( ) for every q < 2. We know that this regularity result is sharp.

8 Since the improvement result is known, we can go back to the removability problem. Then we say that E is removable for bounded µ-quasiregular mappings if every bounded function f : C → C, µ-quasiregular to C \ E, is actually constant. As a difference, notice that analytic functions are not in general µ-quasiregular for every µ, while they are K-quasiregular for every K. So it means that this problem is refered precisely to the regularity of the PDE itself, more than to the concrete value of the constant K. Among the consequences of the Weyl’s Lemma, we ﬁnd that both BMO and VMO problems of characterizing the removable sets for µ-quasiregular mappings have precisely the same solution that for the ∂ equation. When reading it in terms of qua- 1 1 1 siconformal distortion, we get that M (E) = 0 iff M (φ(E)) = 0, and M∗(E) = 0 iff 1 1,2 C M∗(φ(E)) = 0. In other words, for W ( ) compactly supported Beltrami coefﬁcients, the corresponding quasiconformal mappings preserve the classes of removable sets for BMO (and VMO) analytic functions, just as any bilipschitz mapping does.

Of course, the following natural step is the removability L∞ problem. That is, we want to characterize removable sets for bounded µ-quasiregular mappings. We could also formulate the problem asking which are the compact sets E whose µ-quasiconformal images have always vanishing analytic capacity. This reformulation of the problem directly leds us to the bilipschitz invariance of analytic capacity of Tolsa [52]. More precisely, given a compact set E and a planar homeomorphism f , the analytic capacities γ(E) and γ( f (E)) are comparable if and only if f is bilipschitz. As a consequence, the bilipschitz image of a removable set is also removable. One of the main goals in Chapter 4 is to show that these µ-quasiconformal mappings we are dealing with, although they are not bilipschitz in general, they preserve sets of zero analytic capacity and σ-ﬁnite length.

Theorem 8. Let µ ∈ W1,2(C) be a compactly supported Beltrami coefﬁcient, and let φ be µ-quasiconformal. If E is a compact set such that H1(E) is σ-ﬁnite, then

γ(E) = 0 ⇔ γ(φ(E)) = 0

9 For the proof, it is very important that µ-quasiconformal mappings are diffeomor- phisms at every point in C, except on a set of zero length. The same happens to their inverses. This differentiability allows to show that rectiﬁable sets are mapped to recti- ﬁable sets. The characterization of G. David [15] does the rest.

This dissertation is divided into 4 chapters. In Chapter 1 we give an introduction and some general background. Each of the other 3 chapters contains different problems on removability for quasiregular mappings. Hence, each of them is independent from the others.

In Chapter 2 we consider questions on K-quasiconformal distortion and on removable singularities for bounded K-quasiregular mappings. It is based on the paper [5],

• K. ASTALA, A. CLOP, J. MATEU, J. OROBITG, I. URIARTE-TUERO, Distortion of Hausdorff measures and improved Painleve´ removability for quasiregular mappings, Sub- mitted.

In Chapter 3 we obtain some results related to the problem of characterizing the removable sets for α-Holder¨ continuous K-quasiregular mappings. There we include the results from [12] and [13]:

• A. CLOP, Removable singularities for Holder¨ continuous quasiregular mappings in the plane, To appear in Ann. Acad. Sci. Fenn. Ser. A I Math.

• A. CLOP, Nonremovable sets for Holder¨ continuous quasiregular mappings in the plane, To appear in Michigan Math. J.

Finally, Chapter 4 includes questions on removability for quasiregular mappings with the Beltrami coefﬁcient µ given in the Sobolev space W 1,2. In other words, we study the removable sets for the solutions of a given PDE. Here we have included the results of the preprint [14]

• A. CLOP, J. MATEU, J. OROBITG, Beltrami equations with coefﬁcient in the Sobolev space W1,2, Preprint (2006).

10 I have mostly used the notation I have found in the main references on quasiconformal mappings. In order to give consistence to the full text, I have made some changes with respect to the original versions of the papers.

Acknowledgements

I feel indebted to all those who have offered their support and backed me. For this reason I would like to express my sincere gratitude to them all.

To the Universitat Autonoma` de Barcelona, for its technical support. To Rosa Rodr´ıguez, who made all the drawings in these notes. To Miquel Brustenga, who made my days in Finland easier due to his technological support. To all the staff at the Departament de Matematiques` of the UAB, where I have always found a friendly and inspiring atmo- sphere.

To K. Astala, D. Faraco, P. Koskela, I. Prause, I. Uriarte-Tuero, J. Verdera and X. Zhong, for their time and suggestions.

To my friends. To Gerard, Dani, Juanje, Monica` and Isabel, because with you I have laughed very much, and hope to continue to do so. To my burbujitas, M´ıriam, Marta, Elisenda, Joan, Estrella, Carme and Laura. To Anna, Neus, Agust´ı, Pau, Mart´ı and Nausica, for being there. To my parents, who have always been at my side. To Xavi.

Finally to my advisors: Joan Mateu and Joan Orobitg, to which this dissertation is ded- icated. For all the many hours of discussion and for all the patience they have shown. They proposed the problems that are dealt with here. Their ideas, their advice, their constant support and their good sense of humor have been fundamental for the com- pletion of this thesis.

Contents

1 Preliminaries 17 1.1 Function spaces ...... 17 1.2 Hausdorff measures and contents ...... 21 1.3 The problem of Painleve´ ...... 23 1.4 Quasiconformal mappings ...... 25

2 Removable singularities for bounded quasiregular mappings 33 2.1 Introduction ...... 33 2.2 Absolute continuity ...... 36 2.3 Improved Painleve´ theorems ...... 48 2.4 Extremal distortion and nonremovable sets ...... 52

3 Removable singularities for Holder¨ continuous quasiregular mappings 61 3.1 Introduction ...... 61

3.2 Sufﬁcient conditions for Lipα removability ...... 64 3.3 Extremal distortion and nonremovable sets ...... 69 3.4 Removability results for ﬁnite distortion mappings ...... 81

4 Beltrami equations with coefﬁcient in the Sobolev space W1,2 91 4.1 Introduction ...... 91 4.2 Regularity of µ-quasiconformal mappings ...... 94 4.3 Distributional Beltrami equation ...... 98 4.4 µ-quasiconformal distortion of Hausdorff measures ...... 101 4.5 µ-quasiconformal distortion of analytic capacity ...... 106

1 Preliminaries

1.1 Function spaces

If Ω is any open set, we denote by D(Ω) the algebra of functions ϕ : Ω → C of class C ∞ with compact support contained in Ω. For these functions, and given a pair α = (α1, α2) of nonnegative integers, we denote

∂|α| ϕ ∂α ϕ = ∂xα1 ∂yα2

1 1 where |α| = α1 + α2. In particular, ∂ = 2 ∂x − i∂y and ∂ = 2 ∂x + i∂y . By Dϕ(z) we denote the differential matrix of ϕ at the point z, and J(z, ϕ) is the Jacobian determinant. The space of all distributions will be denoted by D0(Ω). We say that a distribution f vanishes on an open set Ω if h f , ϕi = 0 for every ϕ ∈ D(Ω). The following result is known as Weyl’s Lemma.

Theorem 1.1. Let f ∈ D0(Ω) be such that ∂ f = 0 on a planar domain Ω ⊂ C. Then, there exists a holomorphic function h : Ω → C such that f = h.

For p ∈ [1, ∞] and k = 1, 2, . . . , the Sobolev space W k,p(Ω) consists of all functions f for which the distribution itself and its distributional derivatives up to order k are represented by Lp(Ω) functions. The formula

1 p α k f kk,p = ∑ k∂ f kp |α|≤k !

17 1 Preliminaries

k,p Ω ∞ Ω k,p Ω deﬁnes a complete norm in W ( ), for which C ( ) is dense. By W0 ( ) we denote Ω k,p Ω k,p C k,p C k,p Ω the closure of D( ) in W ( ). In particular, W ( ) = W0 ( ). By Wloc ( ) we k,p Ω p Ω α p Ω mean the local Sobolev space, that is, f ∈ Wloc ( ) iff f ∈ Lloc( ) and ∂ f ∈ Lloc( ) k,p Ω whenever |α| ≤ k. We say that a sequence { f j}j converges to f in Wloc ( ) if

α α k∂ fj − ∂ f kLp(F) → 0 as j → ∞, for any compact set F ⊂ Ω and every |α| ≤ k.

1 Ω Ω We say that a function f ∈ Lloc( ) has bounded mean oscillation on , and write f ∈ BMO(Ω), if for any disk D there is a constant fD such that

1 k f k∗ = sup | f (z) − f | dA(z) < ∞ |D| D D⊂Ω ZD We say that a function f ∈ BMO(C) has vanishing mean oscillation on C, and write f ∈ VMO(C) if 1 lim | f − f | = 0 |D| D ZD 1 ∞ as |D| + |D| → . Equivalently, VMO may be introduced as the BMO-closure of compactly supported continuous functions.

Ω Given α ∈ (0, 1), the Holder¨ space Lipα( ) is the class of locally Holder¨ continuous func- Ω Ω tions of exponent α on the set , that is, f ∈ Lipα( ) if the quantity

| f (z) − f (w)| k f kα = sup α |z−w|<1,z6=w |z − w|

Ω is ﬁnite. It turns out that for α ∈ (0, 1) a function f ∈ Lipα( ) iff

1 | ( ) − | ( ) < ∞ sup 1+ α f z fD dA z Ω | | 2 D⊂ D ZD

18 1.1 Function spaces

Moreover, for any p ∈ (1, ∞), the quantity

1 1 p sup | f (z) − f |p dA(z) 1+ αp D D⊂Ω |D| 2 ZD ! is comparable to k f k∗ if α = 0, and to k f kα if α ∈ (0, 1).

p k,p The Sobolev Embedding Theorem establish relations between L , Lipα, BMO and W .

Theorem 1.2. Let p ∈ [1, ∞), and let Ω ⊂ C be either a disk or the whole plane.

< 1,p Ω q Ω 1 1 1 (a) If 1 ≤ p 2, then W ( ) ⊂ L ( ), where q = p − 2 .

(b) If p = 2, then W1,p(Ω) ⊂ BMO(Ω).

1,p (c) If 2 < p < ∞, then W (Ω) ⊂ Lip 2 (Ω). 1− p

An Orlicz function is a continuous function P : [0, ∞) → [0, ∞), nondecreasing, such that P(0) = 0 and lim P(t) = ∞. The Orlicz space LP(Ω) consists in all the measurable t→∞ functions f : Ω → C satisfying

P(λ| f (z)|) dA(z) < ∞ ZΩ for some λ > 0. The quantity

1 k f k = inf : P(λ| f (z)|) dA(z) < P(1) P λ ZΩ is called Luxemburg funtional. If P is convex then k · kP deﬁnes a complete norm on LP(Ω). As examples,

α • P(t) = et − 1. In this case, we denote LP(Ω) = Exp(Ω).

• P(t) = tp logα(e + t), α ≥ 1 − p. Now we write LP(Ω) = Lp logα L(Ω).

We remark that if α > 0 then the norm in L p logα L(Ω) can be computed without going

19 1 Preliminaries through the Luxemburg functional,

1 p p α | f (z)| k f k p α ' | f (z)| log e + dA(z) L log L k f k ZΩ p Two Orlicz functions P, Q are mutually conjugated iff there exists a constant C > 0 such that

| f (z)| |g(z)| dA(z) ≤ Ck f kLP (Ω) kgkLQ (Ω) ZΩ for all f ∈ LP(Ω) and g ∈ LQ(Ω). As examples,

p α q −α p < < ∞ > • P(t) = t log (e + t) and Q(t) = t log (e + t), q = p−1 , 1 p , α 0.

1 • P(t) = t logα(e + t) and Q(t) = et α − 1, for α > 0;

A more detailed discussion on Orlicz spaces can be found, for instance, in the books of Krasnosel’ski˘ı and Ruticki˘ı [36], as well as in Rao and Ren [45].

The Beurling Transform is the operator deﬁned on smooth functions as

−1 f (w) B f (z) = lim dA(w) (1.1) ε→0 π (w − z)2 Z|z−w|≥ε Equivalently, B may be deﬁned as

[ (B f )(ξ) = m(ξ) f (ξ)

ξ2 b with m(ξ) = . It is a Calderon-Zygmund´ operator, so that B is bounded in L p(C) |ξ|2 for all p ∈ (1, ∞), and for p = 1 it satisﬁes a weak estimate. If f ∈ W 1,p(C), p > 1, then

B(∂ f ) = ∂ f because by Plancherel we have

ω2 hB(∂ f ), gi = c ω f (ω) g(ω) dA(ω) = c ω f (ω) g(ω) dA(ω) = h∂ f , gi |ω|2 Z Z b b b b

20 1.2 Hausdorff measures and contents

The Beurling Transform can be thought as the derivative of the Cauchy Transform. It is deﬁned for smooth functions as

1 f (w) C f (z) = − dA(w) (1.2) π w − z Z If f ∈ Lp(C) for some p > 1 then C f ∈ W1,p(C) and

∂(C f ) = B f ∂(C f ) = f

One may normalize this operator in such a way that the image function ﬁx the origin. Thus, we deﬁne as Ahlfors [2, p.85],

1 1 1 C˜ f (z) = − f (w) − dA(w) (1.3) π w − z w Z Actually, for smooth enough functions C˜ f (z) = C f (z) − C f (0). Thus, for any p ∈ (1, ∞), if f ∈ Lp(C) then C˜ f ∈ C + W1,p(C). In any case, if p > 2 then

|C f (z) − C f (w)| |C˜ f (z) − C˜ f (w)| = ≤ Cp k f kp 1− 2 1− 2 |z − w| p |z − w| p

1.2 Hausdorff measures and contents

We call measure function to any nondecreasing function h : [0, ∞) → [0, ∞) such that lim h(t) = 0. Given a measure function h, for every δ ∈ (0, ∞] we denote t→∞

∞ h Hδ (F) = inf ∑ h(diam(Dj)) (j=1 )

∞ where the inﬁmum runs over all countable coverings of F, F ⊂ j=1 Dj, by disks Dj of ∞ h h diameter diam(Dj) ≤ δ. If δ = , then we write M (F) = HS∞(F). It is easy to see h h h that Hδ deﬁnes a monotonic and subadditive set function, such that Hδ (F) ≤ He (F) if

21 1 Preliminaries

0 < e ≤ δ. The limit h h H (F) = lim Hδ (F) δ→0 deﬁnes a Borel regular measure (see for instance Mattila [39]), satisfying

h h H (F) = 0 ⇐⇒ Hδ (F) = 0 for any δ ∈ (0, ∞]. In the particular case h(t) = td, d > 0, then we write Mh(F) = Md(F) and Hh(F) = Hd(F), and we call them d-dimensional Hausdorff content and d- dimensional Hausdorff measure of F, respectively. We can give, to every set F, a real number deﬁned by

dim(F) = inf{d > 0 : Md(F) = 0} = sup{d > 0 : Md(F) > 0} which will be called Hausdorff dimension (or simply dimension) of F.

The following result, known as Frostman’s Lema (see e.g. [39, p.112]), characterizes the compact sets with positive d-dimensional Hausdorff content.

Theorem 1.3. Let E be a compact set, and d ∈ (0, 2]. Then, Md(E) > 0 if and only if there exists a positive Radon measure ν, supported on E, such that ν(D(z, r)) ≤ C rs for any z ∈ E and r > 0. Further, ν can be chosen so that ν(E) & Md(E).

The lower d-dimensional Hausdorff content of F is

d h M∗(F) = sup M (F) h∈Fd

d where F = Fd is the set of all measure functions h(t) = t ε(t), 0 ≤ ε(t) ≤ 1, such that d d d < d lim ε(t) = 0. In general, M∗(F) ≤ M (F), but it may happen that M∗(F) = 0 M (F). t→0 1 1 For instance, if F is the line segment [0, 1] on the plane, then M∗(F) = 0 but M (F) = 1. d d d d For a disk D, M∗(D) = M (D), and for open sets U, M∗(U) ' M (U). The following is an old result due to Sion and Sjerve [50].

22 1.3 The problem of Painleve´

Theorem 1.4. C t t Let E ⊂ be a compact set, and let t ∈ (0, 2]. Then, M∗(E) = 0 if and only if H (E) is σ-ﬁnite.

We say that Γ ⊂ C is a rectiﬁable curve if Γ = f (I) for a Lipschitz map f : I → C, where R C R I ⊂ is a bounded interval. A set E ⊂ is called rectiﬁable if there exists subsets Ij ⊂ C and Lipschitz maps fj : Ij → such that the set

∞

E \ fj(Ij) j[=1 has zero length. A set E ⊂ C is called purely unrectiﬁable if

H1(E ∩ Γ) = 0 for any rectifiable curve Γ ⊂ C. It is well known that rectifiable sets have always σ- finite length. They are stable under countable unions, and any subset of a rectifiable set is also rectifiable. The following is a well known result due to Besicovitch (see e.g.[39, p.205]).

Theorem 1.5. If A ⊂ C has ﬁnite length, then A admits a decomposition

A = R ∪ P ∪ Z where R is rectiﬁable, P is purely unrectiﬁable, and H1(Z) = 0.

1.3 The problem of Painleve´

A set E is called removable for bounded analytic functions if for every open set Ω ⊃ E, every function f ∈ L∞(Ω) holomorphic on Ω \ E extends holomorphically to Ω. In this deﬁnition, one may consider Ω = C. The classical problem of Painleve´ consists on giving metric and geometric characterizations of removable sets for bounded analytic

23 1 Preliminaries functions. Related to this problem one has an important set function, called analytic capacity, introduced by Ahlfors in [3] and deﬁned as

0 γ(E) = sup | f (∞)|; f ∈ H(C \ E), k f k∞ ≤ 1, f (∞) = 0 . (1.4) Here H(Ω) is the space of functions which are holomorphic on the open set Ω, f (∞) = lim f (z) and f 0(∞) = lim z ( f (z) − f (∞)). Notice that by means of Stokes Theorem |z|→∞ |z|→∞ one may show that if f ∈ H(C \ E) then

| f 0(∞)| = |h∂ f , 1i|

It turns out that γ(E) = 0 if and only if E is removable for bounded analytic functions. Moreover, the classical Painleve´ Theorem asserts that γ(E) . H1(E), while dim(E) > 1 implies γ(E) > 0. Therefore, only sets E with dim(E) = 1 are interesting.

The following result establishes the solution to Painleve’s´ Problem for those sets with ﬁnite length. It is due to G. David [15].

Theorem 1.6. Let E be compact, and suppose that H1(E) < ∞. Then,

γ(E) = 0 ⇔ E is purely unrectiﬁable

Later on, it comes from Tolsa [51] that this result turns out to be true even for sets E with σ-finite length. At the same time, for sets with Hausdorff dimension 1 but non σ-finite length we have the following nonremovability sufficient condition from P. Mattila [40].

Theorem 1.7. Let h be a measure function, and let E be a compact set such that Hh(E) > 0. If

1 h(t)2 dt < ∞ t3 Z0 then γ(E) > 0.

24 1.4 Quasiconformal mappings

A complete solution to Painleve’s´ problem is given by Tolsa in [51], in terms of Menger curvature. Another main result in [51] is the so called semiadditivity.

Theorem 1.8. There exists an absolute constant C such that

∞ ∞ γ Ej ≤ C ∑ γ(Ej)   j=1 j[=1   for any family of sets Ej.

Recall that a function f : C → C is called bilipschitz if there exists a constant C such that

1 |z − w| ≤ | f (z) − f (w)| ≤ C|z − w| C

The following theorem is due to X. Tolsa [52].

Theorem 1.9. Let f : C → C be a bilipschitz mapping. Then, for any compact set E ⊂ C,

γ(φ(E)) ' γ(E)

1.4 Quasiconformal mappings

Let K ≥ 1. A homeomorphism φ : Ω → Ω0 between planar domains Ω, Ω0 ⊂ C is called 1,2 Ω K-quasiconformal if it belongs to the Sobolev space Wloc ( ) and satisﬁes

max |∂αφ(z)| ≤ K min |∂αφ(z)| (1.5) α α at almost every z ∈ Ω. This inequality is equivalent to the distortion inequality,

|D f (z)|2 ≤ K J(z, f ) (1.6)

25 1 Preliminaries

From this inequality, it follows that at small scales

max |φ(z) − φ(z0)| ≤ C min |φ(z) − φ(z0)| (1.7) |z−z0|=r |z−z0|=r for a constant C that depends only on K. Moreover, homeomorphisms satisfying (1.7) are K-quasiconformal for some K. In some cases, the following equivalent deﬁnition, given in terms of a partial differential equation, will be more convenient. We call Bel- ∞ trami coefﬁcient to any measurable function µ ∈ L (Ω) such that kµk∞ < 1. Then, a Ω Ω 1,2 Ω µ-quasiconformal mapping is any homeomorphism φ : → φ( ) lying in Wloc ( ) that solves the Beltrami equation, ∂φ = µ ∂φ (1.8)

From this point of view, φ is K-quasiconformal if and only if it is µ-quasiconformal K−1 for some µ with kµk∞ ≤ K+1 . By quasiconformal we mean K-quasiconformal for some K ≥ 1. Here we give some examples:

• All the conformal mappings, which are precisely the only 1-quasiconformal ones.

