Phd Dissertation Singular Integral Operators on Sobolev Spaces On

PhD Dissertation

Singular Integral Operators on Sobolev Spaces on Domains and Quasiconformal Mappings

Mart´ı Prats Soler

Memòriapresentada per a obtenir el grau de Doctor en Matemàtiquesper la Universitat Autònomade Barcelona

2015

Advisor: Dr. Xavier Tolsa Domenech`

Facultat de Ciencies` Departament de Matematiques`

Plus Ultra - Versi´odi`adica

Allàdellàde l’espai he vist somriure una fulla ben amunt del lledoner com grumet en veient terra, com al pregon de l’afrau una ef´ımeralluerna. – Cub de Whitney – jo li he dit–, de la mar margada gemma, de les fites del sender series tu la darrera? – No sóla darrera, no, no sómésque una llanterna de la porta del jard´ı que creies tu la frontera. Es´ sols lo comen¸cament lo que prenies per terme. Lo domini ésinfinit, pertot acaba i comen¸ca, i en¸cà,enllà,amunt i avall, la immensitat ésoberta, i a on tu veus l’hiperplà eixams de cubs formiguegen. Dels camins de l’infinit sónlos cubs la polsinera que puja i baixa a sos peus quan Peter Jones s’hi passeja.

Jacint Verdaguer (quasi)

Acknowledgements

Vull agrair en primer lloc a en Xavier Tolsa pel seu suport, des que vaig comen¸carel màsterfins ara ha estat un far en el cam´ı,donant-me la corda necessàriaper tirar endavant i mostrant-me una manera intuitiva i geomètricad’entendre l’anàlisimatemàtica.Vull fer extensiu aquest agraiment a l’Albert Clop, en Joan Mateu, en Joan Orobitg, en Joan Verdera, companys de grup de recerca i la resta del personal de la unitat d’anàlisiper aquest ambient tant agradable que han aconseguit crear. I, evidenment, als meu germans i cosins matemàtics,companys de despatx i de cafèsdurant tots aquests anys, amb especial mencióal Daniel i l’Antonio per les vides paral¤leles i en Vasilis i l’Albert pels consells i les visions de futur. Seuraavaksi, haluaisin lähettääparhaimmat kiitokset Eero Saksmanille, joka oli Helsingin vierai-luni ajan yhteistyökumppanini, opettajani ja ystäväni.Holistisella näkökulmallaanmatem- atiikkaan, hänmuistutti minua siitä,ettäparhaiten eteenpäinpääsee,kun on hieman sivussa oikealta polulta. Marraskuiseen Muumilaaksoon väriätoivat Riikka, Mikko, Timo, Matteo, Miren, Teemu ja Marco sekämahtavat Töölöntornit ja tietysti Helsingin yliopiston Matematiikan ja tilastotieteen laitos, jonka käytävänjokaisessa ovessa on tunnetun kirjan kirjoittajan nimi. My acknowledgement also to Dr. Antti Vähäkangas, who gave me the track to [Dyd06], giving rise to Lemma 2.20, to Dr. Hans Triebel, whose e-mails shed light to Corollary 2.12 and to Dr. Ritva Hurri-Syrjänenfor her advice on uniform domains. A special mention to Dr. Victor Cruz, whose work inspired most of the results of my thesis. Quiero dar las gracias tambiénal IMUS, por organizar un seminario extraordinario en Sevilla y Málaga,con estupendas partidas de ping-pong y esas cañasal lado del campo del mejor equipo de fútbol del sur de la pen´ınsula(a quien espero que esa tesis sea de ayuda), extensivo a la organización del II Congreso de Jóvenes Investigadores de Sevilla, especialmente a Mari Carmen Reguera, as´ı como a los sucesivos equipos organizadores del EARCO, en Teruel, Girona y Sevilla. Next, my regards to Doctors Christoph Thiele and Diogo Oliveira e Silva for the incredible Summer School in Kopp, the almost invisible village in google maps, as well as the organizing committee of NAFSA in the non-linear Tˇreˇst’,the Barcelona Graduate School of Mathematics and their wonderful teachers, and to the organizers of MAnET, at the risk of repeating myself. Vull fer esment tambéal professorat de la Universitat de Barcelona, on vaig obtenir la llicen- ciatura, amb especial mencióa la Pilar Bayer, en Carles Casacuberta i l’Eduardo Casas pel seu entusiasme i a l’Anton Aubanell per ensenyar-me a ensenyar; i als col¤egues d’estudis a Callús, Moià,Sant Feliu del Racói Saur´ı(Lloren¸c,Jordi-Llu´ıs,Joan, Miquel, Sara, Maria, Margarida, Oleguer, Judits, Ariadna, Gemma i tants d’altres) i a les canonades d’aquell pis durant aquest per´ıode, el mes inoblidable de la meva vida matemàtica,aix´ıcom en Grané,i el Jose Luis Diaz per la seva perseverànciai per no donar-me per perdut quan portava els cabells massa llargs. Finalment, el mésimportant del món,pares, germanes, nebodets, cosins, tiets i àviaCarme per aguantar les meves absències,de cos i d’ànima,durant tots aquests anys i, sobretot, l’Aura, que es mereix la meitat o mésdel t´ıtol. I a la Tieta, l’avi Ramon, la Iaia i el Quim per la seva guia des d’una mica mésenllàde la frontera dels dominis coneguts.

i ii

A Els Navegants, per la teràpia,l’Ateneu i la CUP per la lluita contra corrent, i, ja que ens posem nostàlgics, al guru Kaaraikkudi R. Mani, Sinera, Cantiga, el Virolet, Yarak, l’ARC, Fidelio, secciód’atletisme del Bar¸ca(existeix!), amistats i professorat de l’IES Montserrat, l’EM Reina Violant, gent de la cabra i del porc, ve¨ınatdel poble i del barri de Gràciaper ajudar-me a ser millor persona dia a dia, golpe a golpe, verso a verso. The author was funded by the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013) / ERC Grant agreement 320501. Also, partially supported by grants 2014-SGR-75 (Generalitat de Catalunya), MTM-2010-16232 and MTM-2013-44304-P (Spanish government) and by a FI-DGR grant from the Generalitat de Catalunya (2014FI-B2 00107). Contents

Acknowledgements i

Introduction 1

1 Background 9 1.1 Some examples: the Beurling transform as a model ...... 9 1.2 Notation ...... 14 1.3 Known facts ...... 17 1.4 On uniform domains ...... 19 1.5 Approximating polynomials ...... 23 1.6 Calder´on-Zygmund operators ...... 26

2 T(P) theorems 31 2.1 Classic Sobolev spaces on uniform domains ...... 32 2.2 Fractional Sobolev spaces on uniform domains ...... 36 2.3 Characterization of norms via diﬀerences ...... 48 2.4 Equivalent norms with reduction of the integration domain...... 55

3 Characteristic functions of planar domains 63 3.1 A family of convolution operators in the plane ...... 64 3.2 Besov norm and beta coeﬃcients ...... 64 3.3 The case of unbounded domains ...... 67 3.4 Beta-coeﬃcients step in ...... 70 3.5 Domains which are bounded by the graph of a polynomial ...... 71 3.6 The geometric condition ...... 76 3.7 A localization principle: bounded smooth domains ...... 82 3.8 Bounded smooth domains, supercritical case ...... 85

4 An application to quasiconformal mappings 89 4.1 Some tools ...... 90 4.2 A Fredholm theory argument ...... 91 4.3 Compactness of the commutator ...... 93 4.4 Some technical details ...... 98 4.5 Compactness of the double reﬂection ...... 105

5 Carleson measures on Lipschitz domains 117 5.1 Oriented Whitney coverings ...... 118 5.2 Carleson measures ...... 121 5.3 Integer smoothness: a suﬃcient condition ...... 123

iii iv CONTENTS

5.4 Fractional smoothness: a suﬃcient condition ...... 125 5.5 Smoothness one: a necessary condition ...... 127 5.6 On the complex plane ...... 138

Conclusions 139 Introduction

The present dissertation studies some problems of geometric function theory, which is an area with great impact in mathematical analysis, relating complex analysis, harmonic analysis, geometric measure theory and partial differential equations. In particular it focuses on the relation between Calderón-Zygmund convolution operators and Sobolev spaces on domains. The Sobolev space W s,ppRdq (or simply W s,p) of smoothness s P N and order of integrability 1 ¤ p ¤ 8 is the Banach space of Lp functions with distributional derivatives up to order s in Lp as well. This notion can be extended to 0 s 8 via the so-called Bessel-potential spaces (see P 1 p dq P dz p q Section 1.3). An operator T defined for f Lloc R and x R supp f as » T fpxq Kpx ¡ yqfpyqdy, dzt u R x is called an admissible convolution Calderón-Zygmundoperator of order s P N if it is bounded on s,pp dq 8 P s,1p dzt uq the Sobolev space W R for every 1 p and its kernel K Wloc R 0 satisfies the size and smoothness conditions C |∇jKpxq| ¤ K for every 0 ¤ j ¤ s and x 0 |x|d j (see Section 1.6 for more details). In the complex plane, for instance, the Beurling transform, defined as the principal value » 1 fpwq Bfpzq : ¡ lim dmpwq, Ñ p ¡ q2 π ε 0 |w¡z|¡ε z w is an admissible convolution Calderón-Zygmund oper- p q ¡ 1 ator of any order with kernel K z π z2 . Along this dissertation some properties of this kind of operators restricted to domains will be unravelled. Let Ω Rd be a domain (open and connected) and T an admissible convolution Calderón-Zygmund operator. We are interested in conditions that allow us to infer that the restricted operator defined as TΩpfq χΩ T pχΩ fq is bounded on a certain Sobolev space W s,ppΩq. In that spaces, the case sp d is called critical (see Figure 0.1), since the supercritical case sp ¡ d usually implies continuity of the functions Figure 0.1: Critical, supercritical and sub- involved, and the subcritical case sp d implies only critical indices and corresponding embed- 3 d d some degree of integrability, while the functions are in dings for . Here, ¡ ¦ s2. R p2 p VMO when sp d and the domain is regular enough 2 (see Proposition 1.11).

1 2 CONTENTS

Topics covered in this dissertation

Chapter 1: Background The first chapter provides the reader with the tools which are common to the whole dissertation. It begins with some examples that will help to illustrate the nature of the problems faced in the thesis. The subsequent sections summarize the notation to be used and some well-known facts, including sections devoted to uniform domains, approximating polynomials and Calderón-Zygmund operators. Chapter 2: T(P) theorems In this chapter there are some results for the supercritical case, reducing the boundedness of s,p an operator TΩ on W pΩq to its behavior on test functions, namely polynomials of degree strictly smaller than the considered smoothness. This is in accordance with the pioneering results found in [CMO13]: Theorem ([CMO13]). Let Ω Rd be a bounded C1,ε domain (i.e. a Lipschitz domain with parameterizations of the boundary in C1,ε) for a given ε ¡ 0, let 1 p 8 and 0 s ¤ 1 such that sp ¡ d and let T be an admissible Calderón-Zygmundoperator of order 1 with kernel p q ωpxq P 1p d¡1q K x |x|d where ω C S is homogeneous of degree 0 with zero integral in the unit sphere d¡1 s,p S and even. Then the truncated operator TΩ is bounded on the Sobolev space W pΩq if and s,p only if T pχΩq P W pΩq. This theorem is often called T p1q-theorem because the only condition to be checked in order to s,p s,p show that an operator is bounded on W pΩq is that TΩp1q P W pΩq. The reader will find two main results in that spirit in this chapter. First he or she will find a T pP q-theorem, which deals with W n,ppΩq with n P N and p ¡ d: Theorem. Let Ω Rd be a bounded uniform domain, T an admissible convolution Calderón- ¡ Zygmund operator of order n P N, d p 8 and let Pn 1 stand for the polynomials of degree smaller than n. Then } } 8 P n¡1 ðñ n,pp q TΩP W n,ppΩq for every P P TΩ is bounded on W Ω . This theorem improves the previously known results in the sense that the class of operators considered is wider, the smoothness can be greater than 1 and, moreover, the restrictions on the regularity of the domain are reduced to just asking the domain to be uniform. However, the case 0 s 1 is not covered. The second result of the chapter, Theorem 2.8, solves this gap: Theorem. Let Ω Rd be a bounded uniform domain, T an admissible convolution Calderón- 8 ¡ d Zygmund operator of order 1, 1 p and 0 s 1 with s p . Then } } 8 ðñ s,pp q TΩ1 W s,ppΩq TΩ is bounded on W Ω . Moreover, in Theorem 2.8 the assumptions on the regularity of the kernel are relaxed and it s will be extended to the class of Triebel-Lizorkin spaces Fp,q (see Chapter 2). The novelty in the approach exposed in this chapter is not only on the aforementioned improvements, but also on the techniques used to reach that results, which rely strongly on the properties of uniform domains and their hyperbolic metric. Furthermore, in both cases a Key Lemma is obtained that provides information even if the condition sp ¡ d is not satisfied. To prove the second result, some lemmas are used which are proven in Sections 2.3 and 2.4, results which are intuitive according to the literature, but which cannot be found in the precise shape needed. This includes the definition of an equivalent norm in terms of differences for fractional Triebel-Lizorkin spaces in the spirit of [Ste61] (see Corollary 2.12), an extension theorem for Triebel-Lizorkin spaces on uniform domains following the steps of the celebrated paper CONTENTS 3

[Jon81] by Peter Jones (see Lemma 2.16), and s p q some equivalent norms for Fp,q Ω introduced by Eero Saksman and the author of the present dissertation (see Corollary 2.22). Note that we are far from covering all the supercritical cases (see Figure 0.2). This fact is discussed at the end of the dissertation. The T pP q-theorem reminds the results by Rodolfo H. Torres in [Tor91], where the characterization of some generalized Calderón- Zygmund operators which are bounded on the homogeneous Triebel-Lizorkin spaces in Rd is given in terms of their behavior over polynomials. It is also in place to remark that in [Väh09]Antti V. Vähäkangas obtained some T p1q-theorem for weakly singular integral operators on domains. Roughly speaking, he showed the image of the Figure 0.2: Indices for which the T pP q-theorem characteristic function being in a certain BMO- is valid in W σ,ppΩq for uniform domains in 3. type space to be equivalent to the boundedness R p 9 m,p of TΩ : L pΩq Ñ W pΩq where m is the degree of the singularity of T’s kernel.

Chapter 3: Characteristic functions of planar domains } } According to the results above, estimating TΩ1 W n,ppΩq is crucial to determine if TΩ is bounded on W n,ppΩq or not. This chapter deals with the question of what conditions on the boundary of the domain imply that this norm is finite. The techniques used rely heavily on complex analysis, so all the results of this chapter are for planar domains. The first results pointing in this direction were obtained in [CMO13]. Using a result in [MOV09] 1,ε s,p it is proven that if ε ¡ s and Ω is a C domain then BχΩ P W pΩq. Thus, assuming the conditions in the T p1q-theorem for Ω, s and p to hold, one always has the Beurling transform bounded on W s,ppΩq. Following the thread, Victor Cruz and Xavier Tolsa in [CT12] showed that for 0 s ¤ 1 and 1 p 8 with sp ¡ 1, if the parameterizations of the domain are Lipschitz and the outward s¡1{p s,p unit normal vector N is in the Besov space Bp,p pBΩq, then BχΩ P W pΩq. The Besov spaces of functions are properly defined in 1.2 and 1.3; momentarily the reader unfamiliar with this σ R concept may stick to the right track by noting the fact that Bp,p with σ N can be defined by means of real interpolation between the Sobolev spaces of integer order W tσu,p and W rσs,p. Their appearance in this context is rather natural since the traces of functions in W s,ppΩq are precisely in s¡1{p s¡1{p Bp,p pBΩq if certain regularity conditions are satisfied. The condition N P Bp,p pBΩq implies the s 1¡1{p parameterizations of the boundary of Ω to be in Bp,p and, for sp ¡ 2, the parameterizations are in C1,s¡2{p as well by the Sobolev Embedding Theorem. In that situation, the T p1q result in [CMO13] implies the boundedness of the Beurling transform in W s,ppΩq (see Figure 0.3). Moreover, for s 1, this condition is necessary for Lipschitz domains with small Lipschitz constant such that 1,p BχΩ P W pΩq as shown in [Tol13]. This chapter deals with the case of Sobolev spaces with smoothness n P N. It is shown that ¡ P n¡1{ppB q n¡1,1p q P for p 1, if N Bp,p Ω and the domain has parameterizations in C R , then BχΩ W n,ppΩq, in the same spirit of [CT12]. The T pP q-theorem above will be used to deduce that the Beurling transform is bounded on W n,ppΩq when, in addition, p ¡ 2. Note that in this case again n 1¡ 1 p n¡1,1 Bp,p pRq C pRq. 4 CONTENTS

n¡1{p Theorem. Let p ¡ 2, let n P N and let Ω be a bounded Lipschitz domain with N P Bp,p pBΩq. If f P W n,ppΩq, then

}BpχΩfq} n,pp q ¤ C}N} n¡1{p }f} n,pp q, W Ω Bp,p pBΩq W Ω where C depends on p, n, diampΩq and the Lipschitz character of the domain. The proof will be slightly more tricky than the case of smoothness smaller or equal to 1, since the boundary of the domain must be approximated by polynomials instead of straight lines: the derivative of the Beurling transform of the characteristic function of a half-plane is zero (see [CT12]), but the derivative of the Beurling transform of the characteristic function of a domain bounded by a polynomial of degree greater than one is not zero anymore. Moreover, some extra eﬀort will be done in order to get a quantitative result to be used in the subsequent chapter, involving not only the truncated Beurling transform, but its iterates and other related operators.

s¡1{p n,p Figure 0.3: The normal vector being in Bp,p pBΩq implies that BΩ1 P W pΩq (green and red regions) and, when p ¡ 2 and sp ¡ 2 (green region), the T pP q-theorem applies. This is precisely ¡ 1 2 s 1 p s,1¡ the case where the parameterizations of the boundary belong to Bp,p C p . In the ﬁgure P 1 it is shown the case s n N, but p s 1 follows the same pattern.