• Given 1 ≤ K ≤ K0, every K-quasiconformal mapping is also K0-quasiconformal.

1 K −1 K−1 z • The radial stretching, φ(z) = z |z| , whose Beltrami coefﬁcient is µ(z) = − K+1 z .

z 1 • φ(z) = z (1 − log |z|), with µ(z) = z 1−2 log |z| .

Among the above examples, we remark that the radial stretching is in some sense extremal to obtain regularity of K-quasiconformal mappings. We show in the following example an explicite method for solving the Beltrami equation, under the additional assumption that µ is compactly supported (see e.g.in Ahlfors [2, p.90]).

Example 1.10. We show that if µ is a compactly supported Beltrami coefﬁcient, then there exists exactly one µ-quasiconformal mapping φ such that

φ(z) − z = O(1/|z|) (1.9)

< as |z| → ∞. To construct it, ﬁrst notice that kµBkL2(C)→L2(C) ≤ kµk∞ 1. Hence, the

26 1.4 Quasiconformal mappings formula ∞ Tg = ∑ (µB)n(g) = g + µB(g) + µB(µB(g)) + . . . n=o deﬁnes a bounded linear operator T : L2(C) → L2(C), such that

(I − µB)(Tg) = T(I − µB)(g) = g that is, T = (I − µB)−1. Actually, this holds also in Lp(C) for some values p > 2, since kBkLp(C)→Lp(C) is a continuous function of p [37, p.214]. Deﬁne h = Tµ (then h = µBh + µ) and let φ(z) = z + Ch(z)

1,p C 1,p C Clearly, φ ∈ Wloc ( ), ∂φ = 1 + Bh and ∂φ = h. Thus, φ is a Wloc ( ) solution to the Beltrami equation ∂φ = µ ∂φ. Moreover, an approximation argument [2] shows that in fact the function φ is globally injective. On the other hand, classical topology results ensure that φ is onto. Finally, since µ is compactly supported, h is too, thus it is clear that |Ch(z)| decreases at 1 inﬁnity as |z| . In this construction, different normalizations are also allowed. For instance, there exists exactly one solution φ˜ satisfying

φ˜(0) = 0 and φ˜(z) − (z + a) ∈ W1,p(C) (1.10) for some constant a ∈ C and some p > 2. This may clearly be done just removing a constant,

φ˜(z) = z + C˜h(z) = φ(z) − φ(0)

Notice that normalizations (1.9) and (1.10) differ only in a linear mapping. More precisely, φ and φ˜ satisfy the relation φ˜ ◦ φ−1(z) = z − a. For compactly supported Beltrami coefﬁcient µ, we call both φ and φ˜ principal solution to the Beltrami equation ∂φ = µ ∂φ, and we will specify the normalization (1.9) or (1.10) just when needed.

For general µ, the following result holds not only in the plane, but also on every simply connected domain (see for instance Lehto and Virtanen [37]).

27 1 Preliminaries

Theorem 1.11. Let µ be a Beltrami coefﬁcient. There exists one µ-quasiconformal mapping φ : C → C. More- C C −1 over, if φ1, φ2 : → are µ-quasiconformal, then φ1 ◦ φ2 is conformal.

We are interested not only in homeomorphic, but also in more general solutions to Beltrami equations. Thus we say that a function f : Ω → C is a K-quasiregular mapping Ω C 1,2 Ω in an open set ⊂ if f ∈ Wloc ( ) and satisﬁes inequality (1.5). Of course, we may also talk about µ-quasiregular mappings, whenever a reference to the Beltrami coefﬁcient is needed. As before, a quasiregular map is just a K-quasiregular one, for some K ≥ 1. It is clear that when µ = 0 (or when K = 1) one gets the class of analytic functions. 1,2 For general µ, it turns out that homeomorphic solutions characterize the rest of Wloc solutions. This fact is known as Stoilow’s Theorem.

Theorem 1.12. Let µ be a Beltrami coefﬁcient. Let φ be any µ-quasiconformal mapping.

(a) If h is holomorphic, then f = h ◦ φ is µ-quasiregular.

(b) If f is µ-quasiregular, then h = f ◦ φ−1 is holomorphic.

Therefore, any K-quasiregular function f may be written as a K-quasiconformal change of variables φ of a holomorphic function h. Moreover, both f and φ have the same Bel- trami coefﬁcient. Thus, all regularity properties of K-quasiregular functions are characterized by its K-quasiconformal component.

One of the typical features of quasiconformal mappings is that they exhibit some strong properties of normality (for an exhaustive discussion see the book of Lehto and Virtanen [37]). To keep an idea in mind, assume that F is an arbitrary family of K-quasiconformal mappings φ : C → C, normalized in some way. Normalization for quasiconformal mappings from C to itself can be given by either equation (1.9) or (1.10). Other possi- bilities of normalizing come from the condition diam(φ(D)) = 1, or even letting ﬁxed 3 points in the Riemann sphere. Assume, for instance, that every mapping in F ﬁxes 0, 1 and ∞. Then, one of the following properties implies the other two:

28 1.4 Quasiconformal mappings

• F is an equicontinuous family, when restricted to any compact set.

• F is a compact family, when restricted to any compact set.

• F is uniformly Holder¨ continuous, when restricted to any compact set.

It turns out that we always have these properties, as was shown by Mori [41] in the following theorem. The radial stretching shows that it is sharp.

Theorem 1.13. Let φ : C → C be a normalized K-quasiconformal mapping. Then,

1 |φ(z) − φ(w)| ≤ C |z − w| K where C depends only on K.

A K-quasidisk (resp. a K-quasicircle) is the image of the unit disk D (resp. the unit circle ∂D) under a K-quasiconformal mapping C → C. It follows from (1.7) that for any disk D and any K-quasiconformal mapping φ,

1 diam(φ(D)) ' |φ(D)| 2 with constants that may depend on the normalization. Therefore, quasidisks have area comparable to the square of its diameter. Now, for any disk D, Mori’s Theorem says that 1 |φ(D)| ≤ C |D| K and it is natural to ask wether this holds for a more general class of sets. After several years, this was shown to be true in a celebrated paper of K. Astala [4]. The principal result in this paper is kown as the Area Distortion Theorem .

Theorem 1.14. Let φ be a normalized K-quasiconformal mapping on a domain Ω ⊂ C. For any measurable set E ⊂ Ω, 1 |φ(E)| ≤ C |E| K

29 1 Preliminaries

Along these notes we will see how trascendental is this result. To start, observe that if φ is any homeomorphism with locally integrable jacobian determinant, then higher integrability of J(·, φ) is equivalent to some area distortion theorem. More precisely, 1 ∞ 1−α , J(·, φ) ∈ Lloc for some α ∈ (0, 1) if and only if

|φ(E)| ≤ C |E|α for every measurable set E. However, if we assume now that φ is quasiconformal, then 2 ∞ 1−α , this is equivalent to saying that Dφ ∈ Lloc . As a consequence, it follows the sharp integrability result for K-quasiconformal mappings.

Theorem 1.15. 2K ,∞ C C K−1 Let φ : → be K-quasiconformal mapping. Then, Dφ ∈ Lloc .

The sharpness comes again from the radial stretching. In particular, every K-quasiregular mapping has also this regularity.

It turns out that something better may be said under additional assumptions of con- formality inside some set. More precisely, this sharp integrability exponent may be at- tained in some cases. The following result is due to K. Astala and V. Nesi [10].

Theorem 1.16. Let φ : C → C be a K-quasiconformal mapping, conformal on C \ D, normalized with the 1 ∞ D condition φ(z) − z = O( |z| ) as |z| → . Assume that E ⊂ is such that ∂φ = 0 at almost every point of E. Then, J(z, φ)p dA(z) ≤ 1 ZE K with p = K−1 .

Notice that the normalization (1.9) is not the only one allowed, provided that the jacobian determinant does not change too much. For instance, under normalization (1.10) the above result still holds true.

30 1.4 Quasiconformal mappings

Ω C 1,p Ω We say that f : → is weakly K-quasiregular if and only if f ∈ Wloc ( ) for some p > 1, and f satisﬁes (1.5). The following is a dual version of Theorem 1.15.

Theorem 1.17. 1,p Ω > 2K Let f ∈ Wloc ( ) be weakly K-quasiregular. If p K+1 , then f is K-quasiregular.

It turns out that Theorems 1.15 and 1.17 are strongly related to the problem of ﬁnding the range of real numbers p for which the Beltrami operators

I − µB : Lp(C) → Lp(C)

K−1 are invertible whenever kµk∞ ≤ K+1 . This problem was solved by K. Astala, T. Iwaniec 2K 2K and E. Saksman in [8], where this range was shown to be precisely ( K+1 , K−1 ). More- over, S. Petermichl and A. Volberg [44] showed that one can get injectivity even at the lower borderline. From this, it follows that even weakly K-quasiregular mappings in 2K 1, K+1 Wloc are K-quasiregular.

Another consequence of the Area Distortion Theorem is the so called Dimension Dis- tortion Theorem , also proved by Astala in [4].

Theorem 1.18. Let φ be a K-quasiconformal mapping on a domain Ω. Then,

2K dim(E) dim(φ(E)) ≤ (1.11) 2 + (K − 1) dim(E) for any compact set E ⊂ Ω.

Observe that if φ is K-quasiconformal, then also φ−1 is. Thus, the above bounds may also be read as

1 1 1 1 1 1 1 − ≤ − ≤ K − (1.12) K dim(E) 2 dim(φ(E)) 2 dim(E) 2

The sharpness of these bounds at the level of dimensions is due to Astala [4].

31 1 Preliminaries

Theorem 1.19. Given t ∈ (0, 2) and K ≥ 1, there is a K-quasiconformal mapping φ : C → C and a compact set E of dimension t, such that

2Kt dim(φ(E)) = 2 + (K − 1)t

32 2 Removable singularities for bounded quasiregular mappings

2.1 Introduction

We say that a compact set E is removable for bounded K-quasiregular mappings, or simply K-removable, if for every open set Ω ⊃ E, every bounded K-quasiregular mapping f : Ω \ E → C admits a K-quasiregular extension to Ω. In this deﬁnition, as in the analytic setting, we may replace L∞(Ω) by BMO(Ω). In this chapter we want to give metric and geometric sufﬁcient conditions on E to be K-removable. At the same time, we want non trivial examples of non removable sets E.

The Stoilow factorization Theorem converts bounded (or BMO) quasiregular mappings in bounded (or BMO) analytic mappings. Therefore, a compact set is removable for the K-quasiregular problem if and only if all its K-quasiconformal images are removable for the corresponding analytic problem. One is then led to the celebrated Astala’s paper on quasiconformal distortion [4]. In particular, K-quasiconformal mappings φ send compact sets E of Hausdorff dimension t to compact sets φ(E) of dimension

2Kt dim(φ(E)) ≤ . 2 + (K − 1)t

The ﬁrst question we study in this chapter is if this estimate may be improved to the level of Hausdorff measures Ht. In other words, if φ is a planar K-quasiconformal < < 0 2Kt mapping, 0 t 2 and t = 2+(K−1)t , is it true that

0 Ht(E) = 0 ⇒ Ht (φ(E)) = 0, (2.1)

33 2 Removable singularities for bounded quasiregular mappings

0 that is, φ∗Ht Ht. Classical results of Ahlfors [2] and Mori [41] assert that this is true 2 when t = 0 or t = 2. As a ﬁrst main result of this chapter we prove (2.1) for t = K+1 , i.e. for the case of image dimension t0 = 1.

Theorem. Let φ be a planar K-quasiconformal mapping, and let E be a compact set. Then,

K+1 1 2 2K M (φ(E)) ≤ C M K+1 (E) . (2.2) As a consequence, 2 1 H K+1 (E) = 0 ⇒ H (φ(E)) = 0.

A key point in proving the above result is that we are able to show that for planar quasiconformal mapping h conformal on C \ E and normalized at ∞ the inequality (2.2) can be improved. Such mappings h essentially preserve the 1-dimensional Hausdorff content, 1 1 M (h(E)) ≤ CK M (E) (2.3) where CK depends only on K. For the area the same estimate was shown by Astala in [4]. In fact, as we will see later, a counterpart of (2.3) for the t-dimensional Hausdorff 0 content Mt is the only missing detail for proving the absolute continuity φ∗Ht Ht for general t. Towards solving (2.1) we conjecture that actually

Mt(h(E)) ≤ C Mt(E), 0 < t ≤ 2, whenever E ⊂ C compact and h normalized and conformal in C \ E, with a K-quasiconformal extension to C. For more detailed discussion and other formulations see Section 2.

The reason our methods work in the special case of dimension t = 1 only is that content M1 is equivalent to analytic BMO capacity [54]. For exponents 1 < t < 2, we do have interpolating estimates but unfortunately we have to settle for the Riesz capacities. We have

Cα,t(h(E)) ≤ C Cα,t(E)

34 2.1 Introduction

2 for any t ∈ (1, 2), and where α = t − 1. This fact has consequences towards the absolute continuity for general quasiconformal mappings, but these are not strong enough for (2.1) when 1 < t0 < 2.

Going back to the removability problem, Iwaniec and Martin previously conjectured n [29] that in Rn, n ≥ 2, sets with Hausdorff measure H K+1 (E) = 0 are removable for bounded K-quasiregular mappings. A positive answer for n = 2 was found in [6]. In the present work we show that, surprisingly, for K > 1 one can do even better.

Theorem. Let K > 1 and suppose that E is any compact set with

2 H K+1 (E) σ − ﬁnite.

Then E is removable for all bounded K-quasiregular mappings.

The theorem fails for K = 1, since for instance the line segment E = [0, 1] is not remov- > 2 able. Note that by [4], for any t K+1 there are non-K-removable sets with dim(E) = t. Furthermore, we improve also the converse direction.

Theorem. > 2 Given K 1, there exists compact sets E of dimension K+1 , nonremovable for bounded K- quasiregular mappings.

Via the classical Stoilow factorization, all our main results are closely connected. In- deed, the first step will be to show that for a general K-quasiconformal mapping φ one has 2 1 H K+1 (E) σ-finite ⇒ H (φ(E)) σ-finite.

However, this conclusion will not be enough since there are rectifiable sets of finite length, such as E = [0, 1], that are non-removable for bounded analytic functions. Therefore, we need to know how rectifiable sets are distorted by K-quasiconformal

35 2 Removable singularities for bounded quasiregular mappings mappings. The right answer to this question is given in Theorem 2.16. Essentially,

2 F 1-rectiﬁable ⇒ dim(φ(F)) > K + 1 whenever φ is K-quasiconformal.

This chapter is structured as follows. In Section 2 we deal with the quasiconformal distortion of Hausdorff measures and some Riesz capacities. In Section 3 we apply these results in the removability problems for bounded (or BMO) quasiregular mappings. Finally, Section 4 is devoted to the construction of nonremovable sets.

2.2 Absolute continuity

Here we call principal K-quasiconformal mappings to any K-quasiconformal mappings that are conformal outside a compact set and normalized by equation (1.9), that is, 1 ∞ φ(z) − z = O |z| as |z| → .

In the work [4] of K. Astala, it is shown that for all K-quasiconformal mappings φ : C → C, |φ(E)| ≤ C |E|1/K (2.4) where C is a constant that depends on K and the normalizations. By scaling we may always arrange diam(φ(E)) = diam(E) ≤ 1 (2.5) and then C = C(K) depends only on K. In order to achieve the result (2.4), one ﬁrst reduces to the case where the set E is a ﬁnite union of disks. Secondly, the mapping φ is written as φ = h ◦ φ1, where both h, φ1 : C → C are K-quasiconformal mappings, such that φ1 is conformal on E and h is conformal in the complement of the set F = φ1(E).

Here one obtains the right conclusion for φ1,

1 |φ1(E)| ≤ C |E| K

36 2.2 Absolute continuity

by including φ1 in a holomorphic family of quasiconformal mappings. Further, one shows in [4, p. 50] that under the special assumption that h is conformal outside of F, we have |h(F)| ≤ C |F|, (2.6) where the constant C still depends only on K.

In searching for absolute continuity properties of other Hausdorff measures under quasiconformal mappings, such a decomposition appears unavoidable, and leads one to look for counterparts of (2.6) for Hausdorff measures Ht or Hausdorff contents Mt. Here we establish the result for the dimension t = 1.

Lemma 2.1. Suppose E ⊂ C is a compact set, and let φ : C → C be a principal K-quasiconformal mapping, such that φ is conformal on C \ E. Then,

M1(φ(E)) ' M1(E), with constants depending only on K.

For the proof of this Lemma, we need some previous results. We say that a set E is removable for BMO analytic functions if for every open set Ω ⊃ E, every function f ∈ BMO(Ω), analytic on Ω \ E, admits a holomorphic extension in the whole of Ω. These sets were characterized by Kaufman [31] with the condition M1(E) = 0. The analytic BMO capacity is deﬁned by 0 γ0(E) = sup | f (∞)| where the supremum runs over all functions f ∈ BMO(C) analytic on C \ E, such that k f k∗ ≤ 1. Verdera [54] showed the following estimate.

Lemma 2.2. For every compact set E ⊂ C, 1 M (E) ' γ0(E).

We recall now a result from Reimann [46], asserting that the space of functions of

37 2 Removable singularities for bounded quasiregular mappings bounded mean oscillation, BMO, is invariant under quasiconformal changes of variables. More precisely, if φ is K-quasiconformal and f ∈ BMO(C), then f ◦ φ ∈ BMO(C) and

k f ◦ φk∗ ≤ C(K) k f k∗.

Proof of Lemma 2.1. Due to Lemma 2.2, it is sufﬁcient to show that

γ0(φ(E)) ≤ C(K) γ0(E).

So let f be an admissible function for γ0(E). Then, f ∈ BMO(C) is a holomorphic −1 mapping of C \ E such that k f k∗ ≤ 1 and f (∞) = 0. The function g = f ◦ φ is then also in BMO(C), with

kgk∗ ≤ C(K).

C C g Moreover, since φ is conformal on \ E, g is holomorphic on \ φ(E). Thus, C(K) is an admissible function for γ0(φ(E)). The normalization on φ ensures that g(∞) = 0 and

|g0(∞)| = lim |z g(z)| = lim |φ(w) f (w)| = | f 0(∞)| |z|→∞ |w|→∞

Hence, γ0(E) ≤ C(K) γ0(φ(E)). The converse inequality follows by symmetry.

This lemma is a ﬁrst step towards the results on absolute continuity.

Theorem 2.3. Let E be a compact set and φ : C → C K-quasiconformal, normalized by (2.5). Then

K+1 1 2 2K M (φ(E)) ≤ C M K+1 (E) 2 where the constant C = C(K) depends only on K. In particular, if H K+1 (E) = 0 then H1(φ(E)) = 0.

Proof. There is no restriction if we assume E ⊂ D. We can also assume that φ is a principal K-quasiconformal mapping, conformal outside of D. Now, given ε > 0, there

38 2.2 Absolute continuity

is a ﬁnite covering of our set E by open disks Dj = D(zj, rj), j = 1, ..., n, such that

n 2 2 K+1 K+1 ∑ rj ≤ M (E) + ε. j=1

Ω n Denote by = ∪j=1Dj. As in [4], we use a decomposition φ = h ◦ φ1, where both φ1, h are principal K-quasiconformal mappings. Moreover, we may require that φ1 is conformal in Ω ∪ (C \ D) and that h is conformal outside of φ1(Ω).

By Lemma 2.1, we see that

1 1 1 1 M (φ(E)) ≤ M (φ(Ω)) = M (h ◦ φ1(Ω)) ≤ C M (φ1(Ω)).

1 Hence, the problem has been reduced to estimating M (φ1(Ω)). For this, K-quasidisks have area comparable to the square of the diameter,

1 2 1/2 diam(φ1(Dj)) ' |φ1(Dj)| = J(z, φ1) dA(z) ZDj ! with constants which depend only on K. Thus, using Holder¨ estimates twice, we obtain for p > 1 small enough

1 1 2p 1− 2p n n n p−1 p 2p−1 ∑ diam(φ1(Dj)) ≤ C(K) ∑ J(z, φ1) dA(z) ∑ |Dj| . j=1 j=1 ZDj ! j=1 !

Now we are in the special situation of [10, Lemma 5.2]. Namely, as φ1 is conformal in Ω K the subset , we may take p = K−1 to obtain

n p p ∑ J(z, φ1) dA(z) ' J(z, φ1) dA(z) ≤ 1. Ω j=1 ZDj Z

39 2 Removable singularities for bounded quasiregular mappings

p−1 1 With the above choice of p one has 2p−1 = K+1 . Hence we get

K+1 2 n n 2 K K+1 2 2K K+1 K+1 ∑ diam(φ1(Dj)) ≤ C(K) ∑ rj ≤ C(K) M (E) + ε . (2.7) = = ! j 1 j 1

But ∪jφ1(Dj) is a covering of φ1(Ω), so that actually we have

K+1 1 1 2 2K M (φ(E)) ≤ CM (φ1(Ω)) ≤ C(K) M K+1 (E) + ε Since this holds for every ε > 0, the result follows.