Chapter 4: An application to quasiconformal mappings P 8 } } 1 k Let µ L be compactly supported in C with k : µ L8 1 and consider K : 1¡k . A function f is a K-quasiregular solution to the Beltrami equation

B¯f µ Bf P 1,2 with Beltrami coeﬃcient µ if f Wloc , that is, if f and ∇f are square integrable functions in any compact subset of C, and B¯fpzq µpzqBfpzq for almost every z P C. Such a function f is said to be a K-quasiconformal ¨ mapping if it is a homeomorphism of the complex plane. If, moreover, p q 1 Ñ 8 f z z O z as z , then f is the principal solution to the Beltrami equation. Given a compactly supported Beltrami coeﬃcient µ, the existence and uniqueness of the principal solution is granted by the measurable Riemann mapping Theorem (see [AIM09, Theorem 5.1.2], for instance). The principal solution can be given by means of the Beurling transform. If

h : pI ¡ µBq¡1µ, then the principal solution of the Beltrami equation satisfies that B¯f h and Bf Bh 1. A natural question is to what spaces h belongs. The key point in finding out answers to that question is inverting the operator pI ¡ µBq in some space. Astala showed in [Ast94] that h P Lp CONTENTS 5 for 1 k p 1 1{k (in fact, since h is also compactly supported, one can say the same for every p ¤ 1 k even though pI ¡ µBq may not be invertible in Lp for that values of p, as shown by Astala, Iwaniec and Saksman in [AIS01]). Clop et al. in [CFM 09] and Cruz, Mateu and Orobitg in [CMO13] proved that if µ belongs to the space W s,ppCq with sp ¡ 2 then also h P W s,ppCq. One also finds some results in the same spirit for the critical case sp 2 and the undercritical case sp 2 in [CFM 09] and [CFR10], but here the space to which h belongs is slightly worse than the space to which µ belongs, that is, either some integrability or some smoothness is lost. When it comes to dealing with a Lipschitz domain Ω with supppµq Ω, Mateu, Orobitg and Verdera showed in [MOV09] that, if the parameterizations of the boundary of Ω are in C1,ε with 0 ε 1, then for every 0 s ε one has that

µ P C0,εpΩq ùñ h P C0,spΩq. (0.1)

Furthermore, the principal solution to the Beltrami equation is bilipschitz in that case. The authors allow the domain to have a ﬁnite number of holes with tangent boundaries. In [CF12], Giovanna Citti and Fausto Ferrari proved that, if one does not allow this degenerate situation, then (0.1) holds for s ε. In [CMO13] the authors study also the Sobolev spaces to conclude that for the same kind of domains (i.e., with boundary in C1,ε), when 0 s ε 1 and 1 p 8 with sp ¡ 2 one has that µ P W s,ppΩq ùñ h P W s,ppΩq. (0.2) A key point is proving the boundedness of the Beurling transform in W s,ppΩq, something that they do in the same paper as mentioned in the presentation of the previous chapter. The other key point is the invertibility of I ¡ µB in W s,ppΩq, which is shown using Fredholm theory. Back to the ideas of the presentation of Chapter 3 above, a domain such that its outward unit ¡ { P s0 1 p0 pB q ¡ normal vector N Bp0,p0 Ω with 0 s0 1 and s0p0 1 satisﬁes that the parameterizations s0 1¡1{p0 of its boundary are in Bp0,p0 . The aforementioned result by Cruz and Tolsa in [CT12] implies s0,p0 1,s0¡2{p0 that BχΩ P W pΩq. In addition, when s0p0 ¡ 2, the parameterizations are also in C by the Sobolev Embedding Theorem. This fact, combined with the T p1q-theorem in [CMO13], implies the boundedness of the Beurling transform in W s0,p0 pΩq. However, (0.2) only allows to infer that for every 2{p s s0 ¡ 2{p0 we have that (0.2) holds (see Figure 0.4). Apparently, there is room to improve.

¡ { s0 1 p0 pB q Figure 0.4: Combining the results of [CT12] and [CMO13], if the normal vector is in Bp0,p0 Ω then (0.2) holds for the indices s, p in the yellow region of the second graphic (where 2{p s s0,p0 s0 ¡ 2{p0), although the Beurling transform is bounded on W pΩq. Note that if s0p0 4 then p ¡ p0, that is, not only smoothness, but also some integrability range is lost. 6 CONTENTS

¡ { P s0 1 p0 pB q The natural guess is that if N Bp0,p0 Ω then (0.2) holds for s s0. In this chapter we prove that indeed this is the case for natural values of s0. Theorem. Let n P N, let Ω be a bounded Lipschitz domain with outward unit normal vector N in n¡1{ppB q 8 P n,pp q } } p q Bp,p Ω for some 2 p and let µ W Ω with µ L8 1 and supp µ Ω. Then, the principal solution f to the Beltrami equation is in the Sobolev space W n 1,ppΩq.

Note that this theorem only deals with the natural values of s0, but the restriction s s0 ¡2{p0 is eliminated. For n 1 the author expects this to be a sharp result in view of [Tol13].

Figure 0.5: The gap presented in the previous diagram is not there anymore, so neither smoothness nor integrability is lost. The argument presented is only valid for natural values for the smoothness index by now.

Chapter 5: Carleson measures on Lipschitz domains The T pP q-theorems provide useful tools to check if an operator is bounded on W n,ppΩq as long as p ¡ d. This chapter presents a completely new approach to find a sufficient Carleson condition valid even if p ¤ d in the spirit of the celebrated article [ARS02] by N. Arcozzi, R. Rochberg and E. Sawyer. This condition uses polynomials of degree smaller than n as test functions again. In Section 5.2 the vertical shadows Shvpxq and Shvpxq are defined for every point x in a Lipschitz domain Ω close enough to BΩ. These shadows can be understood as Carleson boxes of the domain (see Figure 1.2). A positive and finite Borel measure µ is an s, p-Carleson measure if for every a P Ω and close enough to the boundary, » pd¡spqp1¡p1q p1 dx distpx, BΩq pµpShvpxq X Shvpaqqq ¤ CµpShvpaqq. p B qd Shvpaq dist x, Ω N. Arcozzi, R. Rochberg and E. Sawyer proved in [ARS02] that in the case when Ω coincides with the unit disk D C, the measure µ is 1, p-Carleson if and only if the trace inequality » » |f|p dµ ¤ C|fp0q|p C |f 1|p dm D D holds for any holomorphic function f on D. It turns out that the notion of 1, p-Carleson measure is also essential for the characterization of the boundedness of Calderón-Zygmund operators of order n in W n,ppΩq when 1 p ¤ d as the next theorem shows. Theorem. Let T be an admissible convolution Calderón-Zygmundoperator of order n, and con- n p sider a bounded Lipschitz domain Ω and 1 p ¤ d. If the measure |∇ TΩP pxq| dx is a 1, p- Carleson measure for every polynomial P of degree at most n ¡ 1, then TΩ is a bounded operator on W n,ppΩq. CONTENTS 7

d ¡ d In connection with Chapter 2, an equivalent formulation for the fractional case p 2 s 1 is presented. In that case, the gradient notation is explained in Definition 2.9. 8 ¡ d ¡ d Theorem. Let 1 p and 0 s 1 with s p 2 , let T be an admissible convolution Calderón-Zygmundoperator of order 1, and consider a bounded Lipschitz domain Ω. If the measure p q | s p q|p s,pp q µ x ∇2TΩ1 x dx is an s, p-Carleson measure, then TΩ is a bounded operator on W Ω . The Carleson condition of this theorem is in fact necessary for n 1 (see Figure 0.6): Theorem. Let T be an admissible convolution Calderón-Zygmundoperator of order 1, and consider a Lipschitz domain Ω and 1 p 8. Then

1,p p TΩ is a bounded operator on W pΩq ðñ |∇T χΩpxq| dx is a p-Carleson measure for Ω.

s,p Figure 0.6: Indices for which the Carleson condition is suﬃcient for TΩ to be bounded on W pΩq for Lipschitz domains in R3. In blue the case s 1, where the Carleson condition is necessary and suﬃcient.

Final remarks The results presented in this dissertation are fruit of the PhD studies of the author started in February 2012 under the direction of Xavier Tolsa. The contents of Chapter 2 are a combination of a joint article with Xavier Tolsa, adapted in the present text to the framework of uniform domains as suggested by the referee during the publication process (see [PT15]), and a joint work with Eero Saksman (see [PS15]). Chapter 3 contains the material of [Pra15b] together with some original proofs (Section 3.6) which are not included in this preprint for the sake of brevity. The results of Chapter 4 can be found in the preprint [Pra15a]. Finally, Chapter 5 contains also material from [PT15], although Section 5.4 cannot be found in any other paper by now. The chronological order should be Chapter 3, Section 2.1, Chapter 5, Chapter 4 and the remaining sections of Chapter 2.

Chapter 1

Background

We introduce the principles that the other chapters have in common. First of all, in Section 1.1, we provide some basic examples to help the reader to understand the nature of the problems we are facing. Next we set up the notation of the dissertation in Section 1.2 and then we list some well-known facts in Section 1.3. In Section 1.4 we introduce uniform domains. They will appear only in Chapter 2 and, after that chapter, we will restrict our study to Lipschitz domains, which are also uniform. However, we introduce here concepts as “admissible chain” or “shadow”, which will be extremely useful in the subsequent chapters. The reader should be aware that, in the last chapter, we will make some modifications on these concepts to introduce the Carleson measures. Section 1.5 is about Meyers’ approximating polynomials on cubes, which are used in all the chapters to discretize Sobolev functions on domains. Finally, Section 1.6 is devoted to defining admissible CalderónZygmund operators and proving that, up to a certain order, the weak derivatives of these operators restricted to a domain make sense in our setting.

1.1 Some examples: the Beurling transform as a model P 8p q Given a compactly supported smooth function φ Cc C , we define its Cauchy transform as » 1 φpwq Cφpzq : dmpwq for all z P , π z ¡ w C C where m stands for the two-dimensional Lebesgue measure. In other words, the Cauchy transform 1 of φ is the convolution of φ with the kernel z . The Beurling transform of φ will be defined as the 1 convolution with z2 , but this kernel is no longer locally integrable. Thus, we must define it via the so-called Cauchy Principal Value, that is, » » p q p q p q ¡ 1 φ z p q ¡ 1 φ z p q P Bφ z : p.v. 2 dm w lim 2 dm w for all z C. (1.1) π pz ¡ wq π εÑ0 | ¡ |¡ pz ¡ wq C w z ε P 8p q B p q 1 pB ¡ B qp q B¯ p q 1 pB B qp q Given any φ C C , then we write φ z 2 xφ i yφ z and φ z 2 xφ i yφ z . P 8p q It is well known that for every φ Cc C we have that B¯Cφpzq φpzq and BCφpzq Bφpzq (1.2)

(see, for example, [AIM09, Chapter 4]). 8 p q P 2 ¥ P p q α Bα1 B¯α2 For α α1, α2 Z with α1, α2 0 and φ Cc C , we write D φ φ. For any open 1 set U C and every distribution f P D pUq the distributional derivative Dαf is the distribution

9 10 CHAPTER 1. BACKGROUND deﬁned by x α y p¡ q|α|x α y P 8p q D f, φ : 1 f, D φ for every φ Cc U . Given numbers n P N, 1 ¤ p ¤ 8 an open set U C and a locally integrable function f deﬁned n,p in U, we say that f is in the Sobolev space W pUq of smoothness n and order of integrability° p α P p | | ¤ | n | | α | if f has distributional derivatives D f L for every α n. We write ∇ f |α|n D f . Along this section we will use the norm } } } } } n } f W n,ppUq f LppUq ∇ f LppUq.

The Beurling transform is an isometry in L2pCq and it is bounded on LppCq for 1 p 8 (see, [AIM09, Chapter 4]), that is, for every f P LppCq we have that

}Bf} pp q ¤ Cp}f} pp q. L C L C Furthermore, the Beurling transform commutes with derivatives, that is B ¥ B B ¥ B, and B ¥ B¯ B¯ ¥ B, so B is bounded on the Sobolev space W n,ppCq as long as 1 p 8 and n P N. However, this does not apply to W n,ppUq for general open sets U. Given a domain (open and connected) Ω, we write BΩf χΩBpχΩfq. Among other results, in the present dissertation we will see that if a domain Ω is regular enough then BΩ is bounded on W n,ppΩq. Let us see some examples. From (1.2) we can deduce the following property for the Beurling transform: for 1 p 8 and f P W 1,ppCq we have that BpB¯fq BCpB¯fq BB¯Cpfq Bf. (1.3)

Using this result, we can undertake the study of the behavior of the Beurling transform of a polynomial restricted to the unit disk.

Example 1.1. Consider the unit disk D tz P C : |z| 1u. Then, for every polynomial P of ¡ p q degree n 1, its transform BDP χDB χDP agrees with a polynomial of degree smaller or equal ¡ n p q P than n 1 in D so ∇ BDP z 0 for z D. Thus, the suﬃcient conditions of Theorems 2.1 and 5.1 are satisﬁed. Those theorems can be n,pp q P ¡ used to prove the boundedness of BD in W D for any n N and p 1 in one stroke.

Proof. We will follow the ideas of [AIM09, page 96]. Consider a given any multiindex λ pλ1, λ2q, λ λ1 λ2 and let Pλpzq z z z¯ . λ 1 ¡λ ¡1 p q 2 p q 2 c p q If λ1 0, then we deﬁne fλ z z¯ χD z z χD z . This function is continuous and is in some Sobolev space W 1,ppCq. Thus, by (1.3) we have that

p q p qp q pB¯ qp q B p q ¡p q 1 p q λ2 1 B χ Pλ z B fλ z fλ z λ2 1 χ c z , D zλ2 2 D so BDPλ 0. λ p0,1q λ ¡λ ¡1 p q p q 1 2 c p q Analogously, if 0 λ1 λ2 1 we deﬁne fλ z z χD z z χD z , which is 1,pp q ¡ ¡ also continuous and is in some Sobolev space W C because λ1 λ2 1 0. Again by (1.3) we have that

p q p qp q pB¯ qp q B p q λ p¡1,1q p q ¡ p ¡ q 1 p q λ2 1 B χ Pλ z B fλ z fλ z λ1z χ z λ2 λ1 1 ¡ χ c z , D D zλ2 λ1 2 D so λ1 BDPλ Pλ p¡1,1qχD. λ2 1 1.1. SOME EXAMPLES: THE BEURLING TRANSFORM AS A MODEL 11

p q p λ p0,1q ¡ q p q In case, λ1 λ2 1 we take fλ z z 1 χD z . We have that p q p qp q pB¯ qp q B p q λ p¡1,1q p q λ2 1 B χDPλ z B fλ z fλ z λ1z χD z , so again λ1 BDPλ Pλ p¡1,1qχD Pλ p¡1,1qχD. λ2 1

λ p0,1q λ1¡λ2¡1 Finally, when λ1 ¡ λ2 1 we choose fλpzq pz ¡ z qχ pzq and we get that ¡ D © p¡ q ¡ ¡ p q p qp q pB¯ qp q B p q λ 1,1 ¡ p ¡ ¡ q λ1 λ2 2 p q λ2 1 B χDPλ z B fλ z fλ z λ1z λ1 λ2 1 z χD z , so we have that ¢ λ1 λ1 ¡ λ2 ¡ 1 p¡ q ¡ p ¡ ¡ q BDPλ Pλ 1,1 P λ1 λ2 2,0 χD. λ2 1 λ2 1

Example 1.2. The Beurling transform of the characteristic function of a square Q (see Figure 1.1) is not in W n,ppQq neither for any n 1 and p ¥ 2 nor for n ¥ 2 and p ¡ 1. Nevertheless, if 1,p 1 ¤ p 2, then BχQ P W pQq.

Figure 1.1: Plot of Re pχQBχQpx iyqq. There is a logarithmic singularity at every vertex of Q.

t P | p q| | p q| u t ¡ ¡ u Proof. Consider the square Q0 z C : Re z Im z 1 which has vertices 1, i, 1, i . In that case, by [AIM09, (4.122)] one can see that 1 pz ¡ 1qpz 1q Bpχ qpzq log , Q0 π pz iqpz ¡ iq where log¡ stands© for the well-deﬁned branch of the logarithm with argument in p0, 2πq (if z P Q0 z2¡1 then Re z2 1 0). Since 1 4z BBpχ qpzq , (1.4) Q0 π z4 ¡ 1 12 CHAPTER 1. BACKGROUND

|B p qp q| 1 P p q R we have that B χQ0 z |z¡1| when z Q0 is close enough to 1 and it follows that B χQ0 1,pp q ¥ p q p q P 1,pp q ¤ W Q0 for p 2 and, since B χQ0 is analytic, B χQ0 W Q0 for 1 p 2. By the ¥ |Bn p qp q| | ¡ |¡n p q R n,pp q same token, for n 2 one has B χQ0 z z 1 and therefore B χQ0 W Q0 for any p ¡ 1. n,pp q Example 1.3. The Beurling transform restricted to the square BQ0 is not bounded on W Q0 neither for n 1 and p ¥ 2 nor for n ¥ 2 and p ¡ 1. However, it is bounded on W 1,ppQq for every 1 p 2. n,pp q ¥ ¡ Proof. Of course (1.2) implies that BQ0 is not bounded on W Q0 when n 2 and p 1 or when n 1 and p ¥ 2. p q P 1,pp q Nevertheless, when n 1 and 1 p 2, we have seen that B χQ0 W Q0 . This condition 1,p does not suﬃce to grant the boundedness of B in W pQ0q for p 2. By Theorem 5.1 we need to check that the measure p q | p q|p µ z ∇BχQ0 z is a p-Carleson measure, as we sketch below (see Deﬁnition 5.16 for the details). Of course since the measure is bounded away from the four vertices,¡ © by symmetry, it is enough to check this condition p | ¡ | 1 p q 1 t P p q ¡ | p q|u for z 1 2 or, equivalently, for µ z |z| in Ω z C : Im z Re z .

Figure 1.2: The vertical shadow ShvpP¡1,2,5q.

Consider the cubes ( P p| | ¡ q i p q | | i p| | q i p q p| | q i Pi,j,k z C : j 1 2 Re z j 2 and j k 2 Im z j k 1 2 (1.5) for i P Z, j P Zzt0u, and k P t4, 5, 6, 7, 8u if j is even, k P t4, 5, 6, 7u if j is odd (see Figure 1.2). This collection of cubes is a Whitney covering of Ω, that is, a collection of disjoint cubes with side-length proportional to their distance to BΩ and such that the union of their closures is Ω. For a cube Pi,j,k in this collection, we deﬁne its vertical shadow as the region of Ω situated beneath it, that is, ( p q P p| | ¡ q i p q | | i p q p| | q i Shv Pi,j,k z C : j 1 2 Re z j 2 and Im z j k 1 2 (see Figure 1.2). 1.1. SOME EXAMPLES: THE BEURLING TRANSFORM AS A MODEL 13

Then we say that the measure is p-Carleson if ¸ 1 p¡2 p ¡ µpShvpQqq `pQq p 1 ¤ CµpShvpP qq,

QShvpP q 1 1 where p p1 1. For every Whitney cube P Pi0,j0,k0 , since µ is almost constant in cubes we have that » » 1 `pP q2 µpP q dµ dmpzq . | | p qp P P z dist P, 0

Note that the Lebesgue measure of the shadow mpShvpP qq is comparable to the Lebesgue measure of the cube itself. Assume that |j0| ¡ 1, that is, assume that P is far from the imaginary axis. Then, all the cubes contained in its shadow are essentially at the same distance of the origin, so ¸ `pQq2 mpSh pP qq `pP q2 µpSh pP qq v . v distpQ, 0qp distpP, 0qp distpP, 0qp QShvpP q

If, instead, P intersects the imaginary axis (equivalently, if |j0| 1), then we classify the cubes in its shadow by their size:

¸i0 ¸ `pQq2 µpSh pP qq . v p qp ¡8 dist Q, 0 i QShvpP q `pQq2i

p q | | p q distpQ,0q For Q Pi,j,k let us deﬁne the index j Q : j k. Then j Q `pQq by (1.5), and

¸i0 ¸ 2ip2¡pq ¸i0 ¸ 1 µpSh pP qq 2p2¡pqi . v p qp p qp ¡8 j Q ¡8 j Q i QShvpP q i QShvpP q `pQq2i `pQq2i

Observe that for a ﬁxed i and n P N, the number of cubes Q with `pQq 2i and jpQq n is bounded by 3. Thus, since p 2, we get

¸i0 ¸8 p ¡ q 1 p ¡ q ¡ µpSh pP qq 2 2 p i 2 2 p i0 `pP q2 p. v np i¡8 n1 Summing up, every Whitney cube P satisﬁes that `pP q2 `pP q2¡p µpP q µpSh pP qq . v distpP, 0qp jpP qp p¡2 ¡ 1 p q ¡ Thus, since p¡1 2 p and j P 1 for all cubes P , we have that

¸ ¸ p ¡ q 1 ¸ p ¡ q 1 ¡ 1 1 ¡ p q 2 p p 1 p q 2 p p 2 p p p 2 ` Q 2¡p ` Q µpSh pQqq `pQq p¡1 `pQq v jpQqpp1 jpQqp QShvpP q QShvpP q QShvpP q so, using that p2 ¡ pqp1 2 ¡ p1 2 ¡ p, we have that ¸ ¸ ¡ 1 ¡ p q2 p p p 2 ` Q µpSh pQqq `pQq p¡1 À µpSh pP qq. v jpQqp v QShvpP q QShvpP q 14 CHAPTER 1. BACKGROUND

1.2 Notation

On inequalities: When comparing two quantities x1 and x2 that depend on some parameters p1, . . . , pj we will write x ¤ C x 1 pi1 ,...,pij 2 if the constant C depends on p , . . . , p . We will also write x À x for short, or pi1 ,...,pij i1 ij 1 pi1 ,...,pij 2 simply x1 À x2 if the dependence is clear from the context or if the constants are universal. We may omit some of these variables for the sake of simplicity. The notation x x will 1 pi1 ,...,pij 2 mean that x À x and x À x . 1 pi1 ,...,pij 2 2 pi1 ,...,pij 1 On polynomials: We write PnpRdq for the vector space of real polynomials of degree smaller or equal than n with d real variables. If it is clear from the context we will just write Pn. On sets: Given two sets A and B, their symmetric difference is A∆B : pA Y BqzpA X Bq and their long distance is DpA, Bq : diampAq diampBq distpA, Bq. (1.6) P d ¡ p q p q Given x R and r 0, we write B x, r or Br x for the open ball centered at x with radius r and Qpx, rq for the open cube centered at x with sides parallel to the axis and side-length 2r. Given any cube Q, we write `pQq for its side-length, and rQ will stand for the cube with the same center but enlarged by a factor r. We will use the same notation for balls and one dimensional cubes, that is, intervals. At some point we need to use segments in Rd: given x, y P Rd, we call the segment with endpoints x and y rx, ys : tp1 ¡ tqx ty : t P r0, 1su. We may use the “open” segment sx, yr: rx, ysztx, yu. On finite diferences: Given a function f :Ω Rd Ñ C and two values x, h P Rd such that rx, x hs Ω, we call 1 p q p q p q ¡ p q ∆hf x ∆hf x f x h f x . Moreover, for any natural number i ¥ 2 we define the iterated difference ¢ ¸i i ∆i fpxq ∆i¡1fpx hq ¡ ∆i¡1fpxq p¡1qi¡j fpx jhq h h h j j0 whenever the segment rx, x ihs Ω. On domains: We call domain an open and connected subset of Rd. ¡ Definition 1.4. Given n ¥ 1, we say that Ω C is a pδ, Rq ¡ Cn 1,1 domain if given any z PBΩ, P n¡1,1p q r¡ s there exists a function Az C R supported in 4R, 4R such that pjq ¤ δ ¤ ¤ Az ¡ for every 0 j n, L8 Rj 1 and, possibly after a translation that sends z to the origin and a rotation that brings the tangent at z to the real line, we have that

Ω X Qp0,Rq tx i y : y ¡ Azpxqu. In case n 1 the assumption of the tangent is removed (we say that Ω is a pδ, Rq-Lipschitz domain). This concept can be extended naturally to every Rd with dimension d ¡ 2. In the present dissertation, we will ﬁnd only Lipschitz domains in Rd with d ¡ 2 in Chapter 5. We call window such a cube. If n 1 and Ω tx i y : y ¡ A0pxqu, we say that Ω is a δ-special Lipschitz domain. 1.2. NOTATION 15

On Whitney coverings: Next we present a construction that will be used in all the chapters. Deﬁnition 1.5. Given a domain Ω, we say that a collection of open dyadic cubes W is a Whitney covering of Ω if they are disjoint, the union of the cubes and their boundaries is Ω, there exists a constant CW such that CW `pQq ¤ distpQ, BΩq ¤ 4CW `pQq, two neighbor cubes Q and R (i.e., Q X R H) satisfy `pQq ¤ 2`pRq, and the family t50QuQPW has ﬁnite superposition. Moreover, we will assume that 1 S 5Q ùñ `pSq ¥ `pQq. (1.7) 2

The existence of such a covering is granted for any open set diﬀerent from Rd and in particular for any domain as long as CW is big enough (see [Ste70, Chapter 1] for instance). d On measure theory: We denote the d-dimensional Lebesgue measure in R by md, or simply m when the dimension is clear from the context. At some point we use m also to denote a natural number. When dealing with line integrals in the complex plane, we will write dz for the form dx i dy and analogously dz¯ dx ¡ i dy, where z x i y. Thus, when integrating a function with respect to the Lebesgue measure of a complex variable z we will always use dmpzq to avoid confusion, or simply dm. For any measurable set A and any measurable function f, we call fA A f dm to the mean of f in A. ﬄ

On indices: In this text N0 stands for the natural numbers including 0. Otherwise we will write N. We will make wide° use of the multiindex notation for± exponents and derivatives. For P d | | d | | d α Z its modulus is α i1 αi and its factorial is α! i1 αi!. Given two multiindices P d ¤ ¤ α, γ Z we write α γ if αi γi for every i. We say α γ if, in addition, α γ. Furthermore, we write #± ¢ ¢ d ¹d αi! P d ~ ¤ ¤ α αi p ¡ q if α N0 and 0 γ α, : i 1 γi! αi γi ! γ γ i1 i 0 otherwise. ± P d P d α αi P 8 For x R and α Z we write x : xi . Given any φ Cc (infintitely many times |α| d P d α ±B differentiable with compact support in R ) and α N0 we write D φ Bαi φ. xi At some point we will use also use roman letter for multiindices, and then, to avoid confusion, we will use the vector notation ~i,~j, . . . On complex notation For z x i y P C we write Re pzq : x and Impzq : y. Note that the symbol i will be used also widely as a index for summations without risk of confusion. The multiindex notation will change slightly: for z P C and α P Z2 we write zα : zα1 z¯α2 . P 8p q We also adopt the traditional Wirtinger notation for derivatives, that is, given any φ Cc C , then Bφ 1 Bφpzq : pzq pB φ ¡ i B φqpzq, Bz 2 x y and Bφ 1 B¯φpzq : pzq pB φ i B φqpzq. Bz¯ 2 x y 8 P p q P 2 α Bα1 B¯α2 Thus, given any φ Cc C and α N0, we write D φ φ. P 1p q P d On Sobolev spaces: For any open set U, every distribution f D U and α N0, the α distributional derivative DU f is the distribution defined by x α y p¡ q|α|x α y P 8p q DU f, φ : 1 f, D φ for every φ Cc U . 16 CHAPTER 1. BACKGROUND

α α Abusing notation we will write D instead of DU if it is clear from the context. If the distribution 1 p q α is regular, that is, if it coincides with an Lloc °function acting on D U , then we say that DU f is a | n | | α | weak derivative of f in U. We write ∇ f |α|n D f . P ¤ ¤ 8 d 1 p q Given numbers n N, 1 p an open set U R and an Lloc U function f, we say that f is in the Sobolev space W n,ppUq of smoothness n and order of integrability p if f has weak α P p P d | | ¤ derivatives DU f L for every α N0 with α n. We will use the norm ¸ } } } } } α } f W n,ppΩq : f LppΩq D f LppΩq. |α|¤n

When Ω is a Lipschitz domain, this norm is equivalent to considering only the higher order derivatives. Namely,

¸d }f} }f} }∇nf} }f} Bnf (1.8) W n,ppΩq LppΩq LppΩq LppΩq j LppΩq j1

(see [Tri78, Theorem 4.2.4]). When Ω is an extension domain for W n,p, that is, when there exists n,pp q Ñ n,pp dq p q| | P n,pp q a bounded operator Λ : W Ω W R such that Λf Ω f Ω for f W Ω , then } } } } f W n,ppΩq inf F W n,pp dq. F :F |Ωf R

From [Jon81], we know that uniform domains (and in particular, Lipschitz domains) are Sobolev extension domains for any indices n P N and 1 ¤ p ¤ 8. One can ﬁnd deeper results in that sense in [Shv10] and [KRZ15]. On Besov and Triebel-Lizorkin spaces: Next we present a generalization of Sobolev spaces. p dq t u8 8p dq Deﬁnition 1.6. Let Φ R be the collection of all the families Ψ ψj j0 Cc R such that " p q supp ψ0 D 0, 2 , p j 1qz p j¡1q ¥ supp ψj D 0, 2 D 0, 2 if j 1,

P d for all multiindex α N there exists a constant cα such that c }Dαψ } ¤ α for every j ¥ 0 j 8 2j|α| and ¸8 p q P d ψj x 1 for every x R . j0

Deﬁnition 1.7. Given any Schwartz function ψ P SpRdq, its Fourier transform is » F ψpζq e¡2πix¤ζ ψpxqdmpxq. d R 1 This notion extends to the tempered distributions SpRdq by duality (see [Gra08, Deﬁnition 2.3.7]).

Deﬁnition 1.8. Let s P R, 1 ¤ p ¤ 8, 1 ¤ q ¤ 8 and Ψ P ΦpRdq. For any tempered distribution 1 f P S pRdq we deﬁne its non-homogeneous Besov norm ! ¡ © ) } }Ψ sj p q f Bs 2 ψjf , p,q Lp lq 1.3. KNOWN FACTS 17

s 1 and we call Bp,q S to the set of tempered distributions such that this norm is finite. 1 Let s P R, 1 ¤ p 8, 1 ¤ q ¤ 8 and Ψ P ΦpRdq. For any tempered distribution f P S pRdq we define its non-homogeneous Triebel-Lizorkin norm ! ¡ © ) } }Ψ sj p q f F s 2 ψjf , p,q lq Lp s 1 and we call Fp,q S to the set of tempered distributions such that this norm is finite.

These norms are equivalent for diﬀerent choices of Ψ. Usually one works with radial ψj and such that ψj 1pxq ψjpx{2q for j ¥ 1. Of course we will omit Ψ in our notation since it plays no role (see [Tri83, Section 2.3]). 8 s Remark 1.9. For q 2 and 1 p the spaces Fp,2 coincide with the so-called Bessel-potential spaces W s,p. In addition, if s P N they coincide with the usual Sobolev spaces, and they coincide with Lp for s 0 (see [Tri83, Section 2.5.6]). In the present text, we call Sobolev space to any W s,p with s ¡ 0 and 1 p 8, even if s is not a natural number. Note that complex interpolation between Sobolev spaces is a Sobolev space (see [Tri78, Section 2.4.2, Theorem 1]).

On conjugate indices: Given 1 ¤ p ¤ 8 we write p1 for its H¨olderconjugate, that is 1 1 p p1 1.

1.3 Known facts

On inequalities: For the sake of completeness, we recall the reader Minkowski’s integral inequality (see [Ste70, Appendix A1]) which we will use every now and then. It states that for 1 ¤ p 8, given a function F : X ¢ Y Ñ C, where X and Y are σ-ﬁnite measure spaces, we have that

¢» ¢» 1 » ¢» 1 p p p |F px, yq| dx dy ¤ |F px, yq|p dy dx. (1.9) Y X X Y We will use also the Young inequality. It states that for measurable functions f and g, we have that } ¦ } ¤ } } } } f g Lq f Lr g Lp (1.10) ¤ ¤ 8 1 1 1 ¡ for 1 p, q, r with q p r 1 (see [Ste70, Appendix A2]). On the Leibniz rule: The Leibniz formula (see [Eva98, Section 5.2.3]) says that given a d P n,pp q P 8p q ¤ P n,pp q domain Ω R , a function f W Ω and φ Cc Ω , we have that φ f W Ω with ¢ ¸ α Dαpφ ¤ fq Dγ φDα¡γ f (1.11) γ γ¤α

P d | | for every multiindex α N0 such that α n. On Green’s formula: The Green Theorem can be written in terms of complex derivatives (see [AIM09, Theorem 2.9.1]). Let Ω be a bounded Lipschitz domain. If f, g P W 1,1pΩq X CpΩq, then » ¢» » ¨ i Bf B¯g dm fpzq dz¯ ¡ gpzq dz . (1.12) Ω 2 BΩ BΩ

On the Residue Theorem: We say that a function f : C Ñ C is meromorphic on an open t u set U C if f is holomorphic on U but a discrete set of points aj j. The singularity of f in 18 CHAPTER 1. BACKGROUND

¥ p qp ¡ qm ¥ aj is called a pole of order mj 1 when limzÑaj f z z aj 0 if and only if m mj 1. The well-known Residue Theorem (see [Con78, Chapter V, Theorem 2.2 and Proposition 2.4], for instance), states that given a meromorphic function f in a connected open set U with poles in t uM aj j1 and no more singularities, and given a closed rectiﬁable curve γ homologous to 0 in U which does not pass through any aj, then the line integral » ¸M 1 1 p ¡ q fpzq dz npγ, a q g mj 1 pa q, (1.13) 2πi j pm ¡ 1q! j γ j1 j where npγ, ajq stands for the winding number of γ around aj (see [Con78, Chapter IV, Deﬁnition 4.2]) and mj is the order of the pole of f in aj. On the Rolle Theorem: We state here also a Complex Rolle Theorem for holomorphic functions [EJ92, Theorem 2.1] that will be a cornerstone of Section 3.5.

Theorem 1.10. [see [EJ92]] Let f be a holomorphic function deﬁned on an open convex set U C. Let a, b P U such that fpaq fpbq 0 and a b. Then there exists z in the segment sa, br such that Re pBfpzqq 0.

On the Sobolev Embedding Theorem: We state a reduced version of the Sobolev Em- bedding Theorem for Lipschitz domains (see [AF03, Theorem 4.12, Part II]). For each Lipschitz domain Ω Rd and every p ¡ d, there is a continuous embedding of the Sobolev space W 1,ppΩq 0,1¡ d into the H¨olderspace C p pΩq. That is, writing

|fpxq ¡ fpyq| }f} }f} 8 sup for 0 s ¤ 1, C0,spΩq L pΩq | ¡ |s x,yPΩ x y xy we have that for every f P W 1,ppΩq,

} } ¤ } } ¤ } } f 8p q f 0,1¡ d CΩ f 1,pp q. (1.14) L Ω C p pΩq W Ω

If Ω is not Lipschitz but it is an extension domain, then the previous estimate remains true. s s On embeddings and equivalent norms for Bp,q and Fp,q: Next we recall some results on the function spaces that we will use. For a complete treatment we refer the reader to [Tri83].

Proposition 1.11 (See [RS96, Section 2.2]). The following properties hold: ¤ ¤ 8 ¤ ¤ 8 P ¡ 1. Let 1 q0, q1 and 1 p , s R and ε 0. Then

Bs ε Bs . p,q0 p,q1

2. Let 1 ¤ p0 p p1 ¤ 8, ¡8 s1 s s0 8, 1 ¤ u ¤ p ¤ v ¤ 8 and 1 ¤ q 8. Then, d d d Bs0 F s Bs1 if s ¡ s ¡ s ¡ . p0,u p,q p1,v 0 1 p0 p p1

¤ 8 d 8 ¤ ¤ 8 3. In particular, given 1 p , p s and 1 q . Then,

s¡ d s¡ d s p s p Fp,q B8,8 and Bp,q B8,8. (1.15) 1.4. ON UNIFORM DOMAINS 19

¡ d d s p s¡ ¡ d R p Remark 1.12. When s p Z, the space B8,8 coincides with the Hölderspace C , so this is a generalization of the Sobolev embedding Theorem to the whole double-scale of Besov and Triebel- s¡ d s ¡ s p q p Lizorkin spaces. Thus, if Ω is an extension domain for Fp,q and sp d, then Fp,q Ω C if s¡ d ¡ε ¡ d R s p q p ¡ d P s p Z and Fp,q Ω C if s p Z. If we set j P Z instead of j P N in Definition 1.6, then we get the homogeneous spaces of 9 s tempered distributions (modulo polynomials) Bp,q. In particular, by [Tri92, Theorem 2.3.3] we have that for s ¡ 0 1 }f} s }f} 9 s }f} p for any f P S , (1.16) Bp,q Bp,q L and the same can be said for Triebel-Lizorkin spaces. In the particular case of homogeneous Besov spaces with 1 ¤ p, q ¤ 8 and s ¡ 0, one can give an equivalent definition in terms of differences of order M ¥ rss 1:

£ 1 £ 1 » q » 8 sup ∆M f q M q p q q |h|¤t h Lp dt ∆h f Lp dm h } } 9 f Bs sq sq d . (1.17) p,q t t d |h| |h| 0 R Consider the boundary of a Lipschitz domain Ω C. When it comes to the Besov space s pB q ÑB | 1p q| Bp,q Ω we can just define it using the arc parameter of the curve, z : I Ω with z t 1 for all t. Note that if the domain is bounded, then I is a finite interval with length equal to the length of the boundary of Ω and we need to extend z periodically to R in order to have a Ñ | sensible definition. We also use an auxiliary bump function ϕΩ : R R such that ϕΩ 2I 1 and ϕΩ|p4Iqc 0. Then, if 1 ¤ p, q 8, we define naturally the homogeneous Besov norm on the boundary of Ω as

}f} s pB q : }pf ¥ zqϕΩ} s p q. (1.18) Bp,q Ω Bp,q R Consider the boundary of a Lipschitz domain Ω C. When it comes to the Besov space s pB q ÑB | 1p q| Bp,q Ω we can just define it using the arc parameter of the curve, z : I Ω with z t 1 for all t. Note that if the domain is bounded, then I is a finite interval with length equal to the length of the boundary of Ω and we need to extend z periodically to R in order to have a Ñ | sensible definition. We also use an auxiliary bump function ϕΩ : R R such that ϕΩ 2I 1 and ϕΩ|p4Iqc 0. Then, if 1 ¤ p, q 8, we define naturally the homogeneous Besov norm on the boundary of Ω as

}f} s pB q : }pf ¥ zqϕΩ} s p q. (1.19) Bp,q Ω Bp,q R 1.4 On uniform domains

There is a considerable amount of literature on uniform domains and their properties, we refer the reader e.g. to [GO79] and [V¨ai88]. Deﬁnition 1.13. Let Ω be a domain and W a Whitney decomposition of Ω and Q, S P W. Given P p qM P M M cubes Q1,...,QM W with Q1 Q and QM S, the M-tuple Q1,...,QM j1 W is a chain connecting Q and S if the cubes Qj and Qj 1 are neighbors for j M. We write r s p qM Q, S Q1,...,QM j1 for short. Let ε P R. We say that the chain rQ, Ss is ε-admissible if • the length of the chain is bounded by

¸M 1 `prQ, Ssq : `pQ q ¤ DpQ, Sq (1.20) j ε j1 (see (1.6)), 20 CHAPTER 1. BACKGROUND

Figure 1.3: A Whitney decomposition of a uniform domain with and an ε-admissible chain. The end-point cubes are colored in red and the central one in blue.