At this point we want to emphasize that for a general quasiconformal mapping φ we p < K K have J(·, φ) ∈ Lloc only for p K−1 . The improved border line integrability (p = K−1 ) under the extra assumption that φ Ω is conformal, was shown in [10, Lemma 5.2]. This phenomenon was crucial for our argument, since we are studying Hausdorff measures rather than dimension. Actually, the same procedure shows that inequality (2.7) works in a much more general setting. That is, still under the special assumption that φ1 is n conformal in ∪j=1Dj, we have for any t ∈ [0, 2]

1 1 1 n d n t K d t ∑ diam(φ1(Dj)) ≤ C(K) ∑ diam(Dj) (2.8) j=1 ! j=1 !

2Kt where d = 2+(K−1)t . On the other hand, another key point in our proof is the estimate

M1(h(E)) ≤ C M1(E) whenever h is a planar K-quasiconformal mapping which is conformal outside of E. We believe that ﬁnding the counterpart to this estimate is crucial for understanding distortion of Hausdorff measures under quasiconformal mappings.

Problem 2.4. Suppose we are given a real number d ∈ (0, 2]. Then for any compact set E ⊂ C and for any

40 2.2 Absolute continuity

K-quasiconformal mapping h which is conformal on C \ E, we have

Md(h(E)) ' Md(E) (2.9) with constants that depend on K and d only.

We already know that (2.9) is true for d = 1 and d = 2. An afﬁrmative answer for general d, combined with the optimal integrability bound proving (2.8), would provide ∗ d t 2Kt the absolute continuity of φ H with respect to H , where d = 2+(K−1)t , 0 ≤ t ≤ 2 and K ≥ 1. Therefore, (2.9) would have important consequences in the theory of quasiconformal mappings.

The positive answer for the dimension d = 1 was based on Lemma 2.2, as well as on the quasiconformal invariance of BMO. However, this is not the only quasiconformal invariant class. Actually, as we will see, the invariance of VMO has interesting consequences. Here we start with the ﬁrst one.

Theorem 2.5. 2 Let E ⊂ C be a compact set, and φ : C → C a K-quasiconformal mapping. If H K+1 (E) is finite (or even σ-finite), then H1(φ(E)) is σ-finite.

1 Due to Theorem 1.4, this result may be equivalently expressed in terms of M∗. Theorem 2.6. 2 K+1 Let E ⊂ C be a compact set, and φ : C → C a K-quasiconformal mapping. If M∗ (E) = 0, 1 then M∗(φ(E)) = 0.

We say that a set E is removable for analytic VMO functions if for every open set Ω ⊃ E, every function f ∈ VMO(Ω), analytic on Ω \ E, extends holomorphically to Ω. These 1 sets were characterized by Verdera [54] with the condition M∗(E) = 0. The analytic VMO capacity is deﬁned by 0 γ∗(E) = sup | f (∞)| where the supremum runs over all VMO(C) functions f , analytic on C \ E, such that k f k∗ ≤ 1. Verdera [54] showed the following Lemma.

41 2 Removable singularities for bounded quasiregular mappings

Lemma 2.7. For every compact set E ⊂ C, 1 M (E)∗ ' γ∗(E).

Notice that VMO is quasiconformally invariant, as BMO. This is due to the fact that VMO can be seen as the BMO closure of compactly supported continuous functions. Consequently, we can prove the following auxiliar result.

Lemma 2.8. Let E be a compact set. For any principal K-quasiconformal mapping φ : C → C, conformal on C \ E, we have 1 1 M∗(φ(E)) ' M∗(E).

Proof. Again, as in Lemma 2.1, we will use the equivalence given at Lemma 2.7 between 1 γ∗ and M∗. Thus, let f be an admissible function for γ∗(φ(E)). Then f ∈ VMO is analytic on C \ φ(E), k f k∗ ≤ 1 and f (∞) = 0. Set g = f ◦ φ. Therefore g ∈ VMO, 0 0 g is analytic on C \ E, kgk∗ ≤ C(K) and |g (∞)| = | f (∞)| since φ is a principal 1 K-quasiconformal mapping. Consequently, C(K) g is admissible for γ∗(E) and hence γ∗(φ(E)) ≤ C(K) γ∗(E).

Proof of Theorem 2.6. Naturally, the argument is similar to that in Theorem 2.3. Without loss of generality, we may assume that E ⊂ D and that φ is a principal K-quasiconformal 2 mapping. Since H K+1 (E) is ﬁnite, for any δ there is a family of disks Di such that 2 2 K+1 K+1 < E ⊂ ∪iDi, ∑i diam(Di) ≤ H (E) + 1 and diam(Di) δ. Set Ω = ∪iDi. Again, we have a decomposition φ = φ2 ◦ φ1, where both φ1 and φ2 are principal K-quasiconformal mappings, and where we may require that φ1 is conformal in (C \ D) ∪ Ω, and φ2 is conformal outside φ1(Ω). Thus,

1 1 M∗(φ(E)) ≤ M∗(φ(Ω)).

By Lemma 2.7, the lower content can be replaced by the VMO capacity,

1 M∗(φ(Ω)) ≤ C γ∗(φ(Ω)).

42 2.2 Absolute continuity

Now, since φ2 is conformal outside of φ1(Ω), by Lemma 2.8 we get

γ∗(φ2 ◦ φ1(Ω)) ' γ∗(φ1(Ω)).

Again, by Lemma 2.7, 1 γ∗(φ1(Ω)) ≤ C M∗(φ1(Ω)).

1 Now it remains to bound M∗(φ1(Ω)). For this, take h ∈ F, h(t) = t ε(t), and argue as in Theorem 2.3. Since K-quasiconformal mappings are Holder¨ continuous with exponent 1/K,

h 1/K M (φ1(Ω)) ≤ ∑ diam(φ1(Dj))ε(diam(φ1(Dj))) ≤ ε(CK δ ) ∑ diam(φ1(Dj)) j j K−1 2K 1/K K 1/2K ≤ ε(CK δ ) ∑ J(z, φ1) K−1 dm(z) |Dj| j ZDj ! K+1 2K K+1 1/K 2 1/K 2 2K ≤ ε(CK δ ) CK ∑ diam(Dj) K+1 ≤ ε(CK δ ) CK H K+1 (E) + 1 . ! j

Finally, taking δ → 0 we get Mh(φ(E)) = 0. This holds for any h ∈ F, and the theorem follows.

2 One might think of extending the preceeding results from the critical index K+1 to arbitrary ones by using other capacities that behave like a Hausdorff content. For instance, the analytic Lipα capacity γα introduced by O’Farrell in [42] satisﬁes

1+α M (E) ' γα(E),

whenever α ∈ (0, 1). Unfortunately, the space Lipα is not quasiconformally invariant. Thus, other procedures are needed. It turns out that the homogeneous Sobolev spaces provide suitable tools, basically since W˙ 1,2(C) is invariant under quasiconformal mappings. Here recall that for 0 < α < 2 and p > 1, the homogeneous Sobolev space

43 2 Removable singularities for bounded quasiregular mappings

W˙ α,p(C) is deﬁned as the space of Riesz potentials

f = Iα ∗ g

p 1 1,p ∈ (C) ( ) = k k α,p = k k = ˙ (C) where g L and Iα z |z|2−α . The norm f W˙ (C) g p. When α 1, W agrees with the space of functions f whose ﬁrst order distributional derivatives are given by Lp(C) functions. Let f ∈ W˙ 1,2(C) and let φ be a K-quasiconformal mapping on C. Deﬁning g = f ◦ φ we have

|Dg(z)|2dA(z) = |D f (φ(z))|2 |Dφ(z)|2 dA(z) C C Z Z ≤ K |D f (φ(z))|2 J(z, φ) dA(z) C Z = K |D f (w)|2 dA(w) C Z so that g ∈ W˙ 1,2(C). In other words, every K-quasiconformal mapping φ induces a bounded linear operator

T : W˙ 1,2(C) → W˙ 1,2(C), T( f ) = f ◦ φ (2.10) with norm depending only on K. As we have mentioned before, this operator T is also bounded on BMO(C) ([46]). Moreover, Reimann and Rychener [47, p.103] proved that 2 ,q W˙ q (C), q > 2, may be represented as a complex interpolation space between BMO(C) 2 ,q and W˙ 1,2(C). It follows that T is bounded on the Sobolev spaces W˙ q (C), q > 2. More precisely, there exists a constant C = C(K, q) such that

k f ◦ φk 2 ,q ≤ Ck f k 2 ,q , (2.11) W˙ q (C) W˙ q (C) for any K-quasiconformal mapping φ on C. These invariant function spaces provide us with related invariant capacities. Recall (e.g. [1, pp.34 and 46]) that for any pair α > 0, p > 1 with 0 < αp < 2, one deﬁnes the Riesz capacity of a compact set E by

p C˙ α,p(E) = sup{µ(E) }

44 2.2 Absolute continuity

where the supremum runs over all positive measures µ supported on E, such that kIα ∗ 1 1 µkq ≤ 1, p + q = 1. We get an equivalent capacity if we replace positive measures µ by p distributions T supported on E, kIα ∗ Tkq ≤ 1, and take the supremum of |hT, 1i| .

To see the connection with equation (2.11) consider the set functions

0 ∞ C ∞ γ1−α,q(E) = sup{| f ( )|; f ∈ H( \ E), k f kW˙ 1−α,q ≤ 1, f ( ) = 0}.

Observe again that | f 0(∞)| = |h∂ f , 1i| where this action must be understood in the sense of distributions. With this terminology we have the analogous result to Lemmas 2.2 and 2.7.

Lemma 2.9. Suppose that E is a compact subset of the plane. Then, for any p ∈ (1, 2),

C˙ α,p(E) ' γ1−α,q(E),

2 p where α = p − 1 and q = p−1 .

q Proof. On one hand, take µ an admissible measure for C˙ α,p. Then, Iα ∗ µ is in L with 1 ∞ norm at most 1. Deﬁne f = z ∗ µ. Clearly, f is analytic outside E, f ( ) = 0 and | f 0(∞)| = µ(E). Moreover, proceeding roughly,

1 ξ 1 f (ξ) ' µ(ξ) = µ(ξ) = R I µ ξ |ξ| |ξ| 1 b b b and consequently we can write b b b

1 f = ∗ µ = R(I ∗ µ) = I − ∗ R(I ∗ µ) z 1 1 α α where R is a Calderon-Zygmund´ operator and k f kW˙ 1−α,q = kR(Iα ∗ µ)kq . kIα ∗ µkq.

For the converse, let f = I1−α ∗ g be an admissible function for γ1−α,q. We have that

T = ∂ f is an admissible distribution for C˙ α,p because

t Iα ∗ T = R (g)

45 2 Removable singularities for bounded quasiregular mappings

t 1/p 0 where R is the transpose of R. Thus, C˙ α,p(E) ≥ |hT, 1i| = | f (∞)| and the proof is complete.

We end up with new invariants built on the Riesz capacities.

Theorem 2.10. Let φ : C → C be a principal K-quasiconformal mapping of the plane, which is conformal on C < < 2 \ E. Let 1 p 2 and α = p − 1. Then

C˙ α,p(φ(E)) ' C˙ α,p(E), with constants that depend only on K and p.

Proof. By the preceding Lemma it sufﬁces to show that γ1−α,q (φ(E)) ≤ γ1−α,q(E).

Let f be an admissible function for γ1−α,q(φ(E)). This means that f is holomorphic on C ∞ \ φ(E), f ( ) = 0 and that k f kW˙ 1−α,q ≤ 1. Then, we consider the function g = f ◦ φ. ∞ 2 Clearly, ∂( f ◦ φ) is conformal outside E, and g( ) = 0. Moreover, for α = p − 1 we 2 have 1 − α = q . Hence, because of equation (2.11),

kgkW˙ 1−α,q ≤ CKk f kW˙ 1−α,q ≤ CK

1 so that g is an admissible function for γ − (E). Hence, as φ is a principal K- CK 1 α,q quasiconformal mapping, 1 0 γ1−α,q(E) ≥ | f (∞)| CK and we may take supremum over f .

The above theorem has direct consequences towards the absolute continuity of Haus- dorff measures, but unfortunately these are slighly weaker than one would wish for. In h fact, there are compact sets F such that Cα,p(F) = 0 and H (F) > 0 for some measure function h(t) = tp ε(t) [1]. Thus, Theorem 2.10 does not help for Problem 2.4. We have to content with the following setup.

46 2.2 Absolute continuity

Given 1 < d < 2 consider the measure functions h(t) = td ε(t) where

Corollary 2.11. Let E be a compact set on the plane, and φ : C → C a principal K-quasiconformal mapping, conformal outside of E. Let 1 < d < 2. Then,

Mh(φ(E)) ≤ C Md(E) for any measure function h(t) = td ε(t) satisfying (2.12). Moreover, if Hd(E) < ∞ then Hh(φ(E)) = 0 for every such h.

Proof. By [1, Theorem 5.1.13], given a measure function h satisfying (2.12) there is a constant C = C(h) with

2 Mh(φ(E)) ≤ C C (φ(E)), α = − 1 α,d d

By Theorem 2.10, Cα,d(φ(E)) ≤ C Cα,d(E) and using again [1, Theorem 5.1.9] we have d ﬁnally Cα,d(E) ≤ M (E).

Arguing now as in Theorems 2.3 and 2.6, we arrive at the following conclusion.

Corollary 2.12. Let E be a compact set of the plane and φ : C → C a K-quasiconformal mapping. Let t ∈ 2 2Kt ( K+1 , 2) and d = 2+(K−1)t . Then,

1 Mh(φ(E)) ≤ C Mt(E) Kt for any measure function h satisfying (2.12).

2 < < < < Here note that for K+1 t 2 we always have 1 d 2.

47 2 Removable singularities for bounded quasiregular mappings

2.3 Improved Painleve´ theorems

A set E is said to be removable for bounded (resp. BMO) K-quasiregular mappings, if every K-quasiregular mapping in C \ E which is in L∞(C) (resp.BMO(C) ) admits a K- quasiregular extension to C. Obviously, when K = 1, in both situations L∞ and BMO, we recover the original Painleve´ problem (see Section 1.3). Moreover, by means of the Stoilow factorization, one can represent any bounded K-quasiregular function as a composition of a bounded analytic function and a K-quasiconformal mapping. The corresponding result holds true also for BMO since this space, like L∞, is quasiconformally invariant. Therefore, when we ask ourselves if a set E is K-removable, we just need to analyze how it may be distorted under quasiconformal mappings, and then apply the known results for the analytic situation. With this basic scheme, it is shown in [4, Corollary 1.5] 2 that every set with dimension below K+1 is K-removable. Indeed, the precise formulas for the distortion of dimension ensure that such sets will always be mapped to a set with dimension strictly smaller than 1, therefore with vanishing analytic capacity.

2 Iwaniec and Martin [29] had earlier conjectured that sets of zero K+1 -dimensional measure are K-removable. A preliminary answer to this question was found in [6], and actually it was that argument which suggested Theorem 2.3. Using our results from 2 Section 2.2 we can now prove that sets of zero K+1 -dimensional measure are removable for BMO K-quasiregular mappings.

Corollary 2.13. 2 Let E be a compact subset of the plane. Assume that H K+1 (E) = 0. Then E is removable for all BMO K-quasiregular mappings.

Proof. Assume that f ∈ BMO(C) is K-quasiregular on C \ E. Denote by µ the Beltrami coefﬁcient of f , and let φ be a principal solution to ∂φ = µ ∂φ. Then, F = f ◦ φ−1 is holomorphic on C \ φ(E) and F ∈ BMO(C). On the other hand, as we showed in Theorem 2.3, H1(φ(E)) = 0. Thus, φ(E) is a removable set for BMO analytic functions. In particular, F admits an entire extension. Hence, f = F ◦ φ extends quasiregularly to

48 2.3 Improved Painleve´ theorems the whole plane.

We believe that Corollary 2.13 is sharp, in the sense that we expect a positive answer to the following question.

Problem 2.14. 2 Does there exist for every K ≥ 1 a compact set E with 0 < H K+1 (E) < ∞ such that E is not removable for all K-quasiregular functions in BMO(C).

> 2 Here we observe that by [4, Corollary 1.5], for every t K+1 there exists a compact set E with dimension t, nonremovable for bounded (and hence for BMO) K-quasiregular mappings.

Next we return back to the L∞ problem by noticing that Theorem 2.3 proves the con- 2 jecture of Iwaniec and Martin that sets with H K+1 (E) = 0 are removable for bounded K-quasiregular mappings. This is so because those sets E satisfy H1(φ(E)) = 0 whenever φ is K-quasiconformal and, hence, γ(φ(E)) = 0. Nevertheless, analytic capacity is somewhat smaller than length, and hence with Theorem 2.5 we may go even further. To 2 be precise, if a set has finite (or even σ-finite) K+1 -dimensional Hausdorff measure, then all its K-quasiconformal images have at most σ-finite length. Thus, such images might still be removable for analytic bounded functions, provided that they do not contain any rectifiable subset with positive length. For this the following result, which appears in [5], gives the correct tools.

Theorem 2.15. Suppose that φ : C → C is a K-quasiconformal mapping. Let E ⊂ ∂D be such that dim(E) = 1. Then, we have the strict inequality

2 dim(φ(E)) > K + 1

As a direct consequence, one obtains the following version, which applies to any recti- ﬁable set.

49 2 Removable singularities for bounded quasiregular mappings

Corollary 2.16. Suppose that E is a rectiﬁable set, and let φ : C → C be a K-quasiconformal mapping. Then, there exists a subset E0 ⊂ E of zero length such that

2 dim φ(E \ E ) > . 0 K + 1

It points out that the proofs of the above results actually give more than what here is stated. Actually, there exists constants c1, c2 > 0 such that if E ⊂ ∂D, then

2 2 dim(E) ≥ 1 − c1 ε ⇒ dim(φ(E)) ≥ 1 − c2 ε (2.13) whenever φ is (1 + ε)-quasiconformal and ε > 0 is small enough. We end up with the following improved version of Painleves´ Theorem for quasiregular mappings.

Theorem 2.17. 2 Let E be a compact set in the plane, and let K > 1. Assume that H K+1 (E) is σ-ﬁnite. Then E is removable for all bounded K-quasiregular mappings.

Proof. Let f : C → C be bounded, and assume that f is K-quasiregular on C \ E. Let ∂ f µ = ∂ f be the Beltrami coefﬁcient of f and let φ be a principal homeomorphic solution to ∂φ = µ ∂φ. Then F = f ◦ φ−1 is a bounded holomorphic mapping on C \ φ(E). If we can extend F holomorphically to the whole plane, then we are done. Thus, we have to show that φ(E) has zero analytic capacity. By Theorem 2.5, φ(E) has σ-ﬁnite length, 1 that is, φ(E) = ∪nFn where each H (Rn) < ∞. From Besicovitch’s Theorem (Theorem

1.5), each set Fn can be decomposed as

Fn = Rn ∪ Un ∪ Bn

where Rn is a rectiﬁable set, Un is a purely unrectiﬁable set, and Bn is a set of zero length. Because of Tolsa’s Theorem on semiadditivity of analytic capacity [51],

γ(Fn) ≤ C (γ(Rn) + γ(Un) + γ(Bn))

50 2.3 Improved Painleve´ theorems

1 Now, γ(Bn) ≤ H (Bn) = 0 and γ(Un) = 0 since purely unrectifiable sets of finite length have zero analytic capacity [15]. On the other hand, Rn is a rectifiable image, under 2 a K-quasiconformal mapping, of a set of dimension K+1 . Thus according to Theorem −1 1 2.15 and Corollary 2.16, applied to φ , we must have H (Rn) = 0. Therefore we get

γ(Fn) = 0 for each n. Again by semiadditivity of analytic capacity we conclude that γ(φ(E)) = 0.

As pointed out earlier, the above theorem does not hold for K = 1, any rectifiable set such as E = [0, 1], of finite and positive length gives a counterexample. In the above proof the improved distortion of rectifiable sets was the decisive phenomenon allowing the result. A similar procedure to that in the proof of Theorem 2.15 may be used to show the following expansion result (compare with the compression result of equation (2.13)).

Corollary 2.18.

There exist constants c1, c2 > 0 such that if φ : C → C is K-quasiconformal, then

2 2 dim(E) ≤ 1 − c1 ε ⇒ dim(φ(E)) < 1 − c2 ε whenever K = 1 + ε, E ⊂ ∂D and ε > 0 is small enough.

2 This Corollary drives to removable sets with Hausdorff dimension above K+1 . Corollary 2.19. There exist a constant c ≥ 1 such that if E ⊂ ∂D is compact and

K − 1 2 dim(E) ≤ 1 − c K + 1 then E is removable for bounded K-quasiregular mappings, K = 1 + ε, whenever e > 0 is small enough.