• and there exists j0 M such that the cubes in the chain satisfy

`pQjq ¥ εDpQ1,Qjq for all j ¤ j0 and `pQjq ¥ εDpQj,QM q for all j ¥ j0. (1.21)

p q Á p q The j0-th cube, which we call central, satisfies that ` Qj0 d εD Q, S by (1.20) and the triangle inequality. We will write QS Qj0 . Note that this is an abuse of notation because the central cube of rQ, Ss may vary for different ε-admissible chains joining Q and S. r s t uM P We write (abusing notation again) Q, S also for the set Qj j1. Thus, we will write P rQ, Ss if P appears in a coordinate of the M-tuple rQ, Ss. For any P P rQ, Ss we call NrQ,SspP q to the following cube in the chain, that is, for j M we have that NrQ,SspQjq Qj 1. We will write N pP q for short if the chain to which we are referring is clear from the context. Every now and then we will mention subchains. That is, for 1 ¤ j1 ¤ j2 ¤ M, the subchain r s r s p q r s Qj1 ,Qj2 rQ,Ss Q, S is defined as Qj1 ,Qj1 1,...,Qj2 . We will write Qj1 ,Qj2 if there is no risk of confusion.

Next we make some observations on the two subchains rQ, QSs and rQS,Ss. Remark 1.14. Consider a domain Ω with covering W and two cubes Q, S P W with an ε- admissible chain rQ, Ss. From Deﬁnition 1.13 it follows that

DpQ, Sq ε,d `prQ, Ssq ε,d `pQSq ε,d DpQ, QSq ε,d DpQS,Sq. (1.22) 1.4. ON UNIFORM DOMAINS 21

If P P rQ, QSs, by (1.20) we have that

DpQ, P q d,ε `pP q. (1.23) On the other hand, by the triangular inequality, (1.19) and (1.20) we have that DpQ, Sq DpQ, P q DpP,Sq 1 `pP q DpP,Sq DpP,Sq À `prP,Ssq ¤ `prQ, Ssq ¤ À À ε , d ε d ε d ε that is, DpP,Sq ε,d DpQ, Sq. (1.24) Definition 1.15. We say that a domain Ω Rd is a uniform domain if there exists a Whitney covering W of Ω and ε P R such that for any pair of cubes Q, S P W, there exists an ε-admissible chain rQ, Ss (see Figure 1.3). Sometimes will write ε-uniform domain to fix the constant ε. Using (1.23) it is quite easy to see that a domain satisfying this definition satisfies to the one given by Peter Jones in [Jon81] (with δ 8 and changing the parameter ε if necessary). It is somewhat more involved to prove the converse implication, but it can be done using the ideas of Remark 1.14. Of course, the definition above is also equivalent to the one given by Gehring and Osgood in [GO79]. In any case it is not transcendent for the present dissertation to prove this fact, which is left for the reader as an exercise. Now we can define the shadows: Definition 1.16. Let Ω be an ε-uniform domain with Whitney covering W. Given a cube P P W centered at xP and a real number ρ, the ρ-shadow of P is the collection of cubes

SHρpP q tQ P W : Q BpxP , ρ `pP qqu, and its “realization” is the set ¤ ShρpP q Q

QPSHρpP q (see Figure 1.4). By the previous remark and the properties of the Whitney covering, we can deﬁne ρε ¡ 1 such that the following properties hold: P | pB X p qq| p q • For every P W, we have the estimate diam Ω Shρε P ` P . r s P r s P p q • For every ε-admissible chain Q, S , and every P Q, QS we have that Q SHρε P . • Moreover, every cube P belonging to an ε-admissible chain rQ, Ss belongs to the shadow p q SHρε QS . Note that the ﬁrst property comes straight from the properties of the Whitney covering, while the second is a consequence of (1.22) and the third holds because of the fact that if P P rQ, Ss then DpP,QSq Àd `prQ, Ssq DpQ, Sq `pQSq by (1.21).

Remark 1.17. Given an ε-uniform domain Ω we will write Sh for Shρε . We will write also SH for SHρε . For Q P W and s ¡ 0, we have that ¸ `pLq¡s À `pQq¡s (1.25) L:QPSHpLq and, moreover, if Q P SHpP q, then ¸ ¸ `pLqs À `pP qs and `pLq¡s À `pQq¡s. (1.26) LPrQ,P s LPrQ,P s 22 CHAPTER 1. BACKGROUND

Figure 1.4: A Whitney decomposition of a Lipschitz domain with the shadows of three diﬀerent cubes (see Deﬁnition 1.16).

Proof. Considering the definition of shadow we can deduce that there is a bounded number of cubes with given side-length in the left-hand side of (1.24) and, therefore, the sum is a geometric sum. Again by the definition of shadow we know that the smaller cube in that sum has side-length comparable to `pQq. To prove (1.25), first note that `pQP q DpQ, P q `pP q by (1.21) and Definition 1.16. For every L P rQ, P s, although it may occur that L R SHpP q, we still have that by the triangle inequality DpL, P q À `prQ, P sq DpQ, P q and, thus, by the definition of shadow we have that DpL, P q À `pP q, i.e. DpL, P q `pP q. (1.27)

When L P rQ, QP s, (1.22) reads as `pLq DpQ, Lq, and when L P rQP ,P s by (1.22) and (1.26), we have that

`pLq DpL, P q `pP q.

In particular, the number of cubes in rQP ,P s is uniformly bounded. Summing up, for L P rQ, P s we have that `pQq À `pLq À `pP q and all the cubes of a given side-length r contained in rQ, P s are situated at a distance from Q bounded by Cr. so the number of those cubes is uniformly bounded. Therefore, the left-hand side of both inequalities of (1.25) are geometric sums, bounded by a constant times the bigger term. The constant depends on s, but also on the uniformity constant of the domain. We recall the deﬁnition of the non-centered Hardy-Littlewood maximal operator. Given f P 1 p dq P d p q Lloc R and x R , we deﬁne Mf x as the supremum of the mean of f in cubes containing x, that is, » p q 1 p q Mf x sup | | f y dy. QQx Q Q It is a well known fact that this operator is bounded on Lp for 1 p 8. The following lemma will be used repeatedly along the proofs contained in the present text. Lemma 1.18. Let Ω be a bounded uniform domain with an admissible Whitney covering W. Assume that g P L1pΩq and r ¡ 0. For every η ¡ 0, Q P W and x P Rd, the following inequalities hold: 1.5. APPROXIMATING POLYNOMIALS 23

1) The non-local inequality for the maximal operator ³ » ¸ gpyq dy Mgpxq gpyq dy inf P Mgpyq À S À y Q d η d η and d η d η . (1.28) | ¡ |¡ |y ¡ x| r DpQ, Sq r y x r S:DpQ,Sq¡r

2) The local inequality for the maximal operator » ³ gpyq dy ¸ gpyq dy À η p q S À p q η d¡η d r Mg x and d¡η d inf Mg y r . | ¡ | |y ¡ x| DpQ, Sq yPQ y x r S:DpQ,Sq r (1.29)

3) In particular we have » ¸ p qd dy À 1 ` S À 1 d η d η , d η d η (1.30) | ¡ |¡ |y ¡ x| r DpQ, Sq `pQq y x r SPW

and, by Deﬁnition 1.16, ¸ » d gpxq dx Àd,ρ inf Mgpyq `pQq . P S y Q SPSHρpQq

Proof. The left-hand side of both inequalities in (1.27) can just bounded by » gpyq dx C p|x ¡ y| rqd η

(in the second one, this bound holds for every x P Q), and this can be bounded separating the integral region in dyadic annuli. The sum in (1.28) can be bounded by an analogous reasoning and (1.29) follows when considering g 1.

1.5 Approximating polynomials

Before introducing the approximating polynomials, we need a rather trivial observation, which is proven here for the sake of completeness.

Remark 1.19. For every polynomial P P Pn, every cube Q and every r ¡ 1, we have that

} } p qd} } P L1pQq ` Q P L8pQq, and for r ¡ 1, also } } À n} } P L8prQq r P L8pQq, with constants depending only on d and n.

Proof. Without loss° of generality we can assume that the cube Q is centered at 0. Given a polynomial P pxq a xγ of degree n, using the linear map φ that sends the unit cube ¨ |γ|¤n γ ¡ 1 1 d Q0 2 , 2 to Q as a change of coordinates, we have that

}P } 1 |Q|}P ¥ φ} 1 . L pQq L pQ0q 24 CHAPTER 1. BACKGROUND

It is well known that all norms in a ﬁnite dimensional vector space are equivalent (see [Sch02, 1 8 Theorem 4.2], for instance). In particular the L pQ0q norm, the sum of coeﬃcients and the L pQ0q norm are equivalent in Pn, so we have that ¸ 1 |γ| }P } `pQq |a | }P } 8 . `pQqd L1pQq γ L pQq |γ|¤n

By the same token, ¸ } } p p qq|γ| | | À n} } P L8prQq r` Q aγ r P L8pQq. |γ|¤n

Recall that the Poincaréinequality tells us that, given a cube Q and a function f P W 1,ppQq with 0 mean in the cube, we have the estimate } } À p q} } f LppQq ` Q ∇f LppQq with universal constants once we fix d and 1 ¤ p ¤ 8 (this well-known result can be shown combining the proof of [Eva98, Theorem 5.8.1/1], the version of Rellich-Kondrachov Theorem in [AIM09, Theorem A.7.1] and a change of variables for the dilatation constant, for instance). If we want to iterate that inequality, we also need the gradient of f to have 0 mean on Q. That leads us to define the next approximating polynomials. Definition 1.20. Consider a domain Ω and a cube Q Ω. Given f P L1pQq with weak derivatives n p q P n up to order n, we define PQ f P as the unique polynomial (restricted to Ω) of degree smaller or equal than n such that β n β D PQf dm D f dm (1.31) Q Q for every multiindex β P Nd with |β| ¤ n. These polynomials can be understood as a particular case of the projection L : W n 1,ppQq Ñ Pn introduced by Norman G. Meyers in [Mey78]. P n¡1,1p q n¡1 P n¡1 Lemma 1.21. Given a cube Q and f W 3Q , the polynomial P3Q f P exists and is unique. Furthermore, this polynomial has the next properties: n¡1 1. Let xQ be the center of Q. If we consider the Taylor expansion of P3Q f at xQ, ¸ n¡1 p q p ¡ qγ P3Q f y mQ,γ y xQ , (1.32) P d γ N |γ| n

then the coeﬃcients mQ,γ are bounded by n¸¡1 | | À j p qj¡|γ|¡d À } } p p qn¡1q mQ,γ n ∇ f L1p3Qq` Q n f W n¡1,8p3Qq 1 ` Q . (1.33) j|γ|

2. Let us assume that, in addition, the function f is in the Sobolev space W n,pp3Qq for a certain 1 ¤ ¤ 8 ¤ ¤ P 8p q i À p . Given 0 j n, if we have a smooth function ϕ C 3Q satisfying ∇ ϕ L8p3Qq 1 ¤ ¤ `pQqi for 0 i j, then we have the Poincar´einequality ¡¡ © © j ¡ n¡1 ¤ p qn¡j} n } ∇ f P3Q f ϕ C` Q ∇ f Lpp3Qq. (1.34) Lpp3Qq 1.5. APPROXIMATING POLYNOMIALS 25

3. Given a domain with a Whitney covering W, two Whitney cubes Q, S P W, a chain rS, Qs as in Deﬁnition 1.13, and f P W n,ppΩq, we have that ¸ `pSqdDpP,Sqn¡1 ¡ n¡1 À } n } f P3Q f ¡ ∇ f L1p3P q. (1.35) L1pSq `pP qd 1 P PrS,Qs

Proof. Note that (1.30) is a triangular system of equations on the coeﬃcients of the polynomial. Indeed, for γ ﬁxed, if the polynomial exists and has Taylor expansion (1.31), then ¸ β! Dγ Pn¡1fpyq m py ¡ x qβ¡γ . 3Q Q,β pβ ¡ γq! Q β¥γ When we take means on the cube 3Q,

γ γ n¡1 D f dm D P3Q f dm 3Q 3Q ¢ ¸ |β¡γ| β! 3 p q β¡γ mQ,β p ¡ q ` Q y dy ¥ β γ ! 2 Qp0,1q β¸γ |β¡γ| Cβ,γ mQ,β`pQq , β¥γ which is a triangular system of equations on the coeﬃcients mQ,β. Solving for mQ,γ , since Cγ,γ 0 we obtain the explicit expression 1 ¸ m Dγ f dm ¡ C m `pQq|β¡γ|. (1.36) Q,γ C β,γ Q,β γ,γ 3Q β¡γ

γ For |γ| n ¡ 1 this gives the value of mQ,γ in terms of D f,

1 γ mQ,γ D f dm. Cγ,γ 3Q ¡ | | n¡1 Using induction on n γ we get the existence and uniqueness of P3Q f. Taking absolute values we obtain (1.32). The equality (1.30) allows us to iterate the Poincar´einequality ¡ © ¡ n¡1 ¤ p q ¡ n¡1 ¤ ¤ ¤ ¤ ¤ n p qn} n } f P3Q f C` Q ∇ f P3Q f C ` Q ∇ f Lpp3Qq. Lpp3Qq Lpp3Qq Therefore, by the Leibniz rule (1.11) we have that ¡¡ © © ¡ © ¸j j ¡ n¡1 À i j¡i ¡ n¡1 ∇ f P3Q f ϕ ∇ ϕ 8p q ∇ f P3Q f Lpp3Qq L 3Q Lpp3Qq i0 ¡ © ¸j `pQqi À j ¡ n¡1 À p qn¡j} n } ∇ f P3Q f ` Q ∇ f Lpp3Qq, `pQqi Lpp3Qq i0 proving (1.33). To prove (1.34), we consider the chain rQ, Ss to write ¸ ¡ n¡1 ¤ ¡ n¡1 n¡1 ¡ n¡1 f P3Q f f P3S f 1p q P3P f P3N pP qf , (1.37) L1pSq L S L1pSq P PrS,Qq 26 CHAPTER 1. BACKGROUND

where we write N pP q instead of NrS,QspP q from Deﬁnition 1.13. Applying Remark 1.19 to the n¡1 ¡ n¡1 DpP,Sq polynomial P3P f P3N pP qf, the cubes S and P and with r `pP q , it follows that n¡1 ¡ n¡1 n¡1 ¡ n¡1 p qd P3P f P3N pP qf P3P f P3N pP qf ` S L1pSq L8pSq `pSqdDpP,Sqn¡1 À n¡1 ¡ n¡1 P3P f P3N pP qf ¡ L8p3P X3N pP qq `pP qn 1 `pSqdDpP,Sqn¡1 n¡1 ¡ n¡1 P3P f P3N pP qf ¡ . L1p3P X3N pP qq `pP qn 1`pP qd

Using this estimate in (1.36), and then (1.33) with ϕ 1, we get ¢ ¸ `pSqdDpP,Sqn¡1 ¡ n¡1 À n¡1 ¡ ¡ n¡1 f P3Q f P3P f f 1p q f P3N pP qf ¡ L1pSq L 3P L1p3N pP qq `pP qd n 1 P PrS,Qq ¸ `pSqdDpP,Sqn¡1 À f ¡ Pn¡1f 3P L1p3P q `pP qd n¡1 P PrS,Qs ¸ `pSqdDpP,Sqn¡1 À }∇nf} . L1p3P q `pP qd¡1 P PrS,Qs

1.6 Calder´on-Zygmund operators

P 1 p dzt uq Definition 1.22. We say that a measurable function K Lloc R 0 is a convolution Calderón- Zygmund kernel if it satisfies the size condition C |Kpxq| ¤ K for x 0, (1.38) |x|d and the Hörmander condition »

sup |Kpx yq ¡ Kpxq| dx CK , (1.39) y0 |x|¥2|y|

d and that kernel can be extended to a convolution with a tempered distribution WK in R in the sense that for all Schwartz functions f, g P S with supppfq X supppgq H, one has »

xWK ¦ f, gy Kpxq pf¡ ¦ gq pxq dx, (1.40) dzt u R 0 where f¡pxq fp¡xq. Remark 1.23. We are using the notion of distributional convolution. Given Schwartz functions f and g, the convolution coincides with multiplication at the Fourier side, that is, f ¦ gpxq pfp¤ gpqq. Given a tempered distribution W , a function f P S and x P Rd, the tempered distribution W ¦ f is deﬁned as x p x p xW ¦ f, gy : xpW ¤ fqq, gy xW, f ¤ gqy xW, f¡ ¦ gy for every g P S. 1.6. CALDERON-ZYGMUND´ OPERATORS 27 ³ Note that f¡ ¦ gpxq fp¡yqgpx ¡ yq dy, so in case supppfq X supppgq H then f¡ ¦ g 0 in a neighborhood of 0 and, therefore, the integral in (1.39) is absolutely convergent by (1.37). ¦ 1 In any case, the distribution W f is regular (i.e., it can be expressed as an Lloc function) and 8 it coincides with the C function W ¦ fpxq xW, τxf¡y, where τxf¡pyq f¡py ¡ xq (see [SW71, Chapter I, Theorem 3.13]).

There are some cancellation conditions that one can impose to a kernel satisfying the size condition (1.37) to grant that it can be extended to a convolution with a tempered distribution. For instance, if K satisﬁes (1.37) and WK is a principal value operator in the sense that »

xWK , ϕy lim Kpxqϕpxq dx for all ϕ P S, jÑ8 |x|¥δj for a certain sequence δj × 0, then WK satisﬁes (1.39) (see [Gra08, Section 4.3.2]).