Proof. This is a consequence of Corollary 2.18. If ε > 0 is small enough and K = 1 + ε, then the K-quasiconformal images of E will always have dimension below 1, so that γ(φ(E)) = 0 for each K-quasiconformal mapping φ.

51 2 Removable singularities for bounded quasiregular mappings

2.4 Extremal distortion and nonremovable sets

The discussion of Section 2.3 on removable sets for bounded K-quasiregular mappings 2 2 still leaves open the case where dim(E) = K+1 but Edoes not have σ-ﬁnite K+1 -dimensional Hausdorff measure. In view of Theorem 2.17 one might think that maybe everything at the critical dimension is removable for bounded K-quasiregular mappings. In this section we show that our results are sharp in a quite strong sense. The main goal of this section consists in ﬁnding some kind of analog result to Mattila’s Theorem (Theorem 1.7) for the K-quasiregular problem.

Theorem 2.20. Let K ≥ 1. Let h(t) = t ε(t) be such that:

• ε is continuous and nondecreasing, ε(0) = 0 and ε(t) = 1 whenever t ≥ 1. tδ • lim = 0 for all δ > 0. t→0 ε(t)

2 There exists a compact set E ⊂ D, a measure function g(s) = s K+1 η(s) and a K-quasiconformal mapping φ such that:

• Hh(φ(E)) ' 1

• Hg(E) ' 1 sδ • The function η is such that lim η(s) = 0, and lim = 0 for all δ > 0. As a s→0 s→0 η(s) 2 consequence, dim(E) = K+1 .

1 2 1 1+ 1 ε(t) 1 η(s) K • dt ≥ ds whenever this makes sense. t K s Z0 Z0

Proof. We will construct a K-quasiconformal mapping φ as the limit of a sequence φN of K-quasiconformal mappings, and E will be a Cantor-type set. The main idea consists on changing, at every step in the construction of E, both the quantity m j and the size of the generating disks. So we will construct a sequence of integers m j wich will go very fastly to ∞, and two sequences of real numbers σj, Rj ∈ (0, 1) that will converge to 0 very quiclky.

52 2.4 Extremal distortion and nonremovable sets

1 Step 1. Consider m1 disjoint disks of radius R1, centered at zi , i = 1, ..., m1, uniformly distributed inside D, such that 1 c = m R2 > 1 1 1 2

The function f (t) = m1 h(tR1) is continuous in the variable t and f (0) = 0. Moreover, for any t > 0

ε(tc1/2 m−1/2) ( ) = ( ) = 1/2 1/2 ( 1/2 −1/2) = 1 1 2 f t m1tR1 ε tR1 t c1 m1 ε tc1 m1 1/2 −1/2 t c1 tc1 m1 so that lim f (t) = ∞ m1→∞

Hence, for any t < 1, we may choose m1 big enough so that the equation m1 h(σ1 R1) = 1 has a unique solution σ1 ∈ (0, t). Under this situation, an easy computation shows that

2 2K K−1 K K+1 K+1 K+1 m1 (σ1 R1) ε(σ1 R1) (c1) = 1

1 1 K Denote r1 = R1, and for each i = 1, . . . , m1, let ϕi (z) = zi + σ1 R1 z, and

1 = 1(D) = ( ) Di K ϕi D zi, r1 σ1 0 1 D K Di = ϕi ( ) = D(zi, σ1 r1)

0 Notice that Di and Di are concentric disks. We deﬁne

1−K 0 σ1 (z − zi) + zi z ∈ Di 1  K −1 z−zi 0 g1(z) =  (z − zi) + zi z ∈ Di \ D  r1 i  z otherwise

   m1 0 This mapping is K-quasiconformal, conformal outside of i=1(Di \ Di), maps Di onto 0 00 1 itself and Di onto Di = D(zi , σ1r1), while the rest of the Splane remains ﬁxed. Deﬁne then φ1 = g1.

53 2 Removable singularities for bounded quasiregular mappings

Step 2. We have already ﬁxed m1, R1, σ1 and c1. Consider m2 disjoint disks of radius R2, 2 D centered at zj , j = 1, . . . , m2, uniformly distributed inside , such that

1 c = m R2 > 2 2 2 2

Then we repeat the above procedure. So the function f (t) = m1m2h(tR1 R2) is continuous in t, f (0) = 0 and lim f (t) = ∞ m2→∞ for any t > 0. Thus, we may choose m2 big enough, so that the equation

m1m2h(σ1σ2R1R2) = 1 has a unique solution σ2 ∈ (0, 1) as small as we wish. Then,

2 2K K−1 K K K+1 K+1 K+1 m1m2 (σ1 σ2 R1R2) ε(σ1σ2R1R2) (c1c2) = 1

2 2 K Denote r2 = R2 σ1r1 and ϕj (z) = zj + σ2 R2 z, and deﬁne the auxiliary disks

1 1 2 D = φ1 ϕ ◦ ϕ (D) = D(z , r2) ij σK j i ij 2 0 1 2 D 0 K Dij = φ1 ϕj ◦ ϕi ( ) = D (zij, σ2 r2) 0 for i = 1, . . . , m1 and j = 1, . . . , m2. Notice again that Dij and Dij are concentric. Now let 1−K 0 σ2 (z − zij) + zij z ∈ Dij 1 −1  z−zij K 0 g2(z) =  (z − zij) + zij z ∈ Dij \ D  r2 ij  z otherwise

  0 Clearly, g2 is K-quasiconformal, conformal outside of i,j(Dij \ Dij), maps Dij onto itself 0 00 and Dij onto Dij = D(zij, σ2r2), while the rest of the Splane remains ﬁxed. Now deﬁne φ2 = g2 ◦ φ1.

54 2.4 Extremal distortion and nonremovable sets

D D Figure 2.1

Nth Step. We take mN uniformly distributed disjoint disks of radius RN, giving at least half of the area of D, 1 c = m R2 > N N N 2

Since the continuous function f (t) = m1 . . . mN h(tR1 . . . RN) satisﬁes f (0) = 0 and

lim f (t) = ∞, for any t > 0, then we may choose mN big enough so that the equation mN→∞

m1 . . . mN h(σ1 . . . σN R1 . . . RN) = 1

has a unique solution σN as small as we wish. Let us remark that we can take lim σN = N→∞ 0. Then, for these values,

2 2K K−1 K K K+1 K+1 K+1 m1 . . . mN (σ1 . . . σN R1 . . . RN) ε(σ1 . . . σN R1 . . . RN) (c1 . . . cN ) = 1

N N K Denote then ϕj (z) = zj + σN RN z and rN = RN σN−1rN−1. Deﬁne, for any multiindex

J = (j1, ..., jN), 1 ≤ jk ≤ mk, k = 1, ..., N,

1 1 N D DJ = φN−1 ϕ ◦ · · · ◦ ϕ ( ) = D(zJ, rN) σK j1 jN N 0 = φ ϕ1 ◦ · · · ◦ ϕN (D) = 0( , σK ) DJ N−1 j1 jN D zJ NrN

55 2 Removable singularities for bounded quasiregular mappings

0 As before, DJ and DJ are concentric. Let

1−K 0 σN (z − zJ ) + zJ z ∈ DJ  1 −1 z−zJ K 0 gN (z) =  (z − zJ) + zJ z ∈ DJ \ D  rN J  z otherwise

  Clearly, g is K-quasiconformal, conformal outside of (D \ D0 ), maps D N J=(j1,...,jN) J J J 0 00 onto itself and DJ onto DJ = D(zJ, σNrN), while the restS of the plane remains ﬁxed. Now deﬁne φN = gN ◦ φN−1.

D Since each φN is K-quasiconformal, and all agree outside the unit disk , there exists a limit K-quasiconformal mapping

φ = lim φN N→∞

1,p C < 2K with convergence in Wloc ( ) for any p K−1 . On the other hand, φ maps the compact set ∞ E = ϕ1 ◦ · · · ◦ ϕN (D)  j1 jN  N\=1 j1,...,[jN   to the set ∞ φ(E) = ψ1 ◦ · · · ◦ ψN (D)  j1 jN  N\=1 j1[,...,jN   i i where here we deﬁned ψj(z) = zj + σi Ri, j = 1, ..., mi, i = 1, ..., N. To end the proof, denote K K sN = (σ1 R1) . . . (σN RN)

tN = (σ1R1) . . . (σN RN)

With this notation, observe that our parameters RN, mN, σN have been chosen so that

2 2K K−1 K+1 K+1 K+1 m1 . . . mN h(tN ) = 1 and m1 . . . mN sN ε(tN ) (c1 . . . cN) = 1

56 2.4 Extremal distortion and nonremovable sets

2 Set g(s) = s K+1 η(s), where η is deﬁned by

2K K−1 η(s) = ε(tN ) K+1 (c1 . . . cN) K+1 whenever s ∈ [sN, sN−1).

Claim 1. Hh(φ(E)) ' 1. We start by ﬁxing δ > 0 very small. The covering (ψ1 ◦ · · · ◦ ψN (D)) is admissible j1 jN j1,...,jN when computing Hh(φ(E)) if diam(ψ1 ◦ · · · ◦ ψN (D)) ≤ δ. Thus, δ j1 jN

Hh(φ( )) ≤ (diam(ψ1 ◦ · · · ◦ ϕN (D))) = . . . ( ) = 1 δ E ∑ h j1 jN m1 mN h tN j1,...,jN because of our choice of σN. For the converse inequality, take a ﬁnite covering (Uj)j of

φ(E) by open disks of diameter diam(Uj) ≤ δ. Now let

δ0 = inf(diam(Uj)) > 0 j

Denote by N the minimal integer such that t ≤ δ . By construction, the family (ψ1 ◦ 0 N0 0 j1 N0 D h · · · ◦ ψ ( ))j1 ,...,jN is a covering of φ(E) with the M -packing condition. Thus, jN0 0

N0 1 D ∑ h(diam(Uj)) ≥ C ∑ h(diam(ϕ ◦ · · · ◦ ϕj ( ))) = C jN0 1 j j1,...,jN0

h h Hence, Hδ (φ(E)) ≥ C and now letting δ → 0, we get that C ≤ H (φ(E)) ≤ 1 and our ﬁrst claim follows.

Claim 2. Hg(E) ' 1. The proof of this fact is analogous to that of Claim 1.

δ sN Claim 3. One can choose σj so that lim = 0 for every δ > 0. N→∞ η(sN )

57 2 Removable singularities for bounded quasiregular mappings

Fix any δ > 0. We ﬁrst notice that

2K K+1 K+1 K−1 δ δ δ 2K δ sN sN tN (σ1 . . . σN ) = 2K K−1 = 1 η( ) +1 +1  ε( )  +1 sN ε(tN ) K (c1 . . . cN) K tN (c1 . . . cN) K !   tδ Now, since lim = 0 for every δ > 0, it will be sufﬁcient to ﬁnd N big enough such t→0 ε(t) that δ 1 (σ1 . . . σN) < (c1 . . . cN) K+1

However, if δ > 0 is ﬁxed, then 1 1 K+1 σδ < j 2 for every j > N = N(δ, K). Thus, the result follows.

1 2 1 1+ 1 ε(t) 1 η(s) K Claim 4. dt ≥ ds. t K s Z0 Z0

From the deﬁnition of η and since c1 . . . cN < 1 we have

tN−1 2 ε(t) 2 tN−1 1+ 1 sN−1 1 dt ≥ ε(t ) log > η(s ) K log − (K − 1) log t N t N s σ ZtN N N N Now notice that 1 1 1 1 s = ≤ N−1 log log K log σN K σN K sN Therefore t 2 s 1+ 1 N−1 ε(t) 1 N−1 η(s) K dt ≥ ds t K s ZtN ZsN and then Claim 4 follows.

We would like to remark at this point that the construction in the above theorem also works if we start with h(t) = t. In that case, we get a compact set E, a K-quasiconformal

58 2.4 Extremal distortion and nonremovable sets

2 mapping φ and a measure function g(s) = s K+1 η(s) satisfying

sδ lim η(s) = 0 and lim = 0 ∀δ > 0 s→0 s→0 η(s)

g 1 2 and such that H (E) ' 1 and H (φ(E)) ' 1. Again, E has dimension K+1 , also with non σ-ﬁnite measure, but now φ(E) has positive ﬁnite length.

Corollary 2.21. There exists a compact set E, such that:

• E is non removable for bounded K-quasiregular mappings.

2 • Hg(E) ' 1, where g(s) = s K+1 η(s) is such that

sδ lim η(s) = 0 and lim = 0 ∀δ > 0. s→0 s→0 η(s)

2 In particular, E has Hausdorff dimension K+1 .

Proof. Let h(t) = t ε(t) be any measure function in the hypothesis of the above example and such that 1 h(t)2 dt < ∞ t3 Z0 2 Now construct a compact set E of dimension K+1 and a K-quasiconformal mapping φ such that Hh(φ(E)) ' 1. By [40], φ(E) has positive analytic capacity, so there exists a bounded function f , holomorphic off φ(E), which is not a constant. Therefore the composition g = f ◦ φ is K-quasiregular off E, bounded on C and cannot extend quasiregularly to C. Thus E is non removable.

It should be pointed out here that this nonremovable set E supports a positive measure ν (actually one may choose ν = Hg) with growth ν(D(z, r)) ≤ g(r) and such that

1 1 1 1+ 1 g(s) p−1 ds η(s) K = ds < ∞ s2−αp s s Z0 Z0

59 2 Removable singularities for bounded quasiregular mappings

2K 2K+1 where α = 2K+1 and p = K+1 . Therefore, by [1, Th. 5.1.13] we get that for these values of α and p we actually have an estimate for a concrete Riesz capacity,

Cα,p(E) > 0

> > 3 However, K 1 implies p 2 so that in fact what we have is

2 0 < Cα,p(E) ≤ C 2 2 3 (E) ' γ 2 (E) . M K+1 (E) 3 (2− K+1 ), 2 K+1

In the expression above, the equivalence C 2 2 3 (E) ' γ 2 (E) comes from a result 3 (2− K+1 ), 2 K+1 by J. Mateu, L. Prat and J. Verdera [38]. Recall that for any β ∈ (0, 2], γβ is the signed Riesz capacity, associated to the signed Riesz kernels z . It is shown in [38] that if |z|1+β β ∈ (0, 1) then

γβ(E) ' C 2 3 (E) 3 (2−β), 2 β < ∞ 2 β < ∞ In particular, H (F) implies γβ(F) = 0. Notice that for β = K+1 , then if H (F) then F is removable for bounded K-quasiregular mappings. Furthermore, if β = 1 then we obtain the capacity γ1, and it follows from Tolsa’s work [51] that it behaves precisely as analytic capacity,

γ1(E) ' γ(E)

This encourages us to write the following problem.

Problem 2.22.

Let E be compact. Assume that γ 2 (E) = 0. Then, E is removable for bounded K-quasiregular K+1 mappings.

If this were true, then this would give interesting relations between the quasiconformal world and the potential theory of signed Riesz kernels.

60 3 Removable singularities for Holder¨ continuous quasiregular mappings

3.1 Introduction

Let α ∈ (0, 1). Recall that a function f : C → C is said to be locally Holder¨ continuous C with exponent α, that is, f ∈ Lipα( ), if

| f (z) − f (w)| ≤ C |z − w|α (3.1) whenever z, w ∈ C are such that |z − w| < 1.

We say that a compact set E is removable for α-Holder¨ continuous K-quasiregular mappings C C C if every Lipα function f : → , K-quasiregular on \ E, is actually K-quasiregular on the whole plane. In this chapter we study the geometric conditions that make a com- C pact set E ⊂ removable (or non removable) for Lipα K-quasiregular mappings.

Some results are already known related to this subject. For instance, Koskela and Martio 1 α [33] showed that if d = K 1 + K , then every compact set with vanishing d-dimensional

Hausdorff measure is removable for Lipα K-quasiregular mappings. Moreover, they also gave some sufﬁcient conditions for removability in Rn. In particular, they proved that if Hd(E) = 0 for d = min{1, nα}, then E is removable for α-Holder¨ continuous K- quasiregular mappings. From another point of view, while dealing with second order quasilinear elliptic equations, Kilpelainen¨ and Zhong [32] showed that H α(n−1)(E) = 0 is enough. In the present chapter we improve the above results and get in Theorem 3.2 the following.

61 3 Removable singularities for Holder¨ continuous quasiregular mappings

Theorem. Let K ≥ 1 and α ∈ (0, 1), and let E be compact. If

Hd(E) = 0

2 with d = K+1 (1 + αK), then E is removable for α-Holder¨ continuous K-quasiregular mappings.

This index d was suggested to us by K. Astala.

Our techniques do not work in Rn, since there is no factorization theorem when n > 2. However, the above result may be rewritten in the planar class of finite distortion map- Ω C C 1,1 Ω pings. Recall that f : ⊂ → is said to be a mapping of finite distortion if f ∈ Wloc ( ), 1 Ω J(·, f ) ∈ Lloc( ) and |D f (z)|2 ≤ K(z) J(z, f ) at almost every z ∈ Ω. Here K : Ω → [1, ∞] is a measurable function, finite at almost every z ∈ Ω. Equivalently, |µ(z)| < 1 for almost every z ∈ Ω. If K ∈ L∞, we get the class of bounded distortion (that is, quasiregular) functions. During the last years, there has been deep progresses in this field. In particular, removability of singularities for bounded mappings of finite distortion have been studied (see, for instance, [7], [18], [27] or [30]). In general, relatively weak assumptions on K are sufficient to extend some basic quasiregular properties, such as continuity, discreteness and openness. Furthermore, in the plane we have existence of normalized solutions (as Theorem 1.11) and factorization theorems (as Theorem 1.12). Althoug the natural regularity for these 1,P C t2 mappings is the Orlicz-Sobolev space Wloc ( ), P(t) = log(e+t) (which is larger than the 1,2 C usual Wloc ( )), we can study the removability problem for Lipα mappings of finite distortion, which appears as a limiting situation of the quasiregular case.

To look for results in the converse direction, we recall from [16] that any compact set E with H1+α(E) > 0 is nonremovable for holomorphic functions and, thus, neither for

K-quasiregular mappings in Lipα. Hence, our interest is centered on sets of dimension 1+αK between d = 2 1+K and 1 + α. We show in Corollary 3.6 that the index d is sharp in the following sense.

62 3.1 Introduction

Theorem. Given α ∈ (0, 1) and K ≥ 1, for any t > d there exists a compact set E of dimension t, and a C C function f ∈ Lipα( ) which is K-quasiregular in \ E, and with no K-quasiregular extension to C.

> 1+αK In other words, given any dimension t d = 2 1+K , we will construct a compact set E C C C of dimension t and a Lipα( ) function which is K-quasiregular on \ E but not on .

Our procedure starts with a compact set E of dimension t and a K-quasiconformal map- 0 2Kt 0 > ping φ such that dim(φ(E)) = t = 2+(K−1)t (notice that t 1). Then, we will show C > that there are Lipβ( ) functions, for some β 0, analytic outside of φ(E), which in turn induce (by composition) K-quasiregular functions on C \ E with some global Holder¨ continuity exponent. This construction will encounter two obstacles. First, the extremal 1+αK dimension distortion of sets of dimension 2 1+K under K-quasiconformal mappings is not exactly 1 + α, the critical number in the analytic setting (notice that this was so for α = 0, i.e. in the removability problem for BMO K-quasiregular mappings). Second, the composition of β-Holder¨ continuous functions with K-quasiconformal mappings is C only in Lipβ/K( ), so there is some loss of regularity that might be critical. To avoid these troubles, we will construct in an explicit way the mapping φ. This concrete construction allows us to show in Corollary 3.5 that our mapping φ exhibits an exponent of Holder¨ continuity given by t 1 K − 1 = + t t0 K 2K 1 which is larger than the usual K obtained by Mori (Theorem 1.13). This idea was suggested to us by X. Zhong. Observe that if dim(E) = t and dim(φ(E)) = t0 it is natural to expect φ to be Lipt/t0 . This regularity will be sufﬁcient for our purposes.

This chapter is structured as follows. In Section 3.2 we prove Main Theorem 3.1. In Section 3.3, we construct nonremovable sets by means of extremal distortion mappings.

Finally, in Section 3.4 we study the Lipα removability problem for the class of ﬁnite distortion mappings.

63 3 Removable singularities for Holder¨ continuous quasiregular mappings

3.2 Sufﬁcient conditions for Lipα removability

In this section, if µ is compactly supported, we mean by principal solution the unique µ-quasiconformal mapping φ normalized as in (1.10), that is, such that φ(0) = 0 and φ(z) − z ∈ C + W1,p(C). We start with an auxiliary compactness result.

Lemma 3.1.