Deﬁnition 1.24. We say that an operator T : S Ñ S1 is a convolution Calder´on-Zygmund operator with kernel K if

1. K is an admissible convolution Calder´on-Zygmundkernel which can be extended to a convolution with a tempered distribution WK ,

2. T satisﬁes that T f WK ¦ f for all f P S and

3. T extends to an operator bounded on L2.

Remark 1.25. Using the Calder´on-Zygmunddecomposition one can see that T is also bounded on Lp for 1 p 8 (see [Gra08, proof of Theorem 4.3.3]). The Fourier transform of an admissible convolution Calder´on-Zygmundoperator T is a Fourier 2 y 8 multiplier for L , and this implies that WK P L (see [SW71, Chapter I, Theorem 3.18]).

It is a well-known fact that the Schwartz class is dense in Lp for 1 ¤ p 8. Thus, if f P Lp and x R supppfq, then » T fpxq Kpx ¡ yqfpyqdy. (1.41)

Definition 1.26. We say that an operator T : S Ñ S1 is an admissible convolution Calderón- P P n,1p dzt uq Zygmund operator of order n N with kernel K Wloc R 0 if T satisfies Definition 1.24 and, moreover, the kernel K satisfies the higher order smoothness condition

C |∇nKpxq| ¤ K for all x 0. (1.42) |x|d n

Example 1.27. The Beurling transform (1.1) is an admissible Calder´on-Zygmundoperator of order 8.

Let T be an admissible convolution Calder´on-Zygmund operator of order n, let Ω Rd be a domain and let 1 p 8. Given a function f P W n,ppΩq, we want to study in what situations its transform TΩf χΩ T pχΩ fq is in some Sobolev space, so we need to check that its weak derivatives exist up to order n. Indeed that is the case.

n,p Lemma 1.28. Given f P W pΩq, the weak derivatives of TΩf in Ω exist up to order n.

Before proving this, we consider the functions deﬁned in all Rd. 28 CHAPTER 1. BACKGROUND

Remark 1.29. Since T is a bounded linear operator in L2pRdq that commutes with translations, for Schwartz functions the derivative commutes with T (see [Gra08, Lemma 2.5.3]). Using that S is dense in W n,ppRdq (see [Tri83, Sections 2.3.3 and 2.5.6], for instance), we conclude that for every f P W n,ppRdq and α P Nd with |α| n DαT pfq TDαpfq (1.43) and, thus, the operator T is bounded on W n,ppRdq. P n,1p dzt uq P p Deﬁnition 1.30. Let K Wloc R 0 be the kernel of T and consider a function f L , a multiindex α P Nd with |α| ¤ n and x R supppfq. We deﬁne » T pαqfpxq : DαKpx ¡ yqfpyq dy.

Lemma 1.31. Let f P Lp. Then T f has weak derivatives up to order n in Rdzsuppf. Moreover, for every multiindex α P Nd with |α| ¤ n and x R suppf DαT fpxq T pαqfpxq.

P 8p dz q Proof. Take a compactly supported smooth function φ Cc R suppf . We can use Fubini’s Theorem and get » » xT pαqf, φy DαKpx ¡ yqfpyq dy φpxq dx »suppφ »suppf DαKpx ¡ yqφpxq dx fpyq dy. suppf suppφ Using the deﬁnition of distributional derivative, Tonelli’s Theorem again and (1.40) we get » » xT pαqf, φy p¡1q|α| Kpx ¡ yqDαφpxq dx fpyq dy »suppf »suppφ p¡1q|α| Kpx ¡ yqfpyq dy Dαφpxq dx p¡1q|α|xT f, Dαφy. suppφ suppf

Proof of Lemma 1.28. Take a classical Whitney covering of Ω, W, and for every Q P W, deﬁne P 8 ¤ ¤ t u a bump function ϕQ Cc such that χ2Q ϕQ χ3Q. On the other hand, let ψQ QPW be a t 3 P u | | partition of the unity associated to 2 Q : Q W . Consider a multiindex α with α n. Then Q ¤ Q p ¡ Qq take f1 ϕQ f, and f2 f f1 χΩ. One can deﬁne ¸ ¡ © p q p q α Qp q pαq Qp q g y : ψQ y TD f1 y T f2 y . QPW

α This function is defined almost everywhere in Ω and is the weak derivative D TΩf. P 8p q Indeed, given a test function φ Cc Ω , then, since φ is compactly supported in Ω, its support intersects a finite number of Whitney double cubes and, thus, the following additions are finite: ¸ x y x ¤ α Q ¤ pαq Q y g, φ ψQ TD f1 ψQ T f2 , φ P ¸Q W ¸ x α Q y x pαq Q y TD f1 , φQ T f2 , φQ , (1.44) QPW QPW 1.6. CALDERON-ZYGMUND´ OPERATORS 29

where φQ ψQ ¤ φ. In the local part we can use (1.42), so

x α Q y p¡ q|α|x Q α y TD f1 , φQ 1 T f1 ,D φQ .

Q When it comes to the non-local part, bearing in mind that f2 has support away form 2Q and P 8p q φQ Cc 2Q , we can use the Lemma 1.31 and we get

x pαq Q y p¡ q|α|x Q α y T f2 , φQ 1 T f2 ,D φQ . Back to (1.43) we have ¸ ¸ ¸ x y p¡ q|α|x Q α y p¡ q|α|x Q α y p¡ q|α|x α y g, φ 1 T f1 ,D φQ 1 T f2 ,D φQ 1 TΩf, D φQ QPW QPW QPW |α| α p¡1q xTΩf, D φy,

α that is, g D TΩf in the weak sense.

Chapter 2

T(P) theorems

Let Ω Rd be a domain and T a convolution Calderón-Zygmund operator. We are interested in conditions that allow us to infer that the restricted operator TΩ χΩT χΩ is bounded on a certain Sobolev space W s,ppΩq. In this chapter we find some results for the supercritical case in terms of test functions, namely polynomials of degree strictly smaller than the considered smoothness s. This is in accordance with the pioneering results found in [CMO13], which deal with operators with even kernel satisfying some smoothness assumptions, and with spaces W s,ppΩq where 0 s ¤ 1, sp ¡ d and Ω is a Lipschitz domain with parameterizations in C1,σ. In that situation, the authors s,p prove that TΩ is bounded on W pΩq if and only s,p if TΩ1 P W pΩq and an analogous result for s p q Bp,p Ω with s 1. The reader will find two results in that spirit in this chapter. The first, Theorem 2.1, deals with W s,ppΩq with s P N and p ¡ d (see the green segments in Figure 2.1), and the second, Theorem 2.8, deals with the supercritical case for Figure 2.1: Indices studied regarding the T pP q- 0 s 1 (see the triangle in Figure 2.1). These theorems in this chapter. results improve the previously known results in the sense that the class of operators considered is wider, the range of indices s, p is larger and, moreover, the restrictions on the regularity of the domain are reduced to just asking the domain to be uniform. Moreover, in the case 0 s 1, the result is valid for a greater family of Triebel-Lizorkin spaces. The novelty in the approach exposed in this chapter is not only on the aforementioned improvements, but on the technique used to reach that results, which relies strongly on the properties of uniform domains and their hyperbolic metric. Furthermore, in both cases we make use of a Key Lemma that provides information even if the condition sp ¡ d is not satisfied. In Chapter 6 we will see how this lemmas can be used to provide other conditions in the critical and subcritical cases. Sections 2.1 and 2.2 are devoted to Theorems 2.1 and 2.8 respectively. The rest of the chapter

31 32 CHAPTER 2. T(P) THEOREMS serves to provide the tools that the author needs for Theorem 2.8 which he could not ﬁnd in the literature.

2.1 Classic Sobolev spaces on uniform domains

Theorem 2.1. Let Ω Rd be a bounded ε-uniform domain, T an admissible convolution Calder´on- Zygmund operator of order n P N and d p 8. Then the following statements are equivalent:

n,p a) The truncated operator TΩ is bounded on W pΩq.

n,p b) For every polynomial P of degree at most n ¡ 1, we have that TΩpP q P W pΩq. ± d P p q p ¡ qλ p ¡ qλj P d Moreover, if x0 Ω, writing Pλ x : x x0 j1 xj x0,j for λ N we have that ¸ } } À } } } np q} TΩ W n,ppΩqÑW n,ppΩq diampΩq,ε,n,p,d CK T LpÑLp ∇ TΩPλ LppΩq. (2.1) |λ| n

To prove this theorem we need the following lemma, which says that it is equivalent to bound the transform of a function and its approximation by polynomials on Whitney cubes.

Key Lemma 2.2. Let Ω Rd be an ε-uniform domain with Whitney covering W (see Deﬁnition 1.15), T an admissible convolution Calder´on-Zygmundoperator of order n P N and 1 p 8. Then the following statements are equivalent:

i) For every f P W n,ppΩq one has

} } ¤ } } TΩf W n,ppΩq C f W n,ppΩq,

where C depends only on ε, n, p, d and T .

ii) For every f P W n,ppΩq one has ¡ © ¸ p n n¡1 ¤ } }p ∇ TΩ P3Q f C f W n,ppΩq, LppQq QPW

where C depends only on ε, n, p, d and T .

Proof. Let Ω be an ε-uniform domain. Given a multiindex α with |α| n, we will bound the diﬀerence ¡ © ¸ p α ¡ n¡1 À p } } qp } n }p D TΩ f P3Q f CK T LpÑLp ∇ f LppΩq, (2.2) LppQq QPW with constants depending on ε, n, p and d. 8 For each cube Q P W we deﬁne a bump function ϕQ P C such that χ 3 ¤ ϕQ ¤ χ2Q and c 2 Q j p q¡j ¤ ∇ ϕQ 8 ` Q for every j n. Then we can break (2.2) into local and non-local parts as follows: ¡ © ¡ ¡ ©© ¸ p ¸ p α n¡1 α n¡1 D TΩ f ¡ P f À D T ϕQ f ¡ P f (2.3) 3Q pp q 3Q pp q P L Q P L Q Q W Q W ¡ ¡ ©© ¸ p α p ¡ q ¡ n¡1 D T χΩ ϕQ f P3Q f 1 2 . LppQq QPW 2.1. CLASSIC SOBOLEV SPACES ON UNIFORM DOMAINS 33

First of all we will show that the local term in (2.3) satisﬁes ¡ ¡ ©© ¸ p α ¡ n¡1 À} }p } n }p 1 D T ϕQ f P3Q f T LpÑLp ∇ f LppΩq. (2.4) LppQq QPW ¡ © ¡ n¡1 P n,pp dq p To do so, note that ϕQ f P3Q f W R and, by (1.42) and the boundedness of T in L , ¡ ¡ ©© ¡ ¡ ©© p p α n¡1 p α n¡1 D T ϕQ f ¡ P f À }T } pÑ p D ϕQ f ¡ P f 3Q pp q L L 3Q pp dq L Q ¡ ¡ ©©L R p } }p α ¡ n¡1 T LpÑLp D ϕQ f P3Q f . Lpp2Qq

Using the Poincar´einequality (1.33), we get ¡ ¡ ©© p α ¡ n¡1 À} n }p D T ϕQ f P3Q f ∇ f Lpp3Qq. LppQq

Summing over all Q we get (2.4). For the non-local part in (2.3), ¡ ¡ ©© ¸ p α p ¡ q ¡ n¡1 2 D T χΩ ϕQ f P3Q f , LppQq QPW we will argue by duality. We can write » § ¡ © § 1 ¸ § § p § α p ¡ q ¡ n¡1 p q§ p q 2 sup D T χΩ ϕQ f P3Q f x g x dx. (2.5) } } ¤ g Lp1 1 QPW Q

Note that given x P Q P W, by Lemma 1.31 one has ¡ © » ¡ © α p ¡ q ¡ n¡1 p q α p ¡ q p ¡ p qq p q ¡ n¡1 p q D T χΩ ϕQ f P3Q f x D K x y 1 ϕQ y f y P3Q f y dy. Ω Taking absolute values and using the smoothness condition (1.41), we can bound

§ ¡ © § » ¡ § § |fpyq ¡ Pn 1fpyq| §DαT pχ ¡ ϕ q f ¡ Pn¡1f pxq§ ¤ C 3Q dy Ω Q 3Q K | ¡ |n d Ωz 3 Q x y 2 ¡ ¸ f ¡ Pn 1f 3Q 1p q À C L S K DpQ, Sqn d SPW and, by (2.5), we have that » ¡ n¡1 1 ¸ ¸ f P3Q f p À L1pSq p q 2 CK sup n d g x dx. (2.6) } } ¤ DpQ, Sq g Lp1 1 QPW Q SPW

By Deﬁnition 1.15, for every pair of Whitney cubes Q and S there exists an admissible chain rS, Qs and, by (1.34), we have that ¸ `pSqdDpP,Sqn¡1 ¡ n¡1 À } n } f P3Q f ¡ ∇ f L1p3P q. (2.7) L1pSq `pP qd 1 P PrS,Qs 34 CHAPTER 2. T(P) THEOREMS

Thus, plugging (2.7) into (2.6), we get

» ¡ 1 ¸ ¸ ¸ `pSqdDpP,Sqn 1}∇nf} p À p q L1p3P q 2 CK sup g x dx d¡1 n d . (2.8) } } ¤ `pP q DpQ, Sq g p1 1 QPW Q SPW P PrS,Qs

By (2.3), (2.4), (2.8) and Lemma 2.3 below, we have that (2.2) holds.

Lemma 2.3. Consider a uniform domain Ω with Whitney covering W, two functions f P LppΩq 1 and g P Lp pΩq and ρ ¥ 1. Then

¸ ¸ p qd p qρ¡1} } } } ` S D P,S f L1p50P q g L1p50Qq A pf, gq : À}f} }g} 1 . ρ `pP qd¡1DpQ, Sqρ d LppΩq Lp pΩq Q,SPW P PrS,Qs

Proof. Using that P P rS, Qs implies DpP,Sq À DpQ, Sq (see Remark 1.14), we get

¸ ¸ d ¸ ¸ d `pSq }f} 1p q}g} 1p q `pSq }f} 1p q}g} 1p q A pf, gq À L 50P L 50Q L 50P L 50Q ρ p qd¡1 p qd 1 p qd¡1 p qd 1 P ` P D Q, S P ` P D Q, S Q,S W P PrS,SQs Q,S W P PrSQ,Qs Ap1qpf, gq Ap2qpf, gq.

p1q We consider ﬁrst the term A pf, gq where the sum is taken with respect to cubes P P rS, SQs and, thus, by (1.23) the long distance DpQ, Sq DpP,Qq. Moreover, we have S P SHpP q by Deﬁnition 1.16. Thus, rearranging the sum, ¸ ¸ ¸ }f} 1p q }g} 1p q Ap1qpf, gq À L 50P L 50Q `pSqd. `pP qd¡1 DpQ, P qd 1 P PW QPW SPSHpP q

By Deﬁnition 1.16 again ¸ `pSqd `pP qd SPSHpP q and, by (1.27) and the ﬁnite overlapping of the cubes t50QuQPW , we get ¸ } } g 1p q inf P Mgpxq L 50Q À x 50P . DpQ, P qd 1 `pP q QPW

p2q Next we perform a similar argument with A pf, gq. Note that when P P rSQ,Qs, we have DpQ, Sq DpP,Sq and Q P SHpP q, leading to

¸ } } ¸ ¸ d f 1p q `pSq Ap2qpf, gq À L 50P }g} . `pP qd¡1 L1p50Qq DpP,Sqd 1 P PW QPSHpP q SPW

By (1.28) we get ¸ d }g} 1p q À inf Mgpxq `pP q L 50Q xP50P QPSHpP q and, by (1.29), ¸ `pSqd 1 . DpP,Sqd 1 `pP q SPW 2.1. CLASSIC SOBOLEV SPACES ON UNIFORM DOMAINS 35

Thus, ¸ } } ¸ f 1p q inf Mg A pf, gq À L 50P 50P `pP qd À }f ¤ Mg} . ρ `pP qd¡1 `pP q L1p50P q P PW P PW 1 By H¨olderinequality and the boundedness of the Hardy-Littlewood maximal operator in Lp , £ £ 1 ¸ 1{p ¸ 1{p p p1 A pf, gq À }f} }Mg} 1 À}f} }g} 1 . ρ Lpp50P q Lp p50P q LppΩq Lp pΩq P PW P

Proof of Theorem 2.1. The implication aq ñ bq is trivial. To see the converse, we assume by hypothesis that ¸ }TΩpPλq}W n,ppΩq ¤ C. (2.9) |λ| n Let f P W n,ppΩq. By the Key Lemma 2.2, we have to prove that ¸ ¡ © } n n¡1 }p À} }p ∇ TΩ P3Q f LppQq f W n,ppΩq. QPW We can write the polynomials ¸ n¡1 p q p ¡ qγ P3Q f x mQ,γ x xQ , |γ| n where xQ stands for the center of each cube Q. Taking the Taylor expansion in x0 for each monomial, one has ¢ ¸ ¸ γ Pn¡1fpxq m px ¡ x qλpx ¡ x qγ¡λ. 3Q Q,γ λ 0 0 Q |γ| n ~0¤λ¤γ

Thus, ¡ © ¢ ¸ ¸ γ ∇nT Pn¡1f pyq m px ¡ x qγ¡λ∇npT P qpyq. (2.10) Ω 3Q Q,γ λ 0 Q Ω λ |γ| n ~0¤λ¤γ Recall the estimate (1.32) in Lemma 1.21, which implies that n¸¡1 n¸¡1 | | ¤ j p qj¡|γ| À j j¡|γ| mQ,γ C ∇ f L8p3Qq` Q ∇ f L8pΩqdiamΩ . (2.11) j|γ| j|γ| Raising (2.10) to the power p, integrating in Q and using (2.11) we get ¡ © p ¸ ¸ n n¡1 À j p pj¡|λ|qp} np q}p ∇ TΩ P3Q f ∇ f 8p q diamΩ ∇ TΩPλ LppQq. LppQq L Ω j n |λ| j j ¤ j By the Sobolev Embedding Theorem, we know that ∇ f L8pΩq C ∇ f W 1,ppΩq as long as p ¡ d. If we add with respect to Q P W and we use (2.9) we get ¡ © ¸ p ¸ ¸ n n¡1 À j p } np q}p À} }p ∇ TΩ P3Q f ∇ f 1,pp q ∇ TΩPλ LppΩq f W n,ppΩq. LppQq W Ω QPW j n |λ| j Note that the constants depend on the diameter of Ω, ε, n, p and d (the extension norm in [Jon81] and, as a consequence, the Sobolev embedding constant depend on all this parameters). This estimate, together with (2.2), implies (2.1). 36 CHAPTER 2. T(P) THEOREMS

2.2 Fractional Sobolev spaces on uniform domains

In this section we study the counterpart of the T pP q-theorem to the Sobolev spaces with fractional smoothness 0 s 1, obtaining a T p1q-theorem (Theorem 2.8). The methods we use are valid for s s any Triebel-Lizorkin space Fp,q as long as the operator T is bounded on Fp,q. This boundedness is granted in some situations by the Calderón-Zygmund decomposition and interpolation, but in some cases it is not that easy. On the other hand, the smoothness hypothesis (1.41) assumed when n P N can be relaxed. For these reasons we adapt the definition of admissible Calderón-Zygmund operator to this setting.