Suppose that µn, µ are Beltrami coefﬁcients, all of them compactly supported on D, and assume K−1 C C that kµnk∞, kµk∞ ≤ K+1 . Let φn, φ : → be, respectively, the principal solutions to the corresponding Beltrami equations. If µn → µ at almost every point, then:

p K (a) lim J(·, φn) = J(·, φ), with convergence in L , for every p ∈ (1, ). n→∞ loc K−1

(b) lim φn = φ uniformly on compact sets. n→∞

Proof. Since µn, µ are compactly supported on D, we can proceed as in Example 1.10 2p to represent φ and φn as φn(z) = z + C˜hn(z), φ(z) = z + C˜h(z). Here hn, h ∈ L (C) −1 −1 are deﬁned, respectively, by hn = (I − µnB) (µn) and h = (I − µB) (µ), C˜g(z) = Cg(z) − Cg(0) is the variant of the classical Cauchy transform deﬁned in equation (1.3), K and p is any real number in (1, K−1 ). A computation shows that

(I − µnB)(h − hn) = (µ − µn) (1 + Bh)

Astala, Iwaniec and Martin [8, Theorem 1] showed that I − µnB is a bilipschitz isomor- 2p C K−1 phism from L ( ) to itself, whose bilipschitz constant depends only on kµnk ≤ K+1 and p. Therefore,

−1 kh − hnk2p ≤ k(I − µnB) k2p k(µ − µn) (1 + Bh)k2p ≤ CK k(µ − µn) (1 + Bh)k2p

2p Hence limn khn − hk2p = 0, because h ∈ L (C) and µn, µ have support uniformly 2p C bounded. It then follows that Dφn converges in Lloc( ) to Dφ, and hence J(·, φn) con-

64 3.2 Sufﬁcient conditions for Lipα removability

p C verges in Lloc( ) to J(·, φ). Furthermore [2, p. 86], there exists a constant C p for which

1 1− p |C˜ f (z) − C˜ f (w)| ≤ Cp k f k2p |z − w| for any pair of points z, w ∈ C. We also have C˜ f (0) = 0 whenever f ∈ L2p(C). Thus,

1 1− p |φ(z) − φn(z)| = |C˜(h − hn)(z)| ≤ Cp kh − hnk2p |z|

−1 φn and φn → φ uniformly on compact sets. Finally, notice that the mappings ψn = −1 φn (1) form a normal familly, since they all are K-quasiconformal and ﬁx 0, 1 and ∞. Therefore, 1 they are locally uniformly K -Holder¨ continuous. Thus, if φ(w) = z,

−1 −1 −1 −1 −1 −1 |φ (z) − φn (z)| = |φ φ(w) − φn φ(w)| = |φn φn(w) − φn φ(w)| 1 −1 −1 K = |φn (1)| |ψn (φn(w)) − ψn(φ(w))| ≤ CK |φn (1)| |φn(w) − φ(w)|

In particular, if here we take z = 1 then

−1 φ (1) 1 K −1 − 1 ≤ CK |φn(w) − φ(w)| → 0 φn (1)

−1 −1 −1 Since φn(0) = 0 for all n, necessarily limn φn (1) = φ (1). Therefore φn converges locally uniformly to φ−1.

Theorem 3.2. C 2 Let E ⊂ be a compact set, α ∈ (0, 1) and K ≥ 1. Set d = K+1 (1 + αK). If

Hd(E) = 0

then, E is removable for K-quasiregular mappings in Lipα.

C C C C Proof. Assume that f : → is a Lipα( ) mapping, K-quasiregular on \ E. We will see that f is K-quasiregular in the whole plane C. ∂ f Let µ = ∂ f be the Beltrami coefﬁcient of f . Obviously, there is no restriction if we assume that E ⊂ D and that µ is supported on D. Let φ be the principal solution to ∂φ = µ ∂φ. Then the function F = f ◦ φ−1 is holomorphic on C \ φ(E) and Holder¨

65 3 Removable singularities for Holder¨ continuous quasiregular mappings continuous on C with exponent α/K. In particular, ∂F deﬁnes a distribution compactly 2 C supported on φ(E). Indeed, since F is continuous, it is an Lloc( ) function, so that its distributional derivatives deﬁne continuous linear functionals on W 1,2(C) functions. From Weyl’s lemma, it is enough to show that ∂F = 0.

For each ε > 0 we can consider a ﬁnite family of disks Dj = D(zj, rj) such that Ωε =

j Dj covers E and n d S ∑ diam(Dj) ≤ ε j=1

We deﬁne µε = µ χC\Ωε . If φε is the principal solution to ∂φε = µε ∂φε, then by Lemma −1 C 3.1 the functions Fε = f ◦ φε converge uniformly to F as ε → 0, because f ∈ Lipα( ). Thus, for any compactly supported C ∞ function ϕ,

h∂F, ϕi = h∂(F − Fε), ϕi + h∂Fε, ϕi = −h(F − Fε), ∂ϕi − h∂Fε, ϕi

Here the ﬁrst term converges to 0 as ε → 0. Thus, we just care about the second term. ∞ Consider a partition of unity ψj subordinated to the covering Dj, that is, each ψj is a C C n function, compactly supported on 2Dj, |Dψj| ≤ , and ∑ ψj = 1 on Ωε. We diam(Dj) j=1 −1 1,2 C now deﬁne ϕj = ψj ◦ φε . Then, each ϕj is in W ( ), has compact support in φε(2Dj), n and ∑j=1 ϕj = 1 on φε(Ωε). For any constants cj we have

φε

E φε(E)

Dj φε(Dj)

ψj ϕj

n n n h∂Fε, ϕi = h∂Fε, ϕ ∑ ϕji = ∑h∂Fε, ϕ ϕji = ∑h∂(Fε − cj), ϕ ϕji j=1 j=1 j=1 n n (3.2) = − ∑h(Fε − cj), ∂ϕ ϕji − ∑h(Fε − cj), ϕ ∂ϕji = I + II j=1 j=1

66 3.2 Sufﬁcient conditions for Lipα removability and now we have just to bound I and II independently. Notice that both terms may now be written as integrals. After a change of coordinates, for I we get

n ≤ kDϕk∞ ∑ | f (w) − cj| J(w, φε)dA(w). j=1 Z2Dj

K p C Now observe [4] that for any p ∈ (1, K−1 ), J(·, φε) ∈ Lloc( ). Using this, together with the fact that f ∈ Lipα, we obtain for I the bound

1/p n p α+2/q |I| ≤ C kDϕk∞ ∑ J(w, φε) dA(w) diam(2Dj) j=1 Z2Dj ! 1/p 1/q n n p qα+2 ≤ C kDϕk∞ ∑ J(w, φε) dA(w) ∑ diam(2Dj) j=1 Z2Dj ! j=1 !

p where q = p−1 . But both sums converge to 0 as ε tends to 0, so that we get I → 0. For the term II in (3.2),

≤ kϕk∞ ∑ | f (w) − cj| |∂ϕj(φε(w))| J(w, φε )dA(w). j=1 Z2Dj

−1 Now, φε is also K-quasiconformal. Thus, by the chain rule,

−1 |Dψj| |Dϕj(φε)| ≤ |Dψj| |Dφε (φε)| ≤ 1 . J(·, φε) 2

Therefore, the Lipα condition for f and the decay of Dψj give

n 1 α−1 |II| . kϕk∞ ∑ J(w, φε) 2 dA(w) diam(2Dj) . j=1 Z2Dj !

67 3 Removable singularities for Holder¨ continuous quasiregular mappings

1 Now, we recall that J(·, φε) 2 deﬁnes a doubling measure, with constants depending only on K, so that

n 1 α−1 |II| . kϕk∞ ∑ J(w, φε) 2 dA(w) diam(Dj) . j=1 ZDj !

On the other hand, inside the union Ωε = ∪j Dj the mapping φε is conformal, so that K J(·, φε) attains the optimal integrability K−1 , as in Theorem 1.16, and

K J(w, φε) K−1 dA(w) ≤ 1 ZΩε Hence, as before, we use Holder¨ ’s inequality two times (one for the integral and one for the sum) to get

K−1 n 2K K α+ 1 |II| . kϕk∞ ∑ J(w, φε) K−1 dA(w) diam(Dj) K j=1 ZDj ! K−1 K+1 n 2K n 2K K 2(1+αK) ≤ kϕk∞ ∑ J(w, φε) K−1 dA(w) ∑ diam(Dj) K+1 . j=1 ZDj ! j=1 !

From the optimal integrability results, the integral above remains uniformly bounded, while the sum is bounded by ε from our choice of the covering.

We wish to emphasize that the improved integrability properties for K-quasiconformal mappings of Astala and Nesi [10] is precisely what allows us to jump from sets of dimension strictly smaller than d to sets of null d-dimensional Hausdorff measure. Note also that Theorem 3.2 may be stated in a more general sense, in terms of the modulus of continuity of f . Assume that f is K-quasiregular in C \ E and, instead of inequality (3.1), suppose that | f (z) − f (w)| ≤ ω(|z − w|)

C < for any z, w ∈ with |z − w| 1, where ω(r) = ω f (r) is any bounded continuous non-decreasing function ω : [0, 1) → [0, ∞) such that ω(0) = 0. If Hh(E) = 0 for

68 3.3 Extremal distortion and nonremovable sets

2 2K h(t) = t K+1 ω(2t) K+1 , then f is K-quasiregular on the whole plane. As particular cases, α C ω f (r) ω(r) = C r gives the previous theorem, while if f ∈ lipα( ), that is, if rα → 0 as d 1 −α > r → 0, then M∗(E) = 0 is enough. If ω(t) = log t , α 0, then the removability g result has a consequence in terms of quasiconformal distortion: H (φ(E)) = 0, where 2 1 −α K+1 g(t) = t log t . Our argument also shows that if M∗ (E) = 0 then E is removable for continuous K-quasiregular mappings.

The results obtained in Chapter 2 encourage us to make the following question.

Problem 3.3. 1+αK > d Let d = 2 1+K , K 1. Is it true that if E is a compact set, such that H (E) is σ-ﬁnite, then E is removable for α-Holder¨ continuous K-quasiregular mappings?

3.3 Extremal distortion and nonremovable sets

One of the main results in [4] is the sharpness of the dimension distortion equation (1.12). To obtain the equality there, the author moved holomorphically a fixed Cantor- type set F. This movement defined actually a holomorphic motion on F (see for instance [9] to have a quick introduction to holomorphic motions). An interesting extension result, known as the λ-lemma, allowed to extend this motion quasiconformally from F to the whole plane. This procedure avoided most of the technicalities and gave the desired result in a surprisingly direct way. However, since we look both for extremal dimension distortion and higher Holder¨ continuity, we need to construct φ explicitely. 0 2Kt Thus, let t ∈ (0, 2) and K ≥ 1 be fixed numbers, and denote t = 2+(K−1)t . As Astala did in [4], we first give a K-quasiconformal mapping φ that maps a regular Cantor set E of dimension t, to another regular Cantor set φ(E) for which dim(φ(E)) is as close as we want to t0. Later, again as in [4], we will glue a sequence of such mappings in the convenient way.

Proposition 3.4. Given t ∈ (0, 2), K ≥ 1 and ε > 0, there exists a compact E and a K-quasiconformal mapping φ : C → C, with the following properties:

69 3 Removable singularities for Holder¨ continuous quasiregular mappings

(a) φ is the identity mapping on C \ D.

(b) E is a self-similar Cantor set, constructed with m = m(ε) similarities.

p C K (d) J(·, φ) ∈ Lloc( ) if and only if p ≤ K−1 .

1 − 1 t (e) |φ(z) − φ(w)| ≤ C m t t0 |z − w| t0 whenever |z − w| < 1.

Proof. Following the scheme of Astala, Iwaniec, Koskela and martin in [7], we will obtain φ as a limit of a sequence of K-quasiconformal mappings

φ = lim φN N→∞ while E will be a regular Cantor set. The mapping φN will act only on the ﬁrst N steps in the construction of E. Usually, radial stretchings like

1 −1 f (z) = z|z| K are extremal for some typical quasiconformal features. Thus, it is natural to construct the mappings φN using dilations and translations of f . However, we look for mappings C φ living in better Holder¨ spaces than Lip1/K( ) (this fails for f ), so that we will replace f by a linear mapping in a small neighbourhood of its singularity. This change will not affect the integrability index, and at the same time will give some improvement on the Holder¨ exponent.

D Take m ≥ 100, and consider m disjoint disks inside of , D(zi, r), uniformly distributed, all with the same radius r = r(m). By taking m big enough, we may always assume that 2 1 cm = mr ≥ 2 . Given any σ ∈ (0, 1) to be determined later, we can consider m simili- tudes D ϕi(z) = zi + σr z, z ∈

70 3.3 Extremal distortion and nonremovable sets and denote, for every i = 1, . . . , m,

1 D = ϕ (D) = D(z , r ), i σ i i 1 0 D Di = ϕi( ) = D(zi, σr1), where r1 = r. We deﬁne

1 K −1 0 σ (z − zi) + zi z ∈ Di 1  K −1 z−zi 0 g1(z) =  (z − zi) + zi z ∈ Di \ D  r1 i  z otherwise.    It may be easily seen that g1 deﬁnes a K-quasiconformal mapping, which is conformal 0 everywhere except on each ring Di \ Di. Moreover, if we put D D

0 Di 00 Di 1 Di g Di

Figure 3.1

1 K D ψi(z) = zi + σ r z, z ∈

0 00 D then g1 maps every Di to itself, while each Di is mapped to Di = ψi( ), as is shown in Figure 3.1. Now we denote φ1 = g1.

00 At the second step, we repeat the above procedure inside of every Di , and the rest

71 3 Removable singularities for Holder¨ continuous quasiregular mappings

remains ﬁxed. That is, we deﬁne g2 on the target set of φ1, and then construct φ2 as

φ2 = g2 ◦ φ1

More explicitly, 1 D = φ ϕ (D) = D(z , r ) ij σ 1 ij ij 2 0 D Dij = φ1 ϕij( ) = D(zij, σr2)

1 where a computation shows that r2 = σ K r r1. Now we deﬁne

1 K −1 0 σ (z − zij) + zij z ∈ Dij 1 −1  z−zij K 0 g2(z) =  (z − zij) + zij z ∈ Dij \ D  r2 ij  z otherwise.

   2 By construction, g2 is K-quasiconformal on C, conformal outside a union of m rings, 0 00 D and maps Dij to Dij = ψij( ), while every point outside of the disks Dij remains ﬁxed under g2, as Figure 3.2 shows. Thus, the composition φ2 = g2 ◦ φ1 (see Figure 3.3) is still

00 00 Di Di

D Dij 0 ij g2 00 Dij Dij

Figure 3.2

K-quasiconformal, agrees with the identity outside of D, and

D D φ2(ϕij( )) = ψij( )

72 3.3 Extremal distortion and nonremovable sets

for any i, j = 1, . . . , m. After N − 1 steps, we will deﬁne gN on the target side of φN−1. D D

φ2

Figure 3.3

For each multiindex J = (j1, ..., jN) of length `(J) = N we will denote

1 D = φ ϕ (D) = D(z , r ) J σ N−1 J J N 0 D DJ = φN−1 ϕJ( ) = D(zJ, σrN ) 1 where now rN = σ K r rN−1. Then, the mapping

1 K −1 0 σ (z − zJ ) + zJ z ∈ DJ 1 −1  z−zJ K 0 gN (z) =  (z − zJ ) + zJ z ∈ DJ \ D  rN J  z otherwise

  C  N is K-quasiconformal on , conformal outside of a union of m rings. Moreover, gN(DJ ) = 0 00 00 D DJ and gN (DJ) = DJ , where DJ = ψJ( ). Hence, φN = gN ◦ φN−1 is also K-quasiconformal and D D φN(ϕJ ( )) = ψJ( ).

With this procedure, it is clear that the sequence φN is uniformly convergent to a homeomorphism φ. It is also clear that φ has distortion bounded by K almost everywhere. Moreover, φ is a K-quasiconformal mapping, since it comes as a limit of a normal fam-

73 3 Removable singularities for Holder¨ continuous quasiregular mappings ily of quasiconformal mappings (see Section 1.4). By construction, φ maps the regular Cantor set ∞ D E = ϕJ( ) `  N\=1 ([J)=N   to ∞ D φ(E) = ψJ( ) `  N\=1 ([J)=N   which obviously is also a regular Cantor set. If now we choose σ so that

m(σr)t = 1 we directly obtain, on one hand, 0 < Ht(E) < ∞, and on the other,

1 1 1 K − 1 log 2 = + mr . dim(φ(E)) t0 2K log m

1 Since cm ≥ 2 for all m, we may always get

dim(φ(E)) ≥ t0 − ε just increasing m if needed.

Now we have to look at the regularity properties of our mapping φ. To do that, we in- 0 D N N troduce the following notation: put G = , and denote by PJ and by GJ , respectively, the peripheral and generating disks of generation N, that is, for any chain J = (j1, ..., jN),

1 PN = ϕ (D) J σ J N D GJ = ϕJ( ).

N 0 N 00 N With this notation, DJ = φN−1(PJ ), DJ = φN−1(GJ ) and DJ = φN(GJ ).

p C Now take any p such that J(·, φ) ∈ Lloc( ). Of course, we can assume p ≥ 1. Then,

74 3.3 Extremal distortion and nonremovable sets one may decompose the p-mass of J(·, φ) over D in the following way

m m J(z, φ)pdA(z) = J(z, φ)pdA(z) + ∑ J(z, φ)pdA(z) + ∑ J(z, φ)pdA(z) D D\∪ P1 P1\G1 G1 Z Z i i i=1 Z i i i=1 Z i

C 1 and since φ = φ1 on \ ∪iGi ,

p p p p J(z, φ) dA(z) = J(z, φ1) dA(z) + m J(z, φ1) dA(z) + m J(z, φ) dA(z) D D\∪ P1 P1\G1 G1 Z Z i i Z Z where here P1 and G1 denote, respectively, any of the peripheral and generating disks of ﬁrst generation. Now, one may repeat this argument for the last integral, and by a recursive argument we get

∞ ∞ p N p N p J(z, φ) dA(z) = ∑ m J(z, φN+1) dA(z) + ∑ m J(z, φN ) dA(z) D GN\∪ PN+1 PN\GN Z N=0 Z i i N=1 Z where, as before, PN and GN denote any peripheral or, respectively, generating disk of N-th generation. Now we compute separately the integrals in both sums. Firstly, if

J = (j1, ..., jN)

p p p J(z, φN+1) dA(z) = J(φN (z), gN+1) J(z, φN ) dA(z) GN\∪ PN+1 GN\∪ PN+1 Z J i (Ji) Z J i (Ji) p −1 p−1 = J(w, gN+1) J(φN (w), φN ) dA(w) D00\∪ D Z J i (Ji) 2N(p−1) 1 −1 p = σ K J(w, gN+1) dA(w) D00\∪ D Z J i (Ji) 2N(p−1) 1 −1 = σ K 1 dA(w) D00\∪ D Z J i (Ji) 1 2N(p−1) K −1 00 = σ |DJ \ ∪iD(Ji)|

2N Nγ = r σ π(1 − cm).

75 3 Removable singularities for Holder¨ continuous quasiregular mappings

1 where γ = 2p K − 1 + 2. Secondly, p p p J(z, φN ) dA(z) = J(φN−1(z), gN ) J(z, φN−1) dA(z) PN\GN PN\GN Z J J Z J J p −1 p−1 = J(w, gN ) J(φN−1(w), φN−1) dA(w) D \D0 Z J J 2(N−1)(p−1) 1 −1 p = σ K J(w, gN ) dA(w) D \D0 Z J J 2π 1 − σγ = r2N σ(N−1)γ Kp γ

K K under the additional assumption p 6= K−1 . If p = K−1 , we get

K 2π 1 K−1 2N J(z, φN ) dA(z) = r K log . PN\GN K−1 σ Z J J K

< K Thus, if p K−1 we get

γ ∞ p 2π 1 − σ γ N J(z, φ) dA(z) = π(1 − cm) + cm ∑ (cm σ ) . D Kp γ Z N=0

Since p is such that J(·, φ) ∈ Lp (C), we necessarily get σγ < 1 . For m big enough this loc cm > < K K is equivalent to γ 0, i.e. p K−1 . At the critical point, p = K−1 , one gets

∞ K 2π 1 K−1 N J(z, φ) dA(z) = π(1 − cm) + cm K log ∑ (cm) D −1 σ Z K K N=0 which will always converge, for any ﬁxed value of m. This shows that we can choose p D K m big enough so that J(·, φ) ∈ Lloc( ) if and only if p ≤ K−1 .

Finally, it just remains to check that φ is Holder¨ continuous with exponent t/t0. By means of Poincare´ inequality, together with the quasiconformality of φ, it is enough [22, p.64] to show that for any disk D

0 J(z, φ) dA(z) ≤ C diam(D)2t/t . ZD

76 3.3 Extremal distortion and nonremovable sets

N 1 < N−1 Hence, for some ﬁxed disk D, take N such that (σr) ≤ 2 diam(D) (σr) . We have

J(z, φ) dA(z) ≤ J(z, φ) dA(z) + J(z, φ) dA(z) D D\∪GN ∪GN Z Z J Z J

N N N ∅ N where the union ∪GJ runs over all disks GJ such that GJ ∩ D 6= . On D \ ∪GJ , we easily see that 2N 1 1 −1 J(·, φ) = J(·, φ ) ≤ σ K . N K Thus, 2N 2 1 1 −1 1 J(z, φ) dA(z) ≤ σ K π diam(D) D\∪GN K 2 Z J t K−1 2N 2 0 π 2( 1 − 1 ) 1 t ≤ c 2K m t t0 diam(D) . K m 2

1 N N N K On the other hand, recall that φ(GJ ) = φN(GJ ) are disks of radius σ r . Hence, 1 2N N N K J(z, φ) dA(z) = ∑ |φ(GJ )| = ∑ |φN(GJ )| = ∑ π σ r ∪ GN Z J J J J J 0 0 2t/t K−1 2N t/t N 2K 1 (σr) = π cm diam(D) ∑ 2 1 ( ) J 2 diam D !