Definition 2.4. Let 1 p, q 8. We say that an operator T : S Ñ S1 is a p, q-admissible convolution Calderón-Zygmund operator of order s P p0, 1q (or simply admissible when q 2) with 0 kernel K if T satisfies Definition 1.24, T also extends to a bounded operator in Fp,q and K satisfies the smoothness condition C |y|s |Kpx yq ¡ Kpxq| ¤ K for every 0 2|y| ¤ |x|. (2.12) |x|d s

Remark 2.5. Note that the smoothness condition (2.12) implies the Hörmandercondition (1.38). Moreover, being of order s implies being of order σ for every σ s. The Fourier transform of a p, q-admissible convolution Calderón-Zygmundoperator T is a 0 r 8 Fourier multiplier for Fp,q, and for L with 1 r as well by Remark 1.25. We refer the reader to [Tri83, Section 2.6] for a rigorous definition of multiplier and for the results on Fourier multipliers that we sum up next as well. 0 s Being a Fourier multiplier for Fp,q implies being a Fourier multiplier also for Fp,q for every¡ s©, 0 0 1 1 for Fp,p and for Fp1,q1 , and the property is stable under interpolation (i.e., the set of indices p , q 0 such that a T is bounded on Fp,q is a convex set, see Figure 2.2). 0 Therefore, the fact of T being bounded on Fp,q in Definition 2.4 is a consequence of being a 1 1 1 1 Calderón-Zygmundoperator when 0 2p q 2p 2 1 (see Figure 2.2). Example 2.6. The Beurling transform (1.1) is an admissible convolution Calderón-Zygmund operator of any order and, therefore, it is a p, q-admissible convolution Calderón-Zygmundoperator 1 1 1 1 for all indices p and q satisfying 0 2p q 2p 2 1.

Deﬁnition 2.7. Let U Rd be a open set, 1 p 8, 1 q 8 and 0 s 8. Then for every measurable function f : U Ñ C we deﬁne

}f} s p q : inf }g} s p dq. Fp,q U P s p dq | Fp,q R g Fp,q R : g U f

Theorem 2.8. Let Ω be a bounded uniform domain, T a p, q-admissible convolution Calder´on- 8 8 ¡ d Zygmund operator of order 0 s 1, 1 p , 1 q with s p . Then

s }TΩ1} s p q 8 ðñ TΩ is bounded on F pΩq. Fp,q Ω p,q

Furthermore,

}TΩ} s p qÑ s p q À CK }T } s Ñ s }T } pÑ p }T } q Ñ q }TΩ1} s p q, Fp,q Ω Fp,q Ω Fp,q Fp,q L L L L Fp,q Ω with constants independent of T . 2.2. FRACTIONAL SOBOLEV SPACES ON UNIFORM DOMAINS 37

1 1 s Figure 2.2: Indices pr, qr such that a p, q-admissible operator is bounded on Fp,r qr.

1 1 qr qr 1 1 1 1 0 Fp1,q1

0 Fp1,q1

1 1 2 1 1 2 1 1 Lp Lq L2 Lq Lp Lq Lp L2 Lp Lq 0 Fp,q

1 1 1 0 1 8 pr 8 Fp,q pr

1 1 1 1 1 1 8 2 1 8 2 1

1 1 1 1 (a) The case 0 2p q 2p 2 1. (b) The complementary situation.

Similarly to Section 2.1, we will use a lemma which says that it is equivalent to bound the transform of a function and its approximation by constants on Whitney cubes. In the fractional case 0 s 1, however, it is not clear what the ∇s-gradient is.

Deﬁnition 2.9. Given a uniform domain Ω with Whitney covering W and f P LppΩq for certain values 0 s 1 and 1 q 8, the s-th (dyadic) fractional gradient of index q of f in a point x P Q P W is £» 1 |fpxq ¡ fpyq|q q ∇sfpxq : dy . q | ¡ |sq d ShpQq x y

In Sections 2.3 and 2.4 we will prove the following remarkable characterizations in terms of diﬀerences.

Theorem 2.10 (see Corollary 2.12, Corollary 2.17 and Lemma 2.18). Let 1 ¤ p 8, 1 ¤ q ¤ 8 ¡ d ¡ d and 0 s 1 with s p q . Then $ £ , ¢ p 1 & » » p . | p q ¡ p q|q q s p dq P maxtp,qu } } f x f y 8 Fp,q R f L : f Lp sq d dy dx , (2.13) % d d |x ¡ y| - R R

(with the usual modiﬁcation for q 8), in the sense of equivalent norms. Let Ω be a bounded uniform domain with an admissible Whitney covering W. Given 1 p 8, 8 ¡ d ¡ d 1 q and 0 s 1 with s p q , we have that £ ¢ p 1 » » p | p q ¡ p q|q q } } } } f x f y } } s f s p q f pp q dy dx f pp q ∇qf p . (2.14) Fp,q Ω L Ω | ¡ |sq d L Ω L pΩq Ω Ω x y 38 CHAPTER 2. T(P) THEOREMS

Key Lemma 2.11. Let Ω be a uniform domain with Whitney covering W, let T be a p, q-admissible convolution Calder´on-Zygmundoperator of order 0 s 1, 1 p 8 and 1 q 8 with ¡ d ¡ d s p q . The following statements are equivalent: P s p q i) For every f Fp,q Ω one has

}TΩf} s p q ¤ C}f} s p q, Fp,q Ω Fp,q Ω with C independent from f. P s p q ii) For every f Fp,q Ω one has ¸ | |p s p ¤ } }p fQ ∇ T χΩ p C f s p q, q L pQq Fp,q Ω QPW

with C independent from f. Moreover, ¡ © ¸ p s p ¡ qp À } } } } } } } }p ∇ TΩ f fQ p CK T s Ñ s T pÑ p T q Ñ q f s p q. (2.15) q L pQq Fp,q Fp,q L L L L Fp,q Ω QPW

Proof. Let Ω be an ε-uniform domain. The core of the proof is showing that (2.15) holds. Once this is settled, since we have that ¸ ¸ ¸ p p p ∇sT f À ∇sT pf ¡ f q |f |p∇sT 1 , q Ω LppQq p q Ω Q LppQq Q q Ω LppQq QPW QPW QPW and ¸ ¸ ¸ p p p |f |p∇sT 1 À ∇sT pf ¡ fq ∇sT f , Q q Ω LppQq p q Ω Q LppQq q Ω LppQq QPW QPW QPW inequality (2.15) proves that ¸ ¸ s p À} }p ðñ | |p s p À} }p ∇ TΩf p f s p q fQ ∇ TΩ1 p f s p q. q L pQq Fp,q Ω q L pQq Fp,q Ω QPW QPW

p } } À} } On the other hand, by assumption °T is bounded³ on L and we have that TΩf LppΩq f LppΩq. } }p } }p | s p q|p Since TΩf s p q TΩf pp q P ∇ TΩf x dx by (2.14), the lemma follows. Fp,q Ω L Ω Q W Q q Again we use duality. That is, to prove (2.15) it suﬃces to prove that given a positive function 1 1 P p p q p qq } } 1 1 g L L Ω with g Lp pLq pΩqq 1, we have that » » ¸ | p ¡ q p q ¡ p ¡ q p q| TΩ f fQ x TΩ f fQ y p q À } } d g x, y dy dx f F s pΩq. p q s q p,q Q Q Sh Q |x ¡ y|

Given a cube Q P W, we can define a bump function ϕQ such that χ6Q ¤ ϕQ ¤ χ7Q and } } ¤ p q¡1 ∇ϕQ L8 C` Q . Given a cube S 5Q we define ϕQS : ϕQ. Otherwise, take ϕQS : ϕS. Note that in both situations, by (1.7) we have that suppϕQS 23S. Then, we can express the difference between TΩpf ¡ fQq evaluated at x P Q and in y P S as

TΩpf ¡ fQqpxq ¡ TΩpf ¡ fQqpyq TΩ rpf ¡ fQq ϕQs pxq ¡ TΩ rpf ¡ fQq ϕQSs pyq (2.16)

TΩ rpf ¡ fQq p1 ¡ ϕQqs pxq ¡ TΩ rpf ¡ fQq p1 ¡ ϕQSqs pyq. 2.2. FRACTIONAL SOBOLEV SPACES ON UNIFORM DOMAINS 39

Note that the ﬁrst two terms in the right-hand side of (2.16) are ‘local’ terms in the sense that the functions to which we apply the operator TΩ are supported in a small neighborhood of the s point of evaluation (and are globally Fp,q, as we will check later on) and the other two terms are ‘non-local’. What we will prove is that the local part » » ¸ ¸ |T rpf ¡ f q ϕ s pxq ¡ T rpf ¡ f q ϕ s pyq| 1 : Ω Q Q Ω Q QS gpx, yq dy dx, s d | ¡ | q Q Q SPSHpQq S x y and the non-local part » » ¸ ¸ |T rpf ¡ f q p1 ¡ ϕ qs pxq ¡ T rpf ¡ f q p1 ¡ ϕ qs pyq| 2 : Ω Q Q Ω Q QS gpx, yq dy dx, s d | ¡ | q Q Q SPSHpQq S x y are both bounded as

1 2 ¤ C}f} s p q. (2.17) Fp,q Ω

We begin by the local part, that is, we want to prove that 1 À}f} s p q. Note that for x P Q Fp,q Ω and y P S P SHpQq, if y P 3Q then ϕQS ϕQ and, otherwise |x ¡ y| `pQq. Thus, » » ¸ |T rpf ¡ f q ϕ s pxq ¡ T rpf ¡ f q ϕ s pyq| 1 ¤ Q Q Q Q gpx, yq dy dx (2.18) s d Q 3Q |x ¡ y| q Q » » ¸ |T rpf ¡ f q ϕ s pxq| Q Q gpx, yq dy dx s d Q ShpQq `pQq q Q » » ¸ ¸ |T rpf ¡ f q ϕ s pyq| Q QS gpx, yq dy dx : 1.1 1.2 1.3 . s d p q q Q Q SPSHpQq S ` Q

Of course, by H¨older’sinequality we have that

» ¢» p ¸ q q p |T rpf ¡ fQq ϕQs pxq ¡ T rpf ¡ fQq ϕQs pyq| p 1.1 ¤ dy dx}g} 1 1 . |x ¡ y|sq d Lp pLq pΩqq Q Q 3Q

By (2.13) we get ¸ p À } rp ¡ q s}p 1.1 T f fQ ϕQ s p dq. Fp,q R Q

s Now, the operator T is bounded on Fp,q by assumption (see Deﬁnition 2.4 and Remark 2.5). Thus, ¸ p À} }p }p ¡ q }p 1.1 T s Ñ s f fQ ϕQ s p dq. Fp,q Fp,q Fp,q R Q

s ¤ Consider° the characterization of the Fp,q-norm given in (2.13). Since ϕQ χ7Q, the ﬁrst term }p ¡ q }p } } f fQ ϕQ pp dq is bounded by a constant times f p by the ﬁnite overlapping of the Q L R L Whitney cubes and the Jensen inequality, and the second is

» ¢» p ¸ q q |pfpxq ¡ fQq ϕQpxq ¡ pfpyq ¡ fQq ϕQpyq| sq d dy dx, d d |x ¡ y| Q R R 40 CHAPTER 2. T(P) THEOREMS where the integrand vanishes if both x, y R 8Q. Therefore, we can write

» ¢» p ¸ q p |pfpxq ¡ f q ϕ pxq ¡ pfpyq ¡ f q ϕ pyq| q 1.1 À}f} Q Q Q Q dy dx Lp |x ¡ y|sq d Q 8Q 8Q » ¢» p ¸ |p p q ¡ q p q|q q f y fQ ϕQ y sq d dy dx (2.19) dz |x ¡ y| Q R 8Q 7Q » £» p ¸ |p p q ¡ q p q|q q f x fQ ϕQ x } } sq d dy dx : f Lp 1.1.1 1.1.2 1.1.3 , dz |x ¡ y| Q 7Q R 8Q

p where the constant depends linearly on the operator norm }T } s Ñ s . Fp,q Fp,q Adding and subtracting pfpxq ¡ fQq ϕQpyq in the numerator of the integral in 1.1.1 we get that » ¢» p ¸ q q |fpxq ¡ f | |ϕ pxq ¡ ϕ pyq| q 1.1.1 À Q Q Q dy dx |x ¡ y|sq d Q 8Q 8Q » ¢» p ¸ q |fpxq ¡ fpyq| q dy dx. |x ¡ y|sq d Q 8Q 8Q } }p The second term is bounded by a constant times f s p q, so Fp,q Ω

» ¢» p ¸ q q q }∇ϕ } 8 |x ¡ y| À Q L | p q ¡ |p } }p 1.1.1 dy f x fQ dx f F s pΩq. |x ¡ y|sq d p,q Q 8Q 8Q } } À 1 Using ∇ϕQ L8 `pQq and the local inequality for the maximal operator (1.28) we get that » ¸ | p q ¡ |p À p qp1¡sqp f x fQ } }p 1.1.1 ` Q dx f s p q (2.20) p qp Fp,q Ω 8Q ` Q Q £ » ³ p ¸ |fpxq ¡ fpξq| dξ À Q } }p dx f F s pΩq. `pQqs d p,q Q 8Q

³ ¡³ © 1 1 | p q ¡ p q| À 1 | p q ¡ p q|q q By Jensen’s inequality `pQqd Q f x f ξ dξ Q `pQqd f x f ξ dξ and, therefore, À} }p 1.1.1 f s p q. (2.21) Fp,q Ω

Now we undertake the task of bounding 1.1.2 in (2.19). Writing xQ for the center of a given cube Q, we have that ¸ » ¢» p dx q 1.1.2 À |fpyq ¡ f |q dy . dp Q dz sp q Q R 8Q |x ¡ xQ| 7Q ¡ d ¡ d dp ¡ Since s p q we have that sp q d. Thus, by (1.29)

¡ ¡ © © p ³ ³ q ¢» p q ¸ q ¸ |fpyq ¡ fpξq| dξ dy 1 q 7Q Q 1.1.2 À |fpyq ¡ fQ| dy ¤ . sp dp ¡d sp dp ¡d dp Q `pQq q 7Q Q `pQq q 2.2. FRACTIONAL SOBOLEV SPACES ON UNIFORM DOMAINS 41

By Minkowski’s inequality (1.9) we have that ¢ ³ ¡³ © 1 p | p q ¡ p q|q q ¸ Q 7Q f y f ξ dy dξ 1.1.2 À , sp dp dpp¡1q Q `pQq q ¡ p and by H¨older’sinequality, using that p 1 p1 we get that

¡ © p ³ ³ dp q q 1 » ¢» p | p q ¡ p q| p q p q ¸ f y f ξ dy dξ` Q ¸ | p q ¡ p q| q À Q 7Q À f y f ξ dy 1.1.2 dp dp sq d dξ sp 1 |y ¡ ξ| Q `pQq q p Q Q 7Q and À } }p 1.1.2 f s p q. (2.22) Fp,q Ω

Dealing with the last term in (2.19) is somewhat easier. Note that by (1.29) we have that

» £» p » ¸ q ¸ | p q ¡ |p ¤ | p q ¡ |p 1 ¤ f x fQ 1.1.3 f x fQ sq d dy dx sp dx dz |x ¡ y| `pQq Q 7Q R 8Q Q 7Q and, since this quantity is bounded by the right-hand side of (2.20), it follows that À} }p 1.1.3 f s p q. (2.23) Fp,q Ω

Summing up, by (2.19), (2.21), (2.22) and (2.23) we get

1.1 À }T } s Ñ s }f} s p q. (2.24) Fp,q Fp,q Fp,q Ω

Back to (2.18), it remains to bound 1.2 and 1.3 . Recall that » » ¸ | rp ¡ q s p q| T f fQ ϕQ x p q 1.2 d g x, y dy dx. s q p q Q Q `pQq Sh Q

p q } p ¤q} 1 Writing G x g x, Lq pΩq and using H¨older’s inequality we get

» £» 1 q1 1 1 d p q ¤ p qq | p q| q À p q p q q g x, y dy g x, y dy Sh Q ρε,d G x ` Q , ShpQq ShpQq and using again H¨older’sinequality it follows that

£ » 1 ¸ p p |T rpf ¡ fQq ϕQs pxq| 1.2 À dx }G} 1 . `pQqsp Lp pΩq Q Q

} } 1 ¤ p Of course, G Lp pΩq 1. Now, by Deﬁnition 1.24 we can use the boundedness of T in L to ﬁnd that £ 1 £ 1 ¸ }p ¡ q }p p ¸ } ¡ }p p f fQ ϕQ pp dq f fQ pp q 1.2 À }T } L R À L 7Q , LpÑLp `pQqsp `pQqsp Q Q 42 CHAPTER 2. T(P) THEOREMS and we can argue again as in (2.20) to prove that

1.2 À}T } pÑ p }f} s p q. (2.25) L L Fp,q Ω

Finally, for the last term in (2.18), that is, for » » ¸ ¸ |T rpf ¡ f q ϕ s pyq| 1.3 Q QS gpx, yq dy dx, s d p q q Q Q SPSHpQq S ` Q by H¨older’s inequality we have that

¤ 1 » » q ¸ ¸ |T rpf ¡ f q ϕ s pyq|q 1.3 ¤ ¥ Q QS dy Gpxq dx. `pQqsq d Q Q SPSHpQq S

The boundedness of T in Lq leads to

¤ 1 » q ¸ ¸ |p p q ¡ q p q|q ¤ } } ¥ f y fQ ϕQS y p qd 1.3 T Lq ÑLq sq d dy ` Q inf MG. p q `pQq Q Q SPSHpQq supp ϕQS

Given a cube Q, the finite overlapping of the family t50SuSPW (see Definition 1.5) implies the finite overlapping of the supports of the family tϕQSu (recall that supppϕQSq 23S), so there is 2p q p q a certain ratio ρ2 such that naming Sh Q : Shρ2 Q we have that

£» 1 ¸ | p q ¡ |q q À f y fQ 1.3 sq d¡dq dy inf MG 2p q `pQq Q Q Sh Q £ £ 1 » » q ¸ | p q ¡ p q| q ¤ f y f ξ d ¡ dξ dy inf MG. 2p q s q d d Q Q Sh Q Q `pQq

1.3 À}T } q Ñ q }f} s p q. (2.26) L L Fp,q Ω

Now, by (2.18), (2.24), (2.25) and (2.26) we have that ¡ ©

1 À }T } s Ñ s }T } pÑ p }T } q Ñ q }f} s p q, (2.27) Fp,q Fp,q L L L L Fp,q Ω and we have ﬁnished with the local part. Now we bound the non-local part in (2.17). Consider x P Q P W. By (1.40), since x is not in the support of pf ¡ fQq p1 ¡ ϕQq, we have that »

TΩ rpf ¡ fQq p1 ¡ ϕQqs pxq Kpx ¡ zq pfpzq ¡ fQq p1 ¡ ϕQpzqq dmpzq, Ω 2.2. FRACTIONAL SOBOLEV SPACES ON UNIFORM DOMAINS 43 and by the same token for y P S P SHpQq »

TΩ rpf ¡ fQq p1 ¡ ϕQSqs pyq Kpy ¡ zq pfpzq ¡ fQq p1 ¡ ϕQSpzqq dmpzq. Ω To shorten the notation, we will write

λQSpz1, z2q Kpz1 ¡ z2q pfpz2q ¡ fQq p1 ¡ ϕQSpz2qq , for z1 z2. Then we have that §» § § § § § § § § § §TΩ rpf ¡ fQq p1 ¡ ϕQqs pxq ¡ TΩ rpf ¡ fQq p1 ¡ ϕQSqs pyq§ § pλQQpx, zq ¡ λQSpy, zqq dmpzq§ , Ω that is, §³ § ¸ » ¸ » § § pλQQpx, zq ¡ λQSpy, zqq dz 2 Ω gpx, yq dy dx. s d | ¡ | q Q Q SPSHpQq S x y 3p q p q p q 3p q p q For ρ3 big enough, Sh Q : Shρ3 Q SPSHpQq Sh S (call SH Q : SHρ3 Q ), we can decompose ³ ¸ » ¸ » | p q ¡ p q| 3p q λQQ x, z λQS y, z dz 2 ¤ Sh Q gpx, yq dy dx (2.28) s d Q p qz S |x ¡ y| q Q S Sh Q 2Q ³ ¸ » ¸ » | p q ¡ p q| z 3p q λQQ x, z λQS y, z dz Ω Sh Q gpx, yq dy dx s d Q p qz S |x ¡ y| q Q S Sh³ Q 2Q ¸ » » |λQQpx, zq ¡ λQQpy, zq| dz Ω gpx, yq dy dx : A B C . s d Q Q 5Q |x ¡ y| q In the ﬁrst term in the right-hand side of (2.28) the variable z is ‘close’ to either x or y, so smoothness does not help. Thus, we will take absolute values, giving rise to two terms separating λQQ and λQS. That is, we use that ³ ¸ » ¸ » p| p q| | p q|q 3p q λQQ x, z λQS y, z dz A ¤ Sh Q gpx, yq dy dx. s d | ¡ | q Q Q SShpQqz2Q S x y

Using the size condition (1.37), |fpzq ¡ f | |λ px, zq| ¤ C Q |1 ¡ ϕ pzq| QQ K |x ¡ z|d Q and |fpzq ¡ f | |λ py, zq| ¤ C Q |1 ¡ ϕ pzq|. QS K |y ¡ z|d QS Summing up, ¸ » » » |fpzq ¡ fQ| |1 ¡ ϕQpzq| dz A ÀC gpx, yq dy dx (2.29) K s d Q ShpQqz2Q Sh3pQq |x ¡ y| q |x ¡ z|d Q» » » ¸ | p q ¡ | | ¡ p q| f z fQ 1 ϕQS z dz p q d g x, y dy dx : 2.1 2.2 , p qz 3p q s q d Q Q Sh Q 2Q Sh Q |x ¡ y| |y ¡ z| 44 CHAPTER 2. T(P) THEOREMS

with constants depending linearly on the Calder´on-Zygmund constant CK . We begin by the shorter part, that is » » » ¸ | p q ¡ | | ¡ p q| f z fQ 1 ϕQ z dz p q 2.1 d g x, y dy dx. p qz 3p q s q d Q Q Sh Q 2Q Sh Q |x ¡ y| |x ¡ z|

Using the fact that 1 ¡ ϕQpzq 0 when z is close to the cube Q, we can say that ¸ » » » À 1 | p q ¡ | p q 2.1 d f z fQ g x, y dy dx dz. s q d 3p qz p qz Q `pQq Sh Q 6Q Q Sh Q 2Q Now, by the H¨olderinequality we have that » d p q À p q p q q g x, y dy ρε,d G x ` Q , ShpQqz2Q

p q } p ¤q} 1 where G x g x, Lq . Thus, » » » » ¸ | p q ¡ | ¸ | p q ¡ p q| À f z fQ p q À f z f ξ p q 2.1 s d G x dx dz s d MG ξ dz dξ. 3p q `pQq 3p q `pQq Q Sh Q Q Q Q Sh Q

1 Finally, by Jensen’s inequality and the boundedness of the maximal operator in Lp we have that » » » £» 1 ¸ | p q ¡ p q| ¸ | p q ¡ p q|q q f z f ξ p q À f z f ξ p q s d MG ξ dz dξ sq d dz MG ξ dξ (2.30) 3p q `pQq 3p q `pQq Q Q Sh Q Q Q Sh Q £ p 1 » ¢» p |fpzq ¡ fpξq|q q À dz dξ }MG} 1 , | ¡ |sq d Lp Ω Ω z ξ that is,

2.1 À}f} s p q. (2.31) Fp,q Ω The second term in (2.29) is the most delicate one. Given cubes Q, S and P and points y P S and z P P with 1 ¡ ϕQSpzq 0, we have that |z ¡ y| DpS, P q. Therefore, we can write » » » ¸ |fpzq ¡ f | |1 ¡ ϕ pzq| dz 2.2 Q QS gpx, yq dy dx s d Q ShpQqz2Q Sh3pQq |x ¡ y| q |y ¡ z|d Q » » » ¸ ¸ ¸ |fpzq ¡ f | dz À Q gpx, yq dy dx. s d p q q p qd Q Q SPSHpQq S P PSH3pQq P ` Q D S, P Next, we change the focus on the sum. Consider an admissible chain connecting two given 3 cubes S and P both in SH pQq. Then DpS, P q `pSP q. Of course, using (1.21) and the fact that S and P are in SH3pQq we get

DpQ, SP q À DpQ, Sq DpS, SP q DpQ, Sq DpS, P q À 2DpQ, Sq DpQ, P q À `pQq p q and, therefore, the cube SP is contained in some SHρ4 Q for a certain constant ρ4 depending on 4 4 d and ε. For short, we write L : SP P SH pQq and Sh pQq : Shρ pQq. Then » » » 4 ¸ ¸ ¸ ¸ | p q ¡ | À f z fQ dz p q 2.2 d g x, y dy dx s d Q P 4p q P p q S P p q P `pQq q `pLq Q L SH »Q S SH L » P SH L » ¸ 1 ¸ 1 |fpzq ¡ fQ| dz gpx, yq dy dx. (2.32) s d d p q q p q `pLq p q Q ` Q Q LPSH4pQq Sh L Sh L 2.2. FRACTIONAL SOBOLEV SPACES ON UNIFORM DOMAINS 45

If we write gxpyq gpx, yq, we have that for any cube L the integral » d gpx, yq dy ¤ `pLq inf Mgx. ShpLq L

P 4p q p q p q Arguing as before, for ρ5 big enough we have that if L SH Q , then Sh L Shρ5 Q : Sh5pQq and therefore » » » | p q ¡ | | p q ¡ | p q À rp ¡ q sp q f z fQ dz f z fQ χSh5pQq z dz M f fQ χSh5pQq ξ dξ. ShpLq ShpLq L

Back to (2.32) we have that » » ¸ 1 ¸ 2.2 À Mrpf ¡ fQqχ 5p qspξqMgxpξq dξ dx s d Sh Q `pQq q Q P 4p q L Q » »L SH Q ¸ 1 rp ¡ q 5 sp q p q d M f fQ χSh pQq ξ Mgx ξ dξ dx s q 4p q Q `pQq Q Sh Q

1 and, by H¨older’sinequality and the boundedness of the maximal operator in Lq and Lq , we have that » £» 1 £» 1 ¸ q q1 1 q q1 À rp ¡ q 5 sp q p q 2.2 d M f fQ χSh pQq ξ dξ Mgx ξ dξ dx s q 4p q 4p q Q `pQq Q Sh Q Sh Q £ 1 ¢ » » q » 1 ¸ q1 À 1 | p q ¡ |q p qq1 q d f ξ fQ dξ g x, ξ dξ dx. s q 5p q Q `pQq Q Sh Q Ω

p q } p ¤q} 1 Again, we write G x g x, Lq and by Minkowski’s integral inequality (1.9) we get that £» ¢» 1 » ¸ q q À 1 | p q ¡ p q| p q 2.2 d f ξ f ζ dζ dξ G x dx s q d 5p q Q `pQq Sh Q Q Q » £» 1 ¸ q À 1 | p q ¡ p q|q p q d f ξ f ζ dξ MG ζ dζ. s q 5p q Q `pQq Q Sh Q

Thus, £ p 1 » ¢» p |fpξq ¡ fpζq|q q 2.2 À dξ dζ }MG} p1 À}f} s p q. (2.33) | ¡ |sq d L Fp,q Ω Ω Ω ξ ζ

Back to (2.28), it remains to bound B and C . For the ﬁrst one, ³ ¸ » ¸ » | p q ¡ p q| z 3p q λQQ x, z λQS y, z dz B Ω Sh Q gpx, yq dy dx, s d | ¡ | q Q Q SShpQqz2Q S x y

3 just note that if x P Q, y P S P SHpQq and z R Sh pQq we have that ϕQQpzq ϕQSpzq 0 and, if ρ3 is big enough, |x ¡ z| ¡ 2|x ¡ y|. Thus, we can use the smoothness condition (2.12), that is, | p q¡ || ¡ |s | p q ¡ p q| ¤ | p ¡ q ¡ p ¡ q| | p q ¡ | ¤ f z fQ x y λQQ x, z λQS y, z K x z K y z f z fQ CK |x¡z|d s . 46 CHAPTER 2. T(P) THEOREMS

In the last term in (2.28), ³ ¸ » » |λQQpx, zq ¡ λQQpy, zq| dz C Ω gpx, yq dy dx, s d Q Q 5Q |x ¡ y| q we are integrating in the region where x P Q, y P 5Q and z R 6Q because otherwise 1 ¡ ϕQpzq would vanish. Also |x ¡ z| ¡ Cd|x ¡ y| and |x ¡ z| |y ¡ z|. Thus, we have again that |λQQpx, zq ¡ | p q¡ || ¡ |s p q| ¤ | p ¡ q ¡ p ¡ q| | p q ¡ | À f z fQ x y λQQ y, z K x z K y z f z fQ CK |x¡z|d s by (2.12) and (1.37) (one may use the last one when 2|x ¡ y| ¥ |x ¡ z| ¡ Cd|x ¡ y|, that is |x ¡ y| |x ¡ z| |y ¡ z|). Summing up, » » » ¸ | p q ¡ || ¡ |s À f z fQ x y dz p q B C CK d g x, y dy dx : 2.3 . (2.34) p q z s q d s Q Q Sh Q Ω 6Q |x ¡ y| |x ¡ z| with constants depending linearly on the Calder´on-Zygmund constant CK . Reordering, » » » ¸ | p q ¡ | p q f z fQ dz g x, y dy 2.3 d s d dx. z |x ¡ z| p q q Q Q Ω 6Q Sh Q |x ¡ y|

The last integral above is easy to bound by the same techniques as before: Given x P Q P W, since d q1 q d, by (1.28), H¨older’sInequality and the boundedness of the maximal operator in L we have that » » p q g x, y dy d¡ d ¡ d À p q q ¤ p q q ¤ } } 1 À p q d ` Q inf Mgx ` Q Mgx Mgx Lq q G x . ShpQq |x ¡ y| q Q Q Thus, » » ¸ ¸ |fpzq ¡ f | dz 2.3 À Q Gpxq dx. DpP,Qqd s Q Q P P

For every pair of cubes P,Q P W, there exists an admissible chain rP,Qs and, writing rP,PQq for the subchain rP,PQsztPQu and rPQ,Qq for rPQ,QsztQu, we get » » ¸ ¸ |fpzq ¡ f | dz 2.3 À P Gpxq dx (2.35) DpP,Qqd s Q Q P P » ¸ ¸ ¸ d |fL ¡ f p q|`pP q N L Gpxq dx DpP,Qqd s Q Q P LPrP,P q » Q ¸ ¸ ¸ d |fL ¡ f p q|`pP q N L Gpxq dx : 2.3.1 2.3.2 2.3.3 . p qd s Q D P,Q Q P LPrPQ,Qq

The ﬁrst term in (2.35) can be bounded by reordering and using (1.27). Indeed, we have that » » » » » ¸ |fpzq ¡ fpξq| dξ dz ¸ Gpxq dx ¸ |fpzq ¡ fpξq| dξMGpzq dz 2.3.1 ¤ À , `pP qd DpP,Qqd s `pP qd s P P P Q Q P P P and, by (2.30) we have that

2.3.1 À}f} s p q. (2.36) Fp,q Ω 2.2. FRACTIONAL SOBOLEV SPACES ON UNIFORM DOMAINS 47

For the second term in (2.35) note that given cubes L P rP,PQs we have that DpP,Qq DpL, Qq by (1.23) and P P ShpLq by Deﬁnition 1.16. Therefore, by (1.27) we have that » » » ¸ 1 ¸ 1 ¸ 2.3.2 À |fpξq ¡ fpζq| dζ dξ Gpxq dx `pP qd `pLq2d DpL, Qqd s L L 5L Q Q P p q » » » » P SH L ¸ 1 MGpζq ¸ |fpξq ¡ fpζq|MGpζq À |fpξq ¡ fpζq| dζ dξ`pLqd dζ dξ, `pLq2d `pLqs `pLqd s L L 5L L L 5L and, again by (2.30), we have that

2.3.2 À}f} s p q. (2.37) Fp,q Ω

Finally, the last term of (2.35) can be bounded analogously: Given cubes L P rPQ,Qs we have that DpQ, P q DpL, P q by (1.23), and » » » ¸ 1 ¸ ¸ `pP qd 2.3.3 À |fpξq ¡ fpζq| dζ dξ Gpxq dx `pLq2d DpP,Lqd s L L 5L QPSHpLq Q P » » » » ¸ `pLqd¡s ¸ |fpξq ¡ fpζq|MGpζq À |fpξq ¡ fpζq|MGpζq dζ dξ dζ dξ, `pLq2d `pLqd s L L 5L L L 5L and

2.3.3 À}f} s p q. (2.38) Fp,q Ω

Now, putting together (2.28), (2.29), (2.34) and (2.35) we have that

À 2 CK 2.1 2.2 2.3.1 2.3.2 2.3.3 , and by (2.31), (2.33), (2.36), (2.37) and (2.38) we have that

2 À CK }f} s p q, (2.39) Fp,q Ω with constants depending on ε, s, p, q and d. Estimates (2.27) and (2.39) prove (2.15). ¡ d ¡ d ¡ d Proof of Theorem 2.8. Let Ω be a bounded ε-uniform domain. Note that since s p p q , we can use the Key Lemma 2.11, that is, we have that TΩ is bounded if and only if for every P s p q f Fp,q Ω we have that ¸ | |p s p ¤ } }p fQ ∇ T χΩ p C f s p q, (2.40) q L pQq Fp,q Ω QPW with C independent from f. Since sp ¡ d, by Deﬁnition 2.7 and Proposition 1.11 we have the s p q 8 | | ¤ } } ¤ continuous embedding Fp,q Ω L . Therefore, given a cube Q we have that fQ f L8pΩq s }f} s p q and (2.40) holds as long as T χΩ P F pΩq. Fp,q Ω p,q More precisely, putting together (2.15) and (2.40), we get ¡ ©

}TΩf} s p q À CK }T } s Ñ s }T } pÑ p }T } q Ñ q }TΩ1} s p q }f} s p q, Fp,q Ω Fp,q Fp,q L L L L Fp,q Ω Fp,q Ω with C depending only on ε, s, p, q and d. 48 CHAPTER 2. T(P) THEOREMS

s 2.3 Characterization of Fp,q via diﬀerences

In this section we prove part of Theorem 2.10. Namely, the estimate (2.13) is a consequence of Corollary 2.12 below, and the ﬁrst estimate in (2.14) is proven in Corollary 2.17. The second estimate in (2.14) is left for Lemma 2.18 in Section 2.4. P 1 P ¤ 8 ¡ P d For a function f Lloc, M N, 0 u , t 0 and x R , we write

£ 1 » u M p q ¡d | M p q|u dt,uf x : t ∆h f x dh , |h|¤t with the usual modiﬁcation for u 8. In [Tri06, Theorem 1.116] we ﬁnd the following result. Theorem (See [Tri06].). Given 1 ¤ r ¤ 8, 0 u ¤ r, 1 ¤ p 8, 1 ¤ q ¤ 8 and 0 s M d ¡ d with mintp,qu r s, we have that $ ¤ , £ p 1 ' p / & » » q . 1 dM fpxqq s p dq P maxtp,ru } } ¥ t,u 8 Fp,q R f L : f Lp sq 1 dt dx ' d t / % R 0 -

(with the usual modiﬁcation for q 8), in the sense of equivalent quasinorms. As an immediate consequence of this result, we get the following corollary. ¤ 8 ¤ 8 ¤ ¡ d ¡ d Corollary 2.12. Let 1 p , 1 q and 0 s 1 M with s p q . Then $ , £ p 1 & » ¢» p . |∆M fpxq|q q s p dq P maxtp,qu } } } } h 8 Fp,q R f L s.t. f As p dq : f Lp sq d dh dx % p,q R d d |h| - R R

(with the usual modiﬁcation for q 8), in the sense of equivalent norms. Proof. Let f P Lmaxtp,qu. Choosing q u r all the conditions in the theorem above are satisﬁed. Therefore, ¤ £ p 1 » » p 1 dM fpxqq q } } } } ¥ t,q f F s p dq f Lp sq 1 dt dx . (2.41) p,q R d t R 0

¡ ³ © 1 M p q ¡d | M p q|q q P d Since dt,qf x t |h|¤t ∆h f x dh for x R , we can change the order of integration to get that » £» p » £» » p 1 dM fpxqq q q t,q dt | M p q|q sq 1 dt dx sq 1 d ∆h f x dh dx d t d | |¤ ¡ ¡| | t R 0 R h 1 1 t h » £» ¢ p |∆M fpxq|q 1 q h ¡ sq d 1 dh dx. d | |¤ sq d |h| R h 1

This shows that }f} s p dq À}f} s p dq and also that Fp,q R Ap,q R » £» p » £» p |∆M fpxq|q q 1 dM fpxqq q h À t,q À} }p sq d dh dx sq 1 dt dx f F s p dq (2.42) d | | 1 |h| d t p,q R R h 2 R 0 2.3. CHARACTERIZATION OF NORMS VIA DIFFERENCES 49

³ ¡³ © p |∆M fpxq|q q p h À} } by (2.41). It remains to see that d | |¡ 1 | |sq d dh dx f s p dq. Using appropriate R h 2 h Fp,q R changes of variables and the triangle inequality, it is enough to check that

» ¢» p | p q|q q f x h À} }p I : sq d dh dx f F s p dq. (2.43) d d p1 |h|q p,q R R R

Let us assume ﬁrst that p ¥ q. Then, since the measure p1 |h|q¡psq dq dh is ﬁnite, we may apply Jensen’s inequality to the inner integral, and then Fubini to obtain » » | p q|p À f x h À} }p I sp d dh dx f Lp , d d p1 |h|q R R and (2.43) follows. d ~ ~ P d If, instead, p q, cover R with disjoint cubes Q~j Q0 `j for j Z . Fix the side-length ` p of these cubes so that their diameter is 1{3. By the subadditivity of x ÞÑ |x| q , we have that

d ¡ d ~ Since s q p , the last sum is ﬁnite and does not depend on j. By (2.42) we have that

£» p » £» p » £» p ¸ q ¸ q ¸ q I À |fpyq|q dy À |fpyq ¡ fpxq|q dy dx |fpxq|q dy dx Q Q Q Q Q ~j ~j ~j ~j ~j ~j ~j ~j À} }p f s p dq. Fp,q R

° ³ ¡³ © p | p q|q q } }p In the last step we have used that ~ f x dy dx f Lp because all the cubes have j Q~j Q~j side-length comparable to 1.