2t/t0 (σr)N and it just remains to bound ∑J 1 . Actually, this is equivalent to ﬁnding 2 diam(D) some constant C such that

N 2t/t0 2t/t0 ∑ diam(GJ ) ≤ C diam(D) . N ∅ GJ ∩D6=

N But the disks GJ come from a self-similar construction, said to give a regular Cantor set of dimension t. In particular, they may be chosen uniformly distributed so that the so-called t-dimensional packing condition is satisﬁed, that is,

N t t ∑ diam(GJ ) ≤ C diam(D) . N ∅ GJ ∩D6=

77 3 Removable singularities for Holder¨ continuous quasiregular mappings

It is easy to show that this condition implies the s-dimensional one for all s > t (in 2t particular, for s = t0 ). Hence, the constant C exists and is independent of m. Thus, what we ﬁnally get is that

0 K−1 2N 2t/t 1 − 1 1 J(z, φ) dA(z) ≤ C m t t0 c 2K diam(D) m 2 ZD and the result follows.

Corollary 3.5. 0 2Kt Let K ≥ 1 and t ∈ (0, 2). Denote t = 2+(K−1)t. There exists a compact set E, of dimension t, and a K-quasiconformal mapping φ : C → C, such that:

(a) Ht(E) is σ-ﬁnite.

(b) dim(φ(E)) = t0.

t (c) |φ(z) − φ(w)| ≤ C |z − w| t0 whenever |z − w| < 1.

Proof. Given ε > 0, K ≥ 1 and t ∈ (0, 2), let φ : C → C and E be as in Proposition 3.4. Then, for any ﬁxed r > 0, the mapping

ψr(z) = r φ(z/r)

and the set Er = rE exhibit the same properties than φ and E, since neither K-quasiconformality nor Hausdorff dimension are modiﬁed through dilations. However, when computing < the new Lipt/t0 constant, if |z − w| r then

1 1 t t − 0 1− 0 0 |ψr(z) − ψr(w)| = r |φ(z/r) − φ(w/r)| ≤ C m t t r t |z − w| t .

D Thus, as in [4], let Dj = D(zj, rj) be a countable disjoint family of disks inside of , and let εj be a sequence of positive numbers, ε j → 0 as j → ∞. For each j, let φj and Ej be 0 as in Proposition 3.4, so that dim(φj(Ej)) ≥ t − εj. In particular, each Ej is a regular z−z Cantor set of m components. Denote then ψ (z) = r φ ( j ) and F = z + r E , and j j j j rj j j j j

78 3.3 Extremal distortion and nonremovable sets deﬁne

ψj(z) z ∈ Dj ψ(z) =  z otherwise.

By construction, ψ is a K-quasiconformal mapping. It maps the set F = ∪ jFj to the set t ψ(F) = ∪jψj(Fj). Moreover, H (F) is σ-ﬁnite, while

0 dim(φ(F)) = sup dim(ψj(Fj)) = t . j

D Finally, assume that z lies inside some ﬁxed Dk and that w ∈ \ ∪j Dj. Then, consider the line segment L between z and w, and denote {zk} = L ∩ ∂Dk. Then, both zk and w are ﬁxed points for ψ, so that

1 1 t t t − 0 1− 0 t t t0 ≤ C mk rk |z − zk| + |zk − w|.

Since we are still free to choose radii rj, we may do it so that the radius rj satisfy

1 − 1 1− t t t0 t0 < mk rk 1

t < or, equivalently, mj rj 1. Under this assumption, we ﬁnally get

t |ψ(z) − ψ(w)| ≤ (C + 1) |z − w| t0

< C whenever |z − w| 1. This clearly shows that ψ ∈ Lipt/t0 ( ).

Although the set in Corollary 3.5 is more critical than the one we constructed in Propo- sition 3.4, in the sense that the ﬁrst gives precisely the extremal dimension distortion, both do the same work when studying non-removable sets for Holder¨ -continuous quasiregular mappings.

Corollary 3.6. > 1+αK < t < Let K ≥ 1 and α ∈ (0, 1). For any t 2 1+K there exists a compact set E with 0 H (E) ∞ , non removable for K-quasiregular mappings in Lipα.

79 3 Removable singularities for Holder¨ continuous quasiregular mappings

Proof. Let E and φ be such that dim φ(E) ≥ t0 − ε > 1 for some ε small enough. Hence, by Frostman’s Lemma, we can construct a positive Radon measure µ supported on φ(E), with growth t0 − 2ε. Its Cauchy transform g = Cµ deﬁnes a holomorphic function on C \ φ(E), not entire, and with a Holder¨ continuous extension to the whole plane, with exponent t0 − 2ε − 1. Set f = g ◦ φ.

Clearly, f is K-quasiregular on C \ E and has no K-quasiregular extension to C. Indeed, if f˜ extends f K-quasiregularly to C, then g˜ = f˜ ◦ φ−1 would provide an entire extension of g, which is impossible. Furthermore, f is Holder¨ continuous with exponent

t t (t0 − 2ε − 1) = t − (2ε + 1) . t0 t0

Thus, we just need ε > 0 small enough so that

t t − (2ε + 1) ≥ α t0 but this inequality is equivalent to

1 + αK 2 t − 2 ≥ ε (2 + (K − 1)t) 1 + K K + 1 and the proof is complete.

1+αK At this point, it should be said that above the critical index 2 1+K one might ﬁnd also some removable set. For instance, due to an unpublished result of S. Smirnov, it is known that if E = ∂D and φ is a K-quasiconformal mapping, then

K − 1 2 dim(φ(E)) ≤ 1 + K + 1 which is better than the usual dimension distortion equation (1.12). Hence, if we choose K ≥ 1 small enough, then there exists α satisfying

K − 1 2 K − 1 K < α < . K + 1 2K

80 3.4 Removability results for ﬁnite distortion mappings

For those values of α, the set E = ∂D is removable for α-Holder¨ continuous K-quasiregular mappings and, however, 1 + αK 2 < dim(E). 1 + K 1+αK This suggests that between 2 1+K and 1 + α everything may happen.

Problem 3.7. 1+αK Let d = 2 1+K . Is there some compact set E of dimension d, nonremovable for α-Holder¨ continuous K-quasiregular mappings?

Problem 3.8. 1+αK < < Is there some real number d0 = d0(α, K), 2 1+K d0 1 + α, such that every compact set E of dimension dim(E) > d0 is nonremovable for α-Holder¨ continuous K-quasiregular 1+αK mappings, and also for every t ∈ (2 1+K , d0) there exist both removable and nonremovable sets of dimension t?

3.4 Removability results for ﬁnite distortion mappings

Recall that if Ω ⊂ C is a domain, then we call mapping of ﬁnite distortion on Ω to any Ω C 1,1 Ω function f : → in the Sobolev class Wloc ( ) with locally integrable jacobian, 1 C Ω ∞ J(·, f ) ∈ Lloc( ), and such that there is a measurable function K f : → [1, ], ﬁnite almost everywhere, called the distortion function of f , for which

2 |D f (z)| ≤ K f (z) J(z, f ) at almost every z ∈ Ω. Equivalently, f satisﬁes a Beltrami equation

∂ f = µ ∂ f where µ is only required to satisfy |µ(z)| < 1 at almost every z ∈ Ω. For instance, ∞ < if K f ∈ L (that is, when kµk∞ 1), and kK f k∞ = K, one recovers the class of K- quasiregular mappings. However, weaker assumptions on K f also give interesting classes of functions. As a ﬁrst particular case, a ﬁnite distortion mapping is said to

81 3 Removable singularities for Holder¨ continuous quasiregular mappings have exponentially integrable distortion if

p C exp K f ∈ Lloc( ) for some p > 0. Secondly, a ﬁnite distortion mapping has subexponentially integrable distortion if K f exp ∈ Lp (C) 1 + log K loc f for some p > 0. An excellent reference for studying mappings of ﬁnite distortion is the book of T. Iwaniec and G. Martin [30].

The removability problem also makes sense for these classes of mappings. Actually, the removable sets for bounded mappings of finite distortion have already been studied (see for instance [7], [18], [26], [27] or [34]). Moreover, it turns out that our arguments in the proof of Theorem 3.2 may be repeated in the setting of finite distortion mappings. For this, we first have to see under which conditions some basic properties (such as existence, uniqueness and factorization of solutions to degenerated Beltrami equations) hold also in this more general situation. The most natural spaces for these 1,P C 1,P Ω results are the Orlicz-Sobolev spaces Wloc ( ). Recall that f ∈ W ( ) if and only if both f and its distributional derivatives D f live in the usual Orlicz space L P(Ω). Most of the cases we will work with Orlicz functions P = P(t) for which

t 7→ P(t5/8) is a convex function (3.3) and such that ∞ P(t) dt = ∞ (3.4) t3 Z1

Examples of such Orlicz functions are P(t) = t2 which gives LP = L2, as well as P(t) = t2 log(e + t), for which locally we have Lq ⊂ LP ⊂ L2 whenever q > 2. Furthermore, under these assumptions we can ﬁnd also weaker spaces than L2. For instance, P(t) = t2 t2 2 P q < log(e+t) and P(t) = log(e+t) log log(3+t) are such that L ⊂ L ⊂ L whenever q 2. The following example is of special interest, since it shows that both conditions (3.3)

82 3.4 Removability results for finite distortion mappings and (3.4) are somewhat necessary and sufficient for developing a rich theory of finite distortion mappings.

Example 3.9. z 1,1 C Let f be deﬁned as f (z) = z + |z| for z 6= 0, f (0) = 0. Then, f ∈ Wloc ( ) with

1 |D f (z)| = |∂ f (z)| + |∂ f (z)| = 1 + |z|

Moreover, the pointwise jacobian J(z, f ) = det(D f (z)) may be explicitely computed,

1 J(z, f ) = |∂ f (z)|2 − |∂ f (z)|2 = 1 + |z|

1 C and clearly J(·, f ) ∈ Lloc( ). Thus, f satisﬁes a distortion inequality

|D f (z)|2 = K(z) J(z, f )

1 with distortion function K(z) = 1 + |z| . Therefore f is a mapping of ﬁnite distortion. However, 1,p C < f ∈ Wloc ( ) for every p 2, but not for p = 2. More precisely,

∞ P(t) f ∈ W1,P(C) ⇐⇒ dt < ∞ loc t3 Z1 This means that if P does not satisfy equation (3.4), then there exists a ﬁnite distortion mapping 1,P C f ∈ Wloc ( ) which is not even continuous.

To explain in just a few words this situation, we should talk about the distributional jacobian 1, 4 Ω C C 3 Ω [30, p.140]. Given a mapping f = ( f1, f2) : ⊂ → of class Wloc ( ), one may deﬁne a distribution J f by

hJ f , ϕi = − f1 dϕ ∧ d f2 Z for any ϕ ∈ D(Ω). We say that f is an orientation preserving mapping if J f is a positive distribution. It turns out that if f is an orientation preserving mapping of class W 1,P for some

83 3 Removable singularities for Holder¨ continuous quasiregular mappings

P satisfying equations (3.3) and (3.4), then

hJ f , ϕi = J(z, f ) ϕ(z) dA(z), Z that is, J f agrees with the usual jacobian J(z, f ) = det(D f (z)). As a matter of fact, the distributional jacobian of our mapping f is

hJ f , ϕi = J(z, f ) ϕ(z) dA(z) + π ϕ(0) Z Hence, (3.4) is necessary for having J(·, f ) = J f .

A precise formulation of what we mean in the above example may be found in [30, Section 7.5]. We summarize it in the following theorem. From now on we only consider Orlicz functions satisfying (3.3) and (3.4).

Theorem 3.10. Ω C C 1,P Ω Suppose that f : ⊂ → belongs to some Wloc ( ).

• If f is an orientation preserving mapping, then f has locally integrable jacobian.

• If f is a ﬁnite distortion mapping, then f is continuous.

The following result (see e.g. [30, Theorem 11.5.1]) establishes the Stoilow factorization 1,P < for Wloc solutions to any Beltrami equation with coefﬁcient such that |µ| 1 a.e.

Theorem 3.11. Let µ be supported on D, such that |µ(z)| < 1 at almost every z ∈ D. If the equation ∂φ = µ ∂φ 1,P Ω has a homeomorphic solution φ in some Wloc ( ), then every other solution f in the same 1,P Ω Ω Wloc ( ) decomposes as f = h ◦ φ with h holomorphic on φ( ).

In fact, it is precisely when looking for existence of homeomorphic solutions in W 1,P where some integrability on µ is needed. Thus, suppose that µ : C → C is a measurable D < 1+|µ(z)| function, supported on , such that |µ(z)| 1 almost everywhere. Let K(z) = 1−|µ(z)| . The following general existence result may be found at [30, Theorem 11.8.3].

84 3.4 Removability results for ﬁnite distortion mappings

Theorem 3.12. p D > If exp(K) ∈ Lloc( ) for some p 0, then the equation

∂φ = µ ∂φ

1,P C t2 admits a unique homeomorphic solution φ such that φ(z) − z ∈ W ( ), with P(t) = log(e+t).

The above two theorems ensure that exponentially integrable distortion mappings live, 1,P t2 at least, in Wloc with P(t) = log(e+t) . However, as in the quasiregular setting, there is some improvement of regularity given in terms of the properties of the distortion function. The precise bounds for this autoimprovement are given in the following theorem, due to D. Faraco, P. Koskela and X. Zhong [18].

Theorem 3.13.

There exists a constant c0 > 0 with the following property. Assume that f : Ω → C is a mapping of ﬁnite distortion, whose distortion function K satisﬁes

p Ω exp (K) ∈ Lloc( )

> 1,P Ω 2 c0 p−1 for some p 0. Then, f ∈ Wloc ( ) with P(t) = t log (e + t).

In particular, exponentially integrable distortion mappings have locally L2 derivatives, p provided that their distortion function K satisﬁes exp (K) ∈ Lloc for p big enough (that is, p > 1 ). More generally, for subexponentially integrable distortion it is known [30, c0 p.267] that the corresponding Beltrami equation has a unique homeomorphic solution 1,P C t2 φ such that φ(z) − z ∈ W ( ) with P(t) = log(e+t) log log(3+t) .

Theorem 3.14. D C Let α ∈ (0, 1). Let E ⊂ be compact. Let f ∈ Lipα( ) be a mapping of ﬁnite distortion on C < \ E. If the distortion function K = K f is subexponentially integrable, and dim(E) 2α, then f is a mapping of ﬁnite distortion on C.

Proof. Under the conditions of the theorem, we have existence of homeomorphic solu- 1,q C < tions φ to the correspondig Beltrami equation, with φ ∈ Wloc ( ) for any q 2. Thus,

85 3 Removable singularities for Holder¨ continuous quasiregular mappings

F = f ◦ φ−1 is holomorphic on C \ E. We will show that

h∂F, ϕi = 0 whenever ϕ ∈ D(C). Let d ∈ (dim(E), 2α). For each ε > 0 we can consider a ﬁnite Ω family of disks Dj = D(zj, rj) such that = j Dj covers E and

n S d ∑ diam(Dj) ≤ ε j=1

As in the proof of Theorem 3.2, let ψj be a partition of unity subordinated to the covering 2D , that is, each ψ is a C∞ function, compactly supported on 2D , |Dψ | ≤ C , and j j j j diam(Dj) n Ω −1 ∑j=1 ψj = 1 on . We now deﬁne ϕj = ψj ◦ φ . It follows from the work of S. Hencl and P. Koskela [24, Theorem 1.1] that any orientation preserving homeomorphism g ∈ 1,1 C −1 1,1 C Wloc ( ) has an inverse g ∈ Wloc ( ) satisfying

|Dg−1(y)|dA(y) ≤ |Dg(x)|dA(x) Zg(D) ZD whenever D ⊂ C is a bounded open set. Moreover, it is also shown [24, Theorem −1 1,2 −1 1.3] that if g has ﬁnite distortion in D, then g ∈ Wloc (g(D)) and g has also ﬁnite 1,2 C distortion. As a consequence, we have that ϕj ∈ W ( ), they are compactly supported n in φ(2Dj), and ∑j=1 ϕj = 1 on φ(Ω). For any constants cj and any testing function ϕ we have n n h∂F, ϕi = − ∑h(F − cj), ∂ϕ ϕji − ∑h(F − cj), ϕ ∂ϕji = I + II (3.5) j=1 j=1 and now we have just to bound both sums independently. For the sum I, after a change of coordinates, the Lipα condition of f and the doubling condition of J(·, φ) give

n n α ∑h(F − cj), ∂ϕ ϕji . kDϕk∞ ∑ J(w, φ)dA(w) diam(2Dj) = = 2Dj ! j 1 j 1 Z n α . kDϕk∞ ∑ J(w, φ)dA(w) diam(Dj) j=1 ZDj !

86 3.4 Removability results for ﬁnite distortion mappings

Therefore

2−α 2 2 α/2 n 2−α n 2 |I| ≤ |Dϕk∞ ∑ J(w, φ)dA(w) ∑ diam(D )   j j=1 ZDj ! j=1 ! α/2  n n 2 ≤ kDϕk∞ ∑ J(w, φ)dA(w) ∑ diam(Dj) j=1 ZDj ! j=1 ! and clearly both factors converge to 0 as ε → 0. To proceed with II, since ϕ j are in W1,2(C), it makes sense to use chain rule to obtain

−1 −1 |Dϕj(z)| ≤ |Dψj(φ (z)| |Dφ (z)|

Hence,

Now, after a change of coordinates, we go back through [24, Theorem 1.1] to get

|Dφ−1(φ(w))| J(w, φ) dA(w) = |Dφ−1(z)| dA(z) ≤ |Dφ(w)| dA(w) Z2Dj Zφ(2Dj) Z2Dj 1 p 1 p q . |Dφ(w)| dA(w) ≤ |Dφ(w)| dA(w) |Dj| ZDj ZDj ! where p, q is any dual pair, 1 < p < 2. Therefore, the following bound for II may be obtained

1 1 n p n q p q(α−1)+2 |II| ≤ kϕk∞ ∑ |Dφ(w)| dA(w) ∑ diam(Dj) j=1 ZDj ! j=1 !

87 3 Removable singularities for Holder¨ continuous quasiregular mappings

Now, let p ∈ (1, 2) be such that q(α − 1) + 2 = d. Then both sums converge to 0 as ε → 0, and the result follows.

A slight modiﬁcation of the above argument gives the following.

Theorem 3.15. Let α ∈ (0, 1), and let E ⊂ D be compact, with ﬁnite H2α(E). If the distortion function K ad- 1,2 C mits a Wloc ( ) homeomorphic solution, then E is removable for α-Holder¨ continuous mappings of ﬁnite distortion K.

Among the distortion functions K that fall into the hypotheses of last theorem, we have those that satisfy p C exp (K) ∈ Lloc( ) for some p > 1 . Here c is the constant in Theorem 3.13. Furthermore, for such dis- c0 0 P C 2 c0 p−1 1,P C tortion functions, Dφ ∈ Lloc( ) with P(t) = t log (e + t). However, Wloc ( ) is a better space (that is, smaller) than W1,2(C), whenever p > 1 . Then, a better set function loc c0 than H2α may be obtained for these mappings. It turns out that L2 logs L acts continu- L2 ously against logs L , and Holder¨ ’s inequality looks like

s kghkL1 ≤ C kgkL2 log L khk L2 logs L

If now h = χD is the characteristic function of a disk D, then khk L2 may be explicitely logs L computed for small values of |D| and then Holder¨ ’s inequality reads as

1 1 |g| 2 |D| 2 |g| dA . |g|2 logs e + dA s 1 D D |g|D log 2 Z Z |D| whenever s > 0. Moreover, if Dj is a quasidisjoint family of disks, a similar argument to that in [7, Lemma 5.1] shows that

2 s |g| 2 s |g| ∑ |g| log e + dA ≤ Cs |g| log e + |g| |g| j ZDj Dj ! Z∪jDj ∪j Dj !

88 3.4 Removability results for ﬁnite distortion mappings whenever g is a nonnegative function for which

1 1 1 2 g2 dA ' g dA |D| |D| Z Z for each disk D. Actually, this is the case if g = |Dφ|. Hence, doing as in the above proof, one gets for s = c0 p − 1

n α−1 II ≤ C kϕk∞ ∑ |Dφ(z)| dA(z) diam(2Dj) j=1 Z2Dj ! 1/2 n α 2 s |Dφ(z)| diam(2Dj) ≤ C kϕk∞ ∑ |Dφ(z)| log e + dA(z) | φ( )| s 1 j=1 2Dj D z 2Dj ! ! log 2 Z diam(2Dj) 1 1 2 n 2 2 s |Dφ(z)| . C kϕk∞ |Dφ(z)| log e + dA(z) ∑ h(diam(2D )) |Dφ(z)| j Z∪j2Dj ∪j2Dj ! ! j=1 !

r2α h where h(r) = s 1 . In other words, finite H (E) is a sufficient condition for E to log r be removable for Lipα finite distortion mappings with distortion function K such that exp (K) ∈ Lp for p > 1 . c0

In the converse direction, one may easily ﬁnd nonremovable sets, using the known results for the quasiregular case.

Corollary 3.16. Let α ∈ (0, 1). For any t > 2α there exists a compact set E of dimension t and a function C C f ∈ Lipα( ), which deﬁnes an exponentially integrable distortion mapping on \ E, and has no ﬁnite distortion extension to C.

> > 1+αK Proof. If t 2α, then there exists K ≥ 1 such that t 2 1+K . Thus, by Corollary 3.6 we C C have a compact set E of dimension t, and a Lipα( ) function f , K-quasiregular on \ E but not on C. Of course, f is an exponentially integrable distortion mapping on C \ E, with distortion function K f essentially bounded by K. If f extended to a ﬁnite distortion C 1 C mapping on , in particular we would have J f ∈ Lloc( ). But then, since K f ≤ K at almost every point, this would imply that actually f extends K-quasiregularly.

4 Beltrami equations with coefﬁcient in the Sobolev space W1,2

4.1 Introduction

We say that a compact set E is removable for bounded µ-quasiregular mappings if every bounded function f , µ-quasiregular on C \ E, is actually constant. Of course, it also ∞ makes sense if we replace L by BMO, VMO or Lipα. In this chapter, we want to give metric and geometric characterizations of these sets.

As we know, the way µ-quasiconformal mappings distort sets is related to some removability problems. Therefore we ﬁrst need some information about µ-quasiconformal distortion. Of course, we have the precise bounds of dimension distortion from Astala [4], which apply to any K-quasiconformal mapping,

2K dim(E) dim(φ(E)) ≤ . 2 + (K − 1) dim(E)

Nevertheless, these bounds depend only on the L∞ norm of µ, and they do not reflect any regularity of the Beltrami coefficient or the homeomorphism itself φ. For instance, if the Beltrami coefficient µ belongs to VMO, then every µ-quasiconformal mapping φ p ∞ has distributional derivatives in Lloc for every p ∈ (1, ) (see for instance [8], [25] or

[23]). Thus, φ ∈ Lipα for every α ∈ (0, 1), and as a consequence,

dim(φ(E)) ≤ dim(E)

91 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2 which is clearly a more precise estimate. Moreover, actually for such µ one has dim(φ(E)) = dim(E).

For different reasons, the assumption µ ∈ VMO is not enough for our purposes. To understand this, we ﬁrst recall that for µ = 0 we have the well known Weyl’s Lemma (Theorem 1.1), which asserts that if f is any Schwarz distribution such that

h∂ f , ϕi = 0 for each test function ϕ ∈ D, then f agrees with a holomorphic function a.e. In other words, distributional solutions to Cauchy-Riemann equation are actually strong solutions. When trying to extend this result to the Beltrami equation, one ﬁrst must de- ﬁne the distribution (∂ − µ ∂) f = ∂ f − µ ∂ f . In general, it can make no sense, since bounded functions in general do not multiply Schwarz distributions nicely. However, if the multiplier is asked to exhibit some regularity, and the domain of the distribution is something bigger than simply D, then something may be done. Namely, one can write h(∂ − µ ∂) f , ϕi = −h f , ∂ϕi + h f , ∂µ ϕi + h f , µ ∂ϕi

p whenever each term makes sense. For instance, this is the case when f ∈ Lloc and 1,q µ ∈ Wloc , pq = p + q. Hence we call ∂ f − µ ∂ f the Beltrami distributional derivative of f , and we can say that a function f ∈ L p is distributionally µ-quasiregular precisely when (∂ − µ ∂) f = 0 as a distribution. Of course, a priori such functions f could be not 1,2 quasiregular, since it is not clear if the distributional equation actually implies f ∈ Wloc . Thus it is natural to ask when does this happen. We show in Theorem 4.5 that if µ is in W1,2 then distributional solutions are actually strong solutions.

Theorem. p Ω > Let f ∈ Lloc( ) for some p 2. Assume that

h(∂ − µ ∂) f , ϕi = 0

Ω 1,2 Ω for any ϕ ∈ D( ). Then, f ∈ Wloc ( ).

92 4.1 Introduction

We must point out that this selﬁmprovement of regularity is even stronger, because of the factorization theorem for µ-quasiregular mappings, as well as the regularity of normalized solutions when the Beltrami coefﬁcient is nice. Namely, when µ ∈ W 1,2 is compactly supported, it can be shown that any µ-quasiconformal mapping is actually 2,q < 2+ε in Wloc whenever q 2. Hence, every Lloc distributional solution to the corresponding 2,q < Beltrami equation is actually a Wloc solution, for every q 2. Moreover, this regularity is sharp.

One may use this selfimprovement to give removability results and study distortion problems for µ-quasiconformal mappings when µ ∈ W 1,2. The conclusions we obtain encourage us to believe that Beltrami equation with W 1,2 Beltrami coefficient is not so far from the classical planar Cauchy-Riemann equation. For instance, we show that if E is compact, then E is removable for BMO µ-quasiregular mappings if and only if H1(E) = 0. Precisely this is what happens for µ = 0 [31]. Moreover, E is removable for VMO µ-quasiregular mappings if and only if H1(E) is σ-finite, again as in the analytic case [54]. Both results may be stated in terms of µ-quasiconformal distortion. In this setting, we get that H1(E) = 0 if and only if H1(φ(E)) = 0, and H1(E) is σ-finite if and only if H1(φ(E)) is.

µ-quasiconformal distortion of analytic capacity is somewhat deeper, since the rectifiable structure of sets plays an important role there. We show in Lemma 4.14 that if µ ∈ W1,2 is compactly supported, then φ maps rectifiable sets to rectifiable sets. As a consequence, purely unrectifiable sets are mapped to purely unrectifiable sets. There- fore, we get from [15] our following main result.

Theorem. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ a µ-quasiconformal mapping. If E has σ-ﬁnite length, γ(E) = 0 ⇔ γ(φ(E)) = 0.

In particular, there are µ-quasiconformal mappings φ, not necessarily bilipschitz, that preserve σ-ﬁnite length sets of vanishing analytic capacity.

93 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

This chapter is structured as follows. In Section 2 we study the regularity of µ-quasiregular mappings. In Section 3 we deal with the distributional Beltrami equation. In Section 4, we consider the removability problems in BMO, VMO and Lipα, and deduce some distortion results for H1. In Section 5 we study µ-quasiconformal distortion of rectiﬁable sets, and the removability problem for L∞ µ-quasiregular mappings.

4.2 Regularity of µ-quasiconformal mappings

It is well known (see for instance [11]) that any K-quasiregular mapping f exhibits more 2K ∞ 1,2 K−1 , regularity than the usual Wloc appearing in its deﬁnition. More precisely, D f ∈ L [4], and this exponent is sharp. If we look not only at K but also at the regularity of the Beltrami coefﬁcient, something better may be said. This situation is given when the

Beltrami coefficient is in Lipα. In this case, every homeomorphic solution φ (and hence the corresponding µ-quasiregular mappings) has first order derivatives also in Lipα. In particular, φ is locally bilipschitz. Another situation is given when µ ∈ VMO. In this p C ∞ case, φ has derivatives in Lloc( ) for every p ∈ (1, ). Something more precise may be said if we assume Sobolev regularity for the Beltrami coefficient.

Proposition 4.1. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ a µ-quasiconformal mapping. 2,q C < Then, φ ∈ Wloc ( ) for every q 2.

Proof. We can assume that µ is compactly supported in D. We follow the ideas from Faraco and Zhong [19]. Let ψ ∈ C ∞(C), 0 ≤ ψ ≤ 1, supported on D, and deﬁne

2 µn(z) = n ψ(nw) µ(z − w) dA(w) Z ∞ Then µn is of class C , has compact support inside of 2D, kµnk∞ ≤ kµk∞ and µn → µ 1,2 in W (C) as n → ∞. As in Lemma 3.1, the corresponding principal solutions φn and φ can be written as φ(z) = z + C˜h(z) and φn(z) = z + C˜hn(z), where h, hn are respectively deﬁned by h = µBh + µ and hn = µnBhn + µn.

94 4.2 Regularity of µ-quasiconformal mappings

Now notice that µn, µ ∈ VMO. Therefore, the operators I − µnB and I − µB are in- vertibles in Lp(C) for all p ∈ (1, ∞) (see Iwaniec [25]). However, the set of bounded operators Lp(C) → Lp(C) deﬁnes a complex Banach algebra, in which the invertible operators are an open set and, moreover, the inversion is continuous. Hence, since we have lim k(I − µnB) − (I − µB)kLp→Lp = lim k(µn − µ)BkLp→Lp n→∞ n→∞

≤ kBkLp→Lp lim kµn − µkp = 0 n→∞ we get −1 −1 lim k(I − µnB) kLp→Lp = k(I − µB) kLp→Lp . n→∞ p C ∞ 1,p C As a consequence, hn → h in L ( ), for all p ∈ (1, ), so that φn → φ in Wloc ( ) for all ∞ p ∈ (1, ∞). Notice now that φn is a C diffeomorphism, conformal outside of 2D. If we take the derivative of the equation ∂φn = µn ∂φn, then we get

(∂ − µn ∂)(∂φn) = ∂µn ∂φn.

But B(∂ϕ) = ∂ϕ if ϕ is regular enough, so that

(I − µnB)(∂∂φn) = ∂µn ∂φn

q Fix any q ∈ (1, 2). Since µn is compactly supported, ∂µn ∂φn is in L (C) so that ∂∂φn = −1 (I − µnB) (∂µn ∂φn). Consequently,

−1 k∂∂φnkq ≤ k(I − µnB) kLq→Lq k∂µn ∂φnkq.

−1 −1 If n is big enough, then k(I − µnB) kLq→Lq ≤ 2 k(I − µB) kLq→Lq . For the other factor, we obtain the bound

1 1 2−q q 2 2q 2q q 2 2−q |∂µn(z) ∂φn(z)| dA(z) ≤ |∂µ(z)| dA(z) |∂φn(z)| D D D Z2 Z2 Z2

2q 2−q C Now, it comes from the convergence in Lloc ( ) of ∂φn that the sequence (∂∂φn)n is q uniformly bounded in L , for all q ∈ (1, 2). Taking a subsequence, we get that φn

95 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

2,q C < < ∆ converges in Wloc ( ), for all 1 q 2, and obviously the limit is φ. Therefore, φ = q C 2,q C c ∂∂φ ∈ L ( ) and hence φ ∈ Wloc ( ).

Philosophically, given a compactly supported Beltrami coefﬁcient µ ∈ W 1,2, we can directly derivate Beltrami equation, at least in the sense of distributions,

∂(∂φ) − µ ∂(∂φ) = ∂µ ∂φ

Now we could feel tempted to rewrite this identity as

(I − µB)(∂∂φ) = ∂µ ∂φ and then use the invertibility of the operator I − µB to obtain the desired conclusion. Unfortunately, we do not know a priori if B(∂∂φ) = ∂∂φ. Further, it is not clear if ∂∂φ is a function. Because of this, we must regularize the equation.

The optimality of the preceeding result can be seen as a consequence of the following example.

Example 4.2. The function φ(z) = z (1 − log |z|) is µ-quasiconformal in a neighbourhood of the origin, with Beltrami coefﬁcient

z 1 µ(z) = z 2 log |z| − 1

1,2 2,q In particular, we have µ ∈ W in a neighbourhood of the origin. Thus, we have φ ∈ Wloc < 2 1 2,2 whenever q 2. However, |D φ(z)| ' |z| so that φ ∈/ Wloc .

In order to study distortion results, we need to know which is the regularity of the inverse of a µ-quasiconformal mapping.

96 4.2 Regularity of µ-quasiconformal mappings

Proposition 4.3. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ a µ-quasiconformal mapping. Then, φ−1 has Beltrami coefﬁcient

∂φ ν(z) = −µ(φ−1(z)) (φ−1(z)) ∂φ

1,2 In particular, ν ∈ W , kµ˜k∞ = kµk∞ and kDµ˜k2 ≤ CK kDµk2.

Proof. An easy computation shows that

−1 ∂φ (z) ∂φ −1 ν(z) = −1 = − µ (φ (z)) ∂φ (z) ∂φ !

On the other hand, if µ ∈ W1,2 then we can write ∂φ = eλ where the function λ = −1 2 log(∂φ) is such that ∂λ = (I − µB) (∂µ). In particular, Dλ ∈ L (C) and kDλk2 ≤ 2k∂µk2 µ˜ λ µ˜ ◦ φ = −µ 2i Im(λ) 1−kµk∞ . Therefore, we can write in terms of as e . If now we take the differential, |D(µ˜ ◦ φ)| ≤ |Dµ| + 2 |µ| |D(Im(λ))|.

k ( ◦ )k ≤ kDµk2 ˙ 1,2(C) so that D µ˜ φ 2 C 1−kµk∞ . Finally, since the homogeneous Sobolev space W is invariant under quasiconformal changes of variables (see equation (2.10)), we ﬁnally obtain kDµ˜k2 ≤ CK kDµk2.

1,2 −1 2,q As a consequence, for compactly supported µ ∈ W , both φ and φ are Wloc functions, < −1 for every q 2. Therefore, φ, φ ∈ Lipα for every α ∈ (0, 1) (notice that this is true under the more general assumption µ ∈ VMO). Thus,

dim(φ(E)) = dim(E) (4.1)

If we combine this identity with Stoilow factorization, then we get that sets E of dimension dim(E) < 1 are removable for bounded (or BMO) µ-quasiregular mappings. This is an easy consequence of equation (4.1) and the quasiconformal invariance of L∞ and BMO. Moreover, if f ∈ Lipα for some α ∈ (0, 1), although in general it is false

97 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

< that f ◦ φ ∈ Lipα, we still have f ◦ φ ∈ Lip β for all β α. Hence, sets of dimension dim(E) < 1 + α are removable for α-Holder¨ continuous µ-quasiregular mappings. Re- K−1 call that for any µ with kµk∞ ≤ K+1 , the critical dimension in the removability problem 2 for bounded (and BMO) µ-quasiregular mappings is K+1 ([4], [5]), while for the Lipα 2 problem it is K+1 (1 + αK) [13]. This means that, in some removability problems, the equation ∂ f − µ ∂ f = 0 (µ ∈ W1,2 compactly supported) has the same critical dimension than the Cauchy-Riemann equation ∂ f = 0. On the other hand, we can ask ourselves if (4.1) translates to the level of Hausdorff measures. This question will be treated in Sections 4.4 and 4.5.

4.3 Distributional Beltrami equation

An important fact in the theory of quasiconformal mappings is the selﬁmprovement 2K 1, K+1 of regularity. Namely, it is known that weakly K-quasiregular mappings in Wloc are actually K-quasiregular. This improvement is highly stronger for K = 1, since in this case we do not need any Sobolev regularity as a starting point. The classical Weyl’s Lemma (Lemma 1.1) establishes that if f is a distribution such that

h∂ f , ϕi = 0 for any test function ϕ ∈ D, then f agrees at almost every point with a holomorphic function. Our following goal is to deduce an extension to this result for the Beltrami operator, provided that it has nice coefﬁcient.

Assume we are given a compactly supported Beltrami coefﬁcient µ, such that µ ∈ W 1,2. p ∞ Let f ∈ Lloc for some p ∈ (2, ). We can deﬁne a linear functional

h∂ f − µ ∂ f , ϕi = −h f , (∂ − ∂µ)ϕi = −h f , ∂ϕi + h f , ∂(µ ϕ)i for each ϕ ∈ D. Clearly, ∂ f − µ ∂ f is a distribution, which will be called the Beltrami distributional derivative of f .

98 4.3 Distributional Beltrami equation

p We say that a function f ∈ Lloc is distributionally µ-quasiregular if its Beltrami distributional derivative vanishes, that is, for every ϕ ∈ D,

h∂ f − µ ∂ f , ϕi = 0.

It turns out that one may take then a bigger class of test functions ϕ.

Lemma 4.4. > p p 1,2 Let p 2, f ∈ Lloc and q = p−1 , and let µ ∈ W be a compactly supported Beltrami coefﬁcient. If h∂ f − µ ∂ f , ϕi = 0 (4.2)

1,q for every ϕ ∈ D, then (4.2) also holds for ϕ ∈ W0 .

1,2 p > Proof. When µ ∈ W is compactly supported and f ∈ Lloc for some p 2, the Beltrami 1,q distributional derivative ∂ f − µ ∂ f acts continuously on W0 functions, since

≤ | f | |∂ϕ| + | f | |∂µ| |ϕ| + | f | |µ| |∂ϕ| Z Z Z ≤ k f kp k∂ϕkq + k f kp k∂µk2 kϕk 2q + k f kp kµk∞ k∂ϕkq 2−q

1,q Hence, if ∂ f − µ ∂ f vanishes when acting on D, it will also vanish on W0 .

Theorem 4.5. p > 1,2 Let f ∈ Lloc for some p 2. Let µ ∈ W be a compactly supported Beltrami coefﬁcient. Assume that h∂ f − µ ∂ f , ψi = 0 for each ψ ∈ D. Then, f is µ-quasiregular.

−1 2,q Proof. Let φ be any µ-quasiconformal mapping, and deﬁne g = f ◦ φ . Since φ ∈ Wloc < q ∞ p−ε > for any q 2, then J(·, φ) ∈ Lloc for every q ∈ (1, ) so that g ∈ Lloc for every ε 0.

99 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

Thus, we may deﬁne ∂g as a distribution. We have for each ϕ ∈ D

h∂g, ϕi = −hg, ∂ϕi

= − g(w) ∂ϕ(w) dA(w) Z = − f (z) ∂ϕ(φ(z)) J(z, φ) dA(z) Z = − f (z) ∂φ(z) ∂(ϕ ◦ φ)(z) − ∂φ(z) ∂(ϕ ◦ φ)(z) dA(z) Z On one hand,

− f (z) ∂φ(z) ∂(ϕ ◦ φ)(z)dA(z) = h∂ f , ∂φ · ϕ ◦ φi + f (z) ∂∂φ(z) ϕ ◦ φ(z) dA(z) Z Z = h∂ f , µ ∂φ · ϕ ◦ φi + f (z) ∂∂φ(z) ϕ ◦ φ(z) dA(z) Z = hµ∂ f , ∂φ · ϕ ◦ φi + f (z) ∂∂φ(z) ϕ ◦ φ(z) dA(z) Z and here everything makes sense. On the other hand,

− f (z) ∂φ(z) ∂(ϕ ◦ φ)(z) dA(z) = h∂ f , ∂φ · ϕ ◦ φi + f (z) ∂∂φ(z) ϕ ◦ φ(z) dA(z) Z Z Thus h∂g, ϕi = h(∂ − µ ∂) f , ϕ ◦ φ · ∂φi

1,q < p But if ϕ ∈ D then ϕ ◦ φ · ∂φ is in W0 for every q 2 and, in particular, for q = p−1 , > 1,q provided that p 2. Thus, since ∂ f − µ ∂ f acts continuously on W0 functions, the right hand side of last equality is zero. Therefore, g is holomorphic, and since f = g ◦ φ then f is µ-quasiregular.

p > From the above theorem, if f is an Lloc function for some p 2 whose Beltrami distributional derivative vanishes, then f may be written as f = h ◦ φ with holomorphic 2,q < h and µ-quasiconformal φ. As a consequence, we get f ∈ Wloc for every q 2, so we jump not only 1 but 2 degrees of regularity.

100 4.4 µ-quasiconformal distortion of Hausdorff measures

4.4 µ-quasiconformal distortion of Hausdorff measures

Let E be a compact set, and let µ be any compactly supported W 1,2 Beltrami coefﬁcient. If φ is µ-quasiconformal, then we already know that dim(φ(E)) = dim(E). However, if dim(E) = 1, we do not know how H1(φ(E)) behaves with respect to H1(E), and the same can be asked with any dimension. In this section we answer this question when dim(E) = 1, but in an indirect way. Our arguments go through some removability problems for µ-quasiregular mappings. For solving these problems, the Weyl’s Lemma for the Beltrami equation (Lemma 4.5) plays an important role.

Lemma 4.6. Let E be a compact set, and µ ∈ W1,2 a Beltrami coefﬁcient, with compact support inside of D. Suppose that f is µ-quasiregular on C \ E, and ϕ ∈ D.

(a) If f ∈ BMO(C), then

1 |h∂ f − µ ∂ f , ϕi| ≤ C (1 + kµk∞ + k∂µk2) (kϕk∞ + kDϕk∞) k f k∗ M (E)

(b) If f ∈ VMO(C), then

1 |h∂ f − µ ∂ f , ϕi| ≤ C (1 + kµk∞ + k∂µk2) (kϕk∞ + kDϕk∞) k f k∗ M∗(E)

C (c) If f ∈ Lipα( ), then

1+α |h∂ f − µ ∂ f , ϕi| ≤ C (1 + kµk∞ + k∂µk2) (kϕk∞ + kDϕk∞) k f kα M (E)

Proof. We consider the function δ = δ(t) deﬁned by

1 1 2 δ(t) = sup | f − f |2 |D| D diam(D)≤2t ZD

101 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2 when 0 < t < 1, and δ(t) = 1 if t ≥ 1. By construction, for each disk D ⊂ C we have

1 1 2 diam(D) | f − f |2 ≤ δ |D| D 2 ZD

Now consider the measure function h(t) = t δ(t). Let Dj be a covering of E by disks, such that h ∑ h(diam(Dj)) ≤ M (E) + ε j ∞ and consider a partition of unity ψj subordinated to the covering Dj. Each ψj is a C C function, compactly supported in 2Dj, |Dψj(z)| ≤ and ∑ ψj = 1 over ∪j Dj. diam(2Dj) j For every test function ϕ ∈ D,

n n −h∂ f − µ ∂ f , ϕi = ∑h f − cj, ∂(ϕ ψj)i − ∑h f − cj, ∂(µ ϕ ψj)i (4.3) j=1 j=1 where cj are any constants. Then we have

n ∑h f − cj, ∂(ϕ ψj)i ≤ ∑ | f − cj| |∂ϕ| |ψj| + |ϕ| |∂ψj| 2D j=1 j Z j 1 2 2 . ∑ | f − cj| k∂ϕk∞ diam(2Dj) + Ckϕk∞ 2Dj ! j Z . ∑ h(diam(Dj)) k∂ϕk∞ diam(2Dj) + Ckϕk∞ j

h and this sum may be bounded by M (E) + ε (kϕk∞ + kDϕk∞). The other sum in (4.3) is divided into two terms,

n ∑h f − cj, ∂(µ ϕ ψj)i ≤ ∑ | f − cj| |∂µ| |ϕ ψj | + ∑ | f − cj| |µ| |∂(ϕ ψj )|. 2D 2D j=1 j Z j j Z j

The second term may be bounded as before,

h ∑ | f − cj| |µ| |∂(ϕ ψj )| . kµk∞ M (E) + ε (kϕk∞ + kDϕk∞) 2D j Z j

102 4.4 µ-quasiconformal distortion of Hausdorff measures

For the ﬁrst term,

1 1 2 2 2 2 ∑ | f − cj| |∂µ| |ϕ ψj | ≤ ∑ | f − cj| |∂µ| kϕk∞ j Z2Dj j Z2Dj ! Z2Dj ! 1 2 1 2 ≤ ∑ δ(diam(Dj)) |2Dj | 2 |∂µ| kϕk∞ D j Z h ' k∂µk2 kϕk∞ M (E) + ε and now it just remains to distinguish in terms of the regularity of f . If f ∈ BMO(C) h 1 then the best we can say is that δ(t) . k f k∗ for all t > 0, so that M (E) ≤ M (E). Secondly, if f ∈ VMO(C) then we also have

h(t) lim = lim δ(t) = 0 t→0 t t→0

h 1 α and hence M (E) ≤ M∗(E). Finally, if f ∈ Lipα, then δ(t) . k f kαt , and therefore Mh(E) ≤ M1+α(E).

Lemma 4.6 has very interesting consequences, related to µ-quasiconformal distortion. First, we show that our µ-quasiconformal mappings preserve sets of zero length.

Corollary 4.7. Let E ⊂ C be a compact set. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ a µ-quasiconformal mapping. Then,

H1(E) = 0 ⇔ H1(φ(E)) = 0

Proof. By symmetry, it will sufﬁce to prove that H1(E) = 0 implies H1(φ(E)) = 0. Assume, thus, that H1(E) = 0. Let f ∈ BMO(C) be holomorphic on C \ φ(E). Then g = f ◦ φ belongs also to BMO(C). Moreover, g is µ-quasiregular on C \ E so that, by Lemma 4.6, h∂g − µ ∂g, ϕi = 0 whenever ϕ ∈ D. As a consequence, by Lemma 4.5, g is µ-quasiregular on the whole of C and hence f admits an entire extension. This says that the set φ(E) is removable for BMO holomorphic functions. But these sets are characterized [31] by the condition H1(φ(E)) = 0.

103 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

Another consequence is the complete solution of the removability problem for BMO µ-quasiregular mappings. Recall that a compact set E is said removable for BMO µ- quasiregular mappings if every function f ∈ BMO(C), µ-quasiregular on C \ E, admits an extension which is µ-quasiregular on C.

Corollary 4.8. Let E ⊂ C be compact. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient. Then, E is removable for BMO µ-quasiregular mappings if and only if H1(E) = 0.

Proof. Assume ﬁrst that H1(E) = 0, and let f ∈ BMO(C) be µ-quasiregular on C \ E. Then, by Lemma 4.6, we have h∂ f − µ ∂ f , ϕi = 0 for every ϕ ∈ D. Now by Lemma 4.5 we deduce that f is µ-quasiregular. Consequently, E is removable. Conversely, if H1(E) > 0, then by Corollary 4.7, H1(φ(E)) > 0, so that φ(E) is not removable for BMO analytic functions. Hence, there exists a function h belonging to BMO(C), holomorphic on C \ φ(E), non entire. But therefore h ◦ φ belongs to BMO(C), is µ-quasiregular on C \ E, and does not admit any µ-quasiregular extension on C. Consequently, E is not removable fors BMO µ-quasiregular mappings.

A second family of consequences of Lemma 4.6 comes from the study of the VMO case. First, we prove that µ-quasiconformal mappings preserve the compact sets with σ-ﬁnite length.

Corollary 4.9. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ any µ-quasiconformal mapping. For every compact set E,

H1(E) is σ-ﬁnite ⇔ H1(φ(E)) is σ-ﬁnite

1 1 Proof. Again, we only have to show that M∗(E) = 0 implies M∗(φ(E)) = 0. Assume, 1 C C thus, that M∗(E) = 0, and let f ∈ VMO( ) be analytic on \ φ(E). If we prove that f extends holomorphically on C, then φ(E) must have σ-ﬁnite length, and we will be done. To do that, we ﬁrst observe that g = f ◦ φ also belongs to VMO(C). Further, g is C 1 µ-quasiregular on \ E, and since M∗(E) = 0, by Lemma 4.6 we get that ∂g − µ ∂g = 0

104 4.4 µ-quasiconformal distortion of Hausdorff measures on D0. Consequently, from Lemma 4.5, g is µ-quasiregular on the whole of C and hence f extends holomorphically on C.

As in the BMO setting, the removability problem for VMO µ-quasiregular functions also gets solved. A compact set E is said to be removable for VMO µ-quasiregular mappings if every function f ∈ VMO(C) µ-quasiregular on C \ E admits an extension which is µ-quasiregular on C.

Corollary 4.10. Let E ⊂ C be compact. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient. Then E is removable for VMO µ-quasiregular mappings if and only if H1(E) is σ-ﬁnite.

1 1 Proof. If H (E) is σ-finite, then M∗(E) = 0, so that from Lemma 4.6 every function f ∈ VMO(C) µ-quasiregular on C \ E satisfies ∂ f = µ ∂ f on D0. By Lemma 4.5, f extends µ-quasiregularly and thus E is removable. If H1(E) is not σ-finite, we have just seen that H1(φ(E)) must not be σ-finite. Thus, there exists a function h ∈ VMO(C), analytic on C \ φ(E), non entire. But therefore h ◦ φ belongs to VMO, is µ-quasiregular on C \ E, and does not extend µ-quasiregularly on C.

The class Lipα has, in comparison with BMO or VMO, the disadvantadge of being not quasiconformally invariant. This means that we cannot read any removability result for 1+α Lipα in terms of distortion of Hausdorff measures, and therefore for H we cannot obtain results as precise as Lemmas 4.7 or 4.9. Thus, the following question remains unsolved.

Problem 4.11. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ µ-quasiconformal. Then,

H1+α(E) = 0 ⇔ H1+α(φ(E)) = 0

However, Lemma 4.5 can be used to study the Lipα removability problem. Recall that a C compact set E is removable for Lipα µ-quasiregular mappings if every function f ∈ Lipα( ), µ-quasiregular on C \ E, has a µ-quasiregular extension on C.

105 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

Corollary 4.12. 1+α Let E be compact, and assume that H (E) = 0. Then, E is removable for Lipα µ-quasiregular mappings.

Proof. As before, if f ∈ Lipα is µ-quasiregular outside of E, then Lemma 4.6 tells us that its Beltrami distributional derivative vanishes. By Lemma 4.5, we get that f is µ-quasiregular.

The above result is sharp, in the sense that if H1+α(E) > 0 then there is a compactly sup- 1,2 ported Beltrami coefﬁcient µ ∈ W such that E is not removable for Lipα µ-quasiregular mappings (take simply µ = 0). In this direction, we expect a positive solution to the following problem.

Problem 4.13. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient. Let E be a compact set, such that 1+α > C C H (E) 0. There exists f ∈ Lipα( ), µ-quasiregular on \ E, that has no µ-quasiregular extension on C.

4.5 µ-quasiconformal distortion of analytic capacity

If µ ∈ W1,2(C) is a Beltrami coefﬁcient, compactly supported on D, and E ⊂ D is compact, we say that E is removable for bounded µ-quasiregular functions, if and only if any bounded function f , µ-quasiregular on C \ E, is actually a constant function. As it is in the BMO case, just 1-dimensional sets are interesting, because of the Stoilow factorization, together with the fact that µ-quasiconformal mappings with µ ∈ W 1,2 do not distort Hausdorff dimension.

As we know from Corollary 4.7, if E is such that H1(E) = 0 then also H1(φ(E)) = 0 whenever φ is µ-quasiconformal. Thus, also γ(φ(E)) = 0. This shows that zero length sets are removable for bounded µ-quasiregular mappings.

106 4.5 µ-quasiconformal distortion of analytic capacity

Now the following step consists on understanding what happens with sets of positive and finite length. It is well known that those sets may be decomposed as the union of a rectifiable set, a purely unrectifiable set, and a set of zero length (Theorem 1.5). Hence, we may study them separately.

Lemma 4.14. C C −1 2,1+ε C > Let φ : → be a planar homeomorphism, such that φ, φ ∈ Wloc ( ) for some ε 0. Suppose that H1(E) = 0 if and only if H1(φ(E)) = 0. Then,

Γ rectiﬁable ⇔ φ(Γ) rectiﬁable

Proof. Since Γ is a rectiﬁable set, then

∞

Γ \ Z = Γi i[=1 1 where Z is a zero length set, and each Γi is a C curve with nonsingular points (i.e. with nonzero tangent vector at each point). Thus, there is no restriction in assuming that Γ is a C1 regular curve. In other words, from now on we will suppose that Γ = {α(t); t ∈ (0, 1)} for some C1 function α : (0, 1) → C such that α0(t) 6= 0 for each t ∈ (0, 1). 2,1+ε Since φ ∈ Wloc , then φ is strongly differentiable C1,1+ε-almost everywhere (see for instance [17]), and the same happens to φ−1. Thus, the set

B = {z ∈ Γ : φ is differentiable at z}

1 −1 is such that C1,1+ε(Γ \ B) = 0. In particular, H (Γ \ B) = 0. Moreover, since also φ ∈ 2,1+ε −1 Wloc , we can apply the chain rule and from φ ◦ φ(z) = z it follows that

Dφ−1(φ(z)) = (Dφ(z))−1

for C1,1+ε-almost every z ∈ B. Thus we may assume that for each z ∈ B we also have

107 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

J(z, φ) 6= 0. Notice then that H1(φ(Γ) \ φ(B)) = 0 and also that H1(φ(Γ)) is σ-ﬁnite.

Fix a point w0 = φ(z0), where z0 ∈ B, and put z0 = α(t0) for some t0 ∈ (0, 1). Then, 0 α (t0) = v 6= 0. We will show that

0 V = hDφ(z0) · α (t0)i

1 is a tangent line to φ(Γ) at w0. Since this argument works at H -almost every w0 ∈ φ(Γ), we will obtain that φ(Γ) is rectiﬁable [39, p. 214, Remark 15.22]. Thus, what we have to show is that for every number s ∈ (0, 1), there is r > 0 such that

φ(Γ) ∩ D(w0, r) ⊂ X(w0, V, s).

Here X(w0, V, s) is the cone of center w0, direction V and amplitude s, that is,

X(w0, V, s) = {w ∈ C : d(w − w0, V) < s |w − w0|}.

First, by the chain rule, the function α˜ = φ ◦ α is differentiable at t0 and Dα˜ (t0) = 0 Dφ(z0) · α (t0). On the other hand, since α is a regular curve, for each r1 > 0 there is r2 > 0 such that

|t − t0| < r1 ⇔ |α(t) − z0| < r2.

Put z = α(t). Then |t − t0| < r1 if and only if z ∈ Γ ∩ D(z0, r2). But if r1 is chosen small enough,

0 d(φ(z) − w0, V) = inf |φ(z) − φ(z0) − Dφ(z0) · α (t0) λ| λ∈R 0 ≤ |φ(z) − φ(z0) − Dφ(z0) · α (t0) (t − t0)|

|α˜ (t) − α˜ (t0) − Dα˜ (t0) (t − t0)| |t − t0| = |α˜ (t)) − α˜ (t0))| |t − t0| |α˜ (t) − α˜ (t0)| 0 1 ≤ s |α˜ (t0)| 0 |φ(z) − φ(z0)|. |α˜ (t0)|

108 4.5 µ-quasiconformal distortion of analytic capacity

Hence, for a given s > 0 there exists r0 > 0 (just take r0 = r2) such that

φ(Γ ∩ D(z0, r0)) ⊂ X(w0, V, s).

Notice also that φ is a homeomorphism, so that if r is small enough, then there we can choose r0 such that

φ(Γ \ D(z0, r0)) ⊂ C \ D(w0, r).

Therefore, given s > 0 we can ﬁnd two real numbers r, r0 > 0 for which the set

φ(Γ) ∩ D(w0, r) = φ(Γ ∩ D(z0, r0)) ∩ D(w0, r)

has all its points in the cone X(w0, V, s). In other words, given s > 0 there is r > 0 such that φ(Γ) ∩ D(w0, r) ⊂ X(w0, V, s).

In this lemma, the regularity assumption is necessary. In the following example, due to J. B. Garnett [21], we construct a homeomorphism of the plane that preserves sets of zero length and, at the same time, maps a purely unrectiﬁable set to a rectiﬁable set.

Example 4.15. 1 Denote by E the planar 4 -Cantor set. Recall that this set is obtained as a countable intersection N of a decreasing family of compact sets EN, each of which is the union of 4 squares of sidelength 1 4N , and where every father has exactly 4 identic children.

Q1 Q2

Q3 Q4

Figure 4.1

At the ﬁrst step, the unit square has 4 children Q1, Q2, Q3 and Q4, as is shown in Figure 4.1.

109 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

The corners of the squares Qj are connected with some parallel lines, as is shown in Figure 4.2.

The mapping φ1 consists on displacing along these lines the squares Q2 and Q3, while Q1 and

Figure 4.2

Q4 remain ﬁxed. This displacement must be done in such a way that de distance between the images of Q2 and Q3 is positive, since φ1 must be a homeomorphism.

Figure 4.3

However, we can do this construction with d0 as small as we wish. The result is shown in

Figure 4.3. Our ﬁnal mapping φ will be obtained as a uniform limit φ = lim φN. The other N→∞ mappings φN are nothing else but copies of φ1 acting on every one of the different squares in all generations. The only restriction is that the sum of distances d0 must be ﬁnite. It is clear that this procedure gives a sequence of homeomorphisms φN which converge uniformly to a

110 4.5 µ-quasiconformal distortion of analytic capacity homeomorphism φ. Further, it can be shown that H1(F) = 0 if and only if H1(φ(F)) = 0. On the other hand, the image of E under the mapping φ is included in a compact connected set, whose length is precisely the sum of the distances d0, which we have chosen to be ﬁnite. Therefore, φ(E) is rectiﬁable.

If µ ∈ W1,2 is a compactly supported Beltrami coefﬁcient, then we know that every 2,q C < µ-quasiconformal mapping belongs to the local Sobolev space Wloc ( ) for all q 2. Furthermore, we also know that φ preserves the sets of zero length (even σ-ﬁnite length are preserved), and the same happens to φ−1. Under these hypotheses, we can use for φ the Lemma 4.14.

Corollary 4.16. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ a µ-quasiconformal mapping.

(a) If E is a rectiﬁable set, then φ(E) is also rectiﬁable.

(a) If E is a purely unrectiﬁable set, then, φ(E) is also purely unrectiﬁable.

Proof. The ﬁrst statement comes from the above lemma. For the second, let Γ be a rectiﬁable curve. Then,

H1(φ(E) ∩ Γ) = 0 ⇔ H1(E ∩ φ−1(Γ)) = 0 but since E is purely unrectiﬁable, all rectiﬁable sets intersect E in a set of zero length. Thus, the result follows.

Theorem 4.17. Let µ ∈ W1,2 be a compactly supported Beltrami coefﬁcient, and φ a µ-quasiconformal mapping. Let E be such that H1(E) is σ-ﬁnite. Then,

γ(E) = 0 ⇐⇒ γ(φ(E)) = 0

Proof. By Corollaries 4.7 and 4.9, if H1(E) is positive and σ-ﬁnite, then H1(φ(E)) is positive and σ-ﬁnite. Hence, from Besicovitch Theorem (Theorem 1.5) we may decompose

111 4 Beltrami equations with coefﬁcient in the Sobolev space W 1,2

φ(E) as

φ(E) = Rn ∪ Nn ∪ Zn n [ with Rn rectiﬁable sets, Nn purely unrectiﬁable sets, and Zn zero length sets. Notice that

γ(Nn) = 0 because purely unrectiﬁable sets of ﬁnite length are removable for bounded 1 analytic functions [15], and also γ(Zn) = 0 since H (Zn) = 0. Thus, due to the semiadditivity of analytic capacity [51], we get

γ(φ(E)) ≤ C ∑ γ(Rn) n

−1 However, each Rn is a rectifiable set, so that φ (Rn) is also rectifiable. Now, since E has σ-finite length, the condition γ(E) = 0 forces that E cannot contain any rectifiable subset 1 −1 1 of positive length, so that H (φ (Rn)) = 0 and hence H (Rn) = 0. Consequently, γ(φ(E)) = 0.

The above theorem is an exlusively qualitative result. Therefore, we must not hope for any improvement in a quantitative sense. Namely, in the bilipschitz invariance of analytic capacity by Tolsa [52], it is shown that a planar homeomorphism φ : C → C satisﬁes γ(φ(E)) ' γ(E) for every compact set E if and only if it is a bilipschitz mapping, while Example 4.2 shows that there exist µ-quasiconformal mappings φ in the above hypotheses, which are not Lipschitz continuous. Nevertheless, we can ask ourselves the following question.

Problem 4.18. Let µ be a compactly supported Beltrami coefﬁcient of class W 1,2, and let φ be µ-quasiconformal. Then, γ(E) = 0 ⇔ γ(φ(E)) = 0 for any compact set E.

112 4.5 µ-quasiconformal distortion of analytic capacity

113

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119

Index

BMO(Ω), 18 distortion inequality, 2 VMO, 18 finite distortion mapping, 81 Wk,p(Ω), 17 exponentially integrable, 82 k,p Ω Wloc ( ), 18 subexponentially integrable, 82 Ω Lipα( ), 18 ˙ α,p C W ( ), 44 Hausdorff 0 D (Ω), 17 content, 22 D(Ω), 17 dimension, 22 lower content, 22 Beltrami measure, 22 coefficient, 26 distributional derivative, 98 Lemma equation, 26 Frostman, 22 operator, 31 Weyl, 17 bilipschitz mapping, 25 Orlicz function, 19 capacity mutually conjugated, 20 analytic, 24 Orlicz space, 19 Orlicz-Sobolev space, 82 analytic Lipα, 43 analytic BMO, 37 principal solution, 27 analytic VMO, 41 purely unrectifiable set, 23 Riesz, 44 signed Riesz, 60 quasicircle, 29

121 Index quasiconformal mapping, 26 K-quasiconformal, 25 µ-quasiconformal, 26 quasidisk, 29 quasiregular mapping, 28 K-quasiregular, 28 µ-quasiregular, 28 distributionally quasiregular, 99 weakly quasiregular, 31 rectiﬁable curve, 23 rectiﬁable set, 23

Theorem area distortion, 29 Astala-Nesi, 30 bilipschitz invariance, 25 David, 24 dimension distortion, 31 Mattila, 24 Mori, 29 semiadditivity, 25 Sobolev, 19 Stoilow, 28

122