¤ 8 ¤ ¤ 8 ¡ d ¡ d Deﬁnition 2.13. Consider 1 p , 1 q and 0 s 1 with s p q . Let U be an d P s p q open set in R . We say that a measurable function f Ap,q U if

• The function f P LppUq.

• The seminorm £ ¢ p 1 » » p | p q ¡ p q|q q } } f x f y f 9 s p q : dy dx (2.44) Ap,q U | ¡ |sq d U U x y

is ﬁnite.

We deﬁne the norm } } } } } } f s p q : f pp q f 9 s p q. Ap,q U L U Ap,q U

¡ d ¡ d 8 s p dq Remark 2.14. The condition s p q ensures that the Cc -functions are in the class Ap,q R . 50 CHAPTER 2. T(P) THEOREMS

P 8p q Proof. Indeed, given a bump function ϕ Cc D , £ ¢ p 1 » » p | p q ¡ p q|q q } } ¥ ϕ x ϕ y ϕ As p dq sq d dy dx p,q R p qc |x ¡ y| 2D D £ ¢ p 1 » » p q 1 |ϕpyq|q dy dx dp p qc sp q 2D D |x| d d which is finite if and only if p s q . The converse implication is an exercise. s p q s p q In some situations, the classical Besov spaces Bp,p U Ap,p U and the fractional Sobolev s,pp q s p q s p q s,pp q spaces W U Ap,2 U . For instance, when Ω is a Lipschitz domain then Ap,2 Ω W Ω (see [Str67]). We will see that this is a property of all the uniform domains. Consider a given ε-uniform domain Ω. In [Jon81] Peter Jones defines an extension operator 1,pp q Ñ 1,pp dq 8 | | Λ0 : W Ω W R for 1 p , that is, a bounded operator such that Λ0f Ω f Ω for every f P W 1,ppΩq. This extension operator is used to prove that the intrinsic characterization of W 1,ppΩq and the infimum norms coincide, that is, } } } } } } inf g W 1,pp dq f LppΩq ∇f LppΩq. g:g|Ωf R s p q Next we will see that the same operator is an extension operator for Ap,q Ω for 0 s 1 ¡ d ¡ d with s p q . To define it we need a Whitney covering W1 of Ω (see Definition 1.5), a Whitney c covering W2 of Ω and we define W3 to be the collection of cubes in W2 with side-lengths small enough, so that for any Q P W3 there is a S P W1 with DpQ, Sq ¤ C`pQq and `pQq `pSq (see [Jon81, Lemma 2.4]). We define the symmetrized cube Q¦ as one of the cubes satisfying these properties. Note that the number of possible choices for Q¦ is uniformly bounded and, if Ω is an unbounded uniform domain, then W2 W3. (2.45)

Lemma 2.15 (see [Jon81]). For cubes Q1,Q2 P W3 and S P W1 we have that • The symmetrized cubes have ﬁnite overlapping: there exists a constant C depending on the ¦ parameters ε and d such that #tQ P W3 : Q Su ¤ C. • The long distance is invariant in the following sense: p ¦ ¦q p q p ¦ q p q D Q1 ,Q2 D Q1,Q2 and D Q1 ,S D Q1,S (2.46)

X H p ¦ ¦q p q • In particular, if Q1 2Q2 (Q1 and Q2 are neighbors by (1.7)), then D Q1 ,Q2 ` Q1 . t u We( deﬁne the family of bump° functions ψQ QPW2 to be a partition of the unity associated to 11 Q , that is, their sum is ψQ 1, they satisfy the pointwise inequalities 0 ¤ ψQ ¤ χ 11 10 QPW2 10 Q } } À 1 and ∇ψQ 8 `pQq . We can deﬁne the operator ¸ 1 p q p q ¦ P p q Λ0f x ψQ x fQ for any f Lloc Ω QPW3

(recall that fU stands for the mean of a function f in a set U). 8 ¡ d ¡ d Lemma 2.16. Let Ω be a uniform domain, let 1 p, q and 0 s 1 with s p q . s p q Ñ s p dq P maxtp,qu Then, Λ0 : Ap,q Ω Ap,q R is an extension operator. Furthermore, Λ0f L for every P s p q f Ap,q Ω . 2.3. CHARACTERIZATION OF NORMS VIA DIFFERENCES 51

Proof. We have to check that £ ¢ p 1 » » p | p q ¡ p q|q q } } } } Λ0f x Λ0f y À} } Λ0f As p dq Λ0f Lp sq d dy dx f As pΩq. p,q R d d |x ¡ y| p,q R R } } ¤ } } } } First, note that Λ0f Lp f LppΩq Λ0f LppΩcq. By Jensen’s inequality, we have that ¢ ¸ ¸ d p p p 1 p 11 }Λ f} À |f ¦ | }ψ } ¤ }f} ¦ `pQq . 0 LppΩcq p Q Q Lp `pQqd LppQ q 10 QPW3 QPW3

By the ﬁnite overlapping of the symmetrized cubes,

} }p À } }p Λ0f LppΩcq f LppΩq.

The same can be said about Lq when q ¡ p. In that case, moreover, one can cover Ω with balls p tBjujPJ with radius one such that |Bj X Ω| 1. Then, using the subadditivity of x ÞÑ |x| q we get

£ » p ¸ q } }p ¤ | p q|q f Lq pΩq f y dy (2.47) B XΩ j ¤ j £» p £» p ¸ q q ¥ q q Àq |fpyq ¡ fpxq| dy dx |fpxq| dy dx X X X X j Bj Ω Bj Ω Bj Ω Bj Ω » ¢» p | p q ¡ p q|q q À f x f y } }p } }p dy dx f pp q f s p q, | ¡ |sq d L Ω Ap,q Ω Ω Ω x y by Deﬁnition 2.13. It remains to check that £ ¢ p 1 » » p q q |Λ0fpxq ¡ Λ0fpyq| } } 9 À} } Λ0f As p dq sq d dy dx f As pΩq. p,q R d d |x ¡ y| p,q R R

More precisely, we will prove that

À} }p a b c f s p q, Ap,q Ω where

» ¢» p » ¢» p |fpxq ¡ Λ fpyq|q q |Λ fpxq ¡ fpyq|q q a : 0 dy dx, b : 0 dy dx and | ¡ |sq d | ¡ |sq d Ω Ωc x y Ωc Ω x y » ¢» p |Λ fpxq ¡ Λ fpyq|q q c : 0 0 dy dx. | ¡ |sq d Ωc Ωc x y

Let us begin with

» ¢» ° p q q |fpxq ¡ P ψSpyqfS¦ | a S W3 dy dx. | ¡ |sq d Ω Ωc x y 52 CHAPTER 2. T(P) THEOREMS

t P u P 11 P Call W°4 : S W3 : all the neighbors of S are° in W3 . Given y 10 S, where S W4, we have that ψ pyq 1 and, otherwise 0 ¤ 1 ¡ ψ pyq ¤ 1. Thus P PW3 P P PW3 P » £ » p ¸ ¸ q q |fpxq ¡ f ¦ | a À S ψ pyq dy dx p qsq d S Q D Q, S 11 S QPW1 SPW3 10 ¤ § ° ¨ § p ¸ » ¸ » § §q q 1 ¡ P ψP pyq fpxq ¥ P W3 dy dx : a1 a2 . p qsq d Q S D Q, S QPW1 SPW2zW4 ³ ¦ d In a1 by the choice of the symmetrized cube we have that 11 ψSpyq dy `pS q . Jensen’s ³ 10 S q | p q ¡ ¦ | ¤ 1 | p q ¡ p q|q inequality implies that f x fS `pS¦qd S¦ f x f ξ dξ. By (2.46) and the ﬁnite overlapping of the symmetrized cubes, we get that » £ » p ¸ ¸ |fpxq ¡ fpξq|q q a1 À dξ dx À}f}p . p ¦qsq d A9 s pΩq Q S¦ D Q, S p,q QPW1 SPW3

To bound a2 just note that for Q P W1 and S P W2zW4, we have that S is far from the boundary, say `pSq ¥ `0, where `0 depends only on diampΩq and ε and, if Ω is unbounded, then

`0 8 and a2 0 by (2.45). Thus, we have that

¤ p ¤ p » » q q ¸ ¸ |fpxq|q ¸ `pSqd a2 À ¥ dy dx À ¥ }f}p . p qsq d p qsq d Lp Q 11 S D Q, S D Ω,S QPW1 SPW2zW4 10 SPW2zW4 Recall that Whitney cubes have side-length equivalent to their distance to BΩ. Moreover, the number° of cubes of a given side-length bigger than `0 is uniformly bounded when Ω is bounded, `pSqd so is a geometric sum. Therefore, SPW2zW4 `pSqsq d

¤ p ¸ q ¥ 1 p ¡sp p a2 À }f} ¤ C p q` }f} . `pSqsq Lp ε,diam Ω 0 Lp SPW2zW4 Next, note that, using the same decomposition as above, we have that » ¢» ° p q q | P ψQpxqfQ¦ ¡ fpyq| b Q W3 dy dx | ¡ |sq d Ωc Ω x y » £ » § § p ¸ ¸ § §q q f ¦ ¡ fpyq À ψ pxqp dx Q dy Q p qsq d 11 Q S D Q, S QPW3 10 SPW1 £ £ p » p » ¸ ¸ ¸ |fpyq|q q 1 ¡ ψ pxq dx dy : b1 b2 . Q p qsq d P S D P,S P PW2zW4 QPW3 SPW1 We have that ¡ © p ¤ ³ q ¸ ¸ » 1 | p q ¡ p q| q `pQqd Q¦ f ξ f y dξ b1 À `pQqd ¥ dy p ¦ qsq d S D Q ,S QPW3 SPW1 2.3. CHARACTERIZATION OF NORMS VIA DIFFERENCES 53 and, thus, by Minkowsky’s integral inequality, we have that ¤ » £ » 1 p ¸ `pQqd ¸ |fpξq ¡ fpyq|q q b1 À ¥ dy dξ . p qdp | ¡ |sq d ` Q Q¦ S ξ y QPW3 SPW1

By H¨older’sinequality and the ﬁnite overlapping of symmetrized cubes, we get that

» ¢» p » ¢» p ¸ q q 1 |fpξq ¡ fpyq| q dp |fpξq ¡ fpyq| q b1 À dy dξ`pQq p1 À dy dξ, dpp¡1q | ¡ |sq d | ¡ |sq d `pQq Q¦ Ω ξ y Ω Ω ξ y QPW3 that is, À} }p b1 f 9 s p q. Ap,q Ω

To bound b2 , note that as before, if Ω is unbounded then b2 0 and, otherwise, we have that

£ » p ¸ ¸ | p q|q q ¸ p qd p qd f y À} }p ` Q b2 ` Q dy f q p q . p qsq d L Ω sp dp S D Q, Ω p q q QPW2zW4 SPW1 QPW2zW4 dist Q, Ω

¡ d ¡ d dp ¡ Now, since s p q we have that sp q d. Therefore,

¸ p qd ¸ ¡ ¡ dp ` Q 1 d sp q ¤ Cε,diampΩq` . sp dp sp dp ¡d 0 p q q p q q QPW2zW4 dist Q, Ω QPW2zW4 ` Q

¤ } } À} } On the other hand, if Ω is bounded and q p then f Lq pΩq f LppΩq by the H¨olderinequality and, if p q then }f} q p q À}f} s p q by (2.47). L Ω Ap,q Ω Let us focus on c . We have that

» ¢» ° ° p q q | P ψP pxqfP ¦ ¡ P ψSpyqfS¦ | c P W3 S W3 dy dx. | ¡ |sq d Ωc Ωc x y

P 11 P P c X p `0 q Given x 10 Q where Q W4 and y Ω °B x, 10 , then neither°x nor y are in the support of any bump function of a cube in W zW , so ψ pyq 1 and ψ pxq 1. Therefore 2 3 P PW3 P P PW3 P ¸ ¸ ¸ ¸ ψP pxqfP ¦ ¡ ψSpyqfS¦ ψP pxqψSpyq pfP ¦ ¡ fS¦ q .

P PW3 SPW3 P X2QH SPW3 ¨ P 1 p q If, moreover, y B x, 10 ` Q , since the points are ‘close’ to each other, we will use the H¨older regularity of the bump functions, so we write ¸ ¸ ¸ ψP pxqfP ¦ ¡ ψSpyqfS¦ pψP pxq ¡ ψP pyqq fP ¦ .

P PW3 SPW3 P PW3 ¨ ¨ P z P 1 p q P `0 This decomposition is still valid if Q W2 W4 and y B x, 10 ` Q , that is, y B x, 10 , but we will treat this case apart since we lose the cancellation of the sums of bump functions but we gain a uniform lower bound on the side-lengths of the cubes involved. Finally, we will group the 54 CHAPTER 2. T(P) THEOREMS

À} }p c1 f 9 s p q. Ap,q Ω

Also combining the arguments used to bound a2 and b2 we get that if Ω is bounded, then ¡ © p À } } } } c4 f LppΩq f Lq pΩq , and it vanishes otherwise. The novelty comes from the fact that we are integrating in Ωc both terms in c , so the variables in the integrals c2 and c3 can get as close as one can imagine. Here we need to use the smoothness of the bump functions, but also the smoothness of f itself. The trick for c2 is ° 11 to use that tψQu is a partition of the unity with ψQ supported in Q, that is, ψSpxq ° 10 SPW3 p q P 11 P SX2QH ψS x 1 if x 10 Q with Q W4. Thus,

» £» ° ¨ p ¸ q q | X H pψSpxq ¡ ψSpyqq fS¦ ¡ fQ¦ | c2 S 2Q dy dx, | ¡ |sq d Q Bpx, 1 `pQqq x y QPW4 10

} } À 1 and using the fact that ∇ψQ 8 `pQq and (1.28), we have that

» £ » p ¸ ¸ § § q q § §q |x ¡ y| 1 c2 À f ¦ ¡ f ¦ dy dx q S Q p qq | ¡ |sq d Q Bpx, 1 `pQqq ` Q x y QPW4 SX2QH 10 £° § § p £ § § p ¸ § §q q ¸ ¸ § §q q X H fS¦ ¡ fQ¦ f ¦ ¡ f ¦ À `pQqd S 2Q `pQqd S Q , s `pQqsq DpQ¦,S¦qsq QPW4 QPW4 SX2QH which can be bounded as c1 . 2.4. EQUIVALENT NORMS WITH REDUCTION OF THE INTEGRATION DOMAIN. 55

Finally, we bound the error term c3 , assuming Ω to be a bounded domain. Here we cannot use the cancellation of the partition of the unity anymore. Instead, we will use the Lp norm of f, the H¨olderregularity of the bump functions and the fact that all the cubes considered are roughly of the same size: » £» ° p q q ¸ | pψ pxq ¡ ψ pyqq f ¦ | SX2QH S S S c3 dy dx ` | ¡ |sq d Q Bpx, 0 q x y QPW2zW4 10 » £» p ¸ ¸ q p 1 1 À |fS¦ | q dy dx ` | ¡ |ps¡1qq d Q Bpx, 0 q `0 x y QPW2 SPW3 10 ¤ p q¤ SX2QH `0 ` Q 2`0¸ À } }p À} }p ε,`0,q,p f LppS¦q f LppΩq. SPW3 `0 ¤ p q¤ 2 ` S `0

Corollary 2.17. Let Ω be a uniform domain with an admissible Whitney covering W. Given 8 8 ¡ d ¡ d s p q s p q 1 p , 1 q and 0 s 1 with s p q , we have that Ap,q Ω Fp,q Ω , and

s }f} s p q }f} s p q for all f P F pΩq. Fp,q Ω Ap,q Ω p,q P s p q Proof. By Corollary 2.12, given f Fp,q Ω we have that

}f} s p q ¤ inf }g} s p dq inf }g} s p dq }f} s p q. Ap,q Ω maxtp,qu Ap,q R Fp,q R Fp,q Ω gPL :g|Ωf g:g|Ωf

P s p q By the Lemma 2.16 we have the converse: for every f Ap,q Ω we have that

}f} s p q inf }g} s p dq ¤ }Λ0f} s p dq }Λ0f} s p dq ¤ C}f} s p q. Fp,q Ω Fp,q Fp,q Ap,q Ap,q Ω g:g|Ωf R R R

2.4 Equivalent norms with reduction of the integration domain.

s p q Next we present an equivalent norm for Fp,q Ω in terms of diﬀerences but reducing the domain }¤} of integration of the inner variable to the shadow of the outer variable in the seminorm 9 s p q Ap,q Ω deﬁned in (2.44). Lemma 2.18. Let Ω be a uniform domain with an admissible Whitney covering W, let 1 p, q 8 ¡ d ¡ d P s p q and 0 s 1 with s p q . Then, f Fp,q Ω if and only if ¤ £ p 1 » » p ¸ q q ¥ |fpxq ¡ fpyq| } } r } } 8 f As pΩq f LppΩq sq d dy dx . (2.48) p,q p q |x ¡ y| QPW Q Sh Q

} }p P This quantity deﬁnes a norm which is equivalent to f s p q and, moreover, we have that f Fp,q Ω Lmaxtp,qupΩq. 56 CHAPTER 2. T(P) THEOREMS

Proof. Let Ω be an ε-uniform domain. Recall that in (2.44) we deﬁned £ ¢ p 1 » » p | p q ¡ p q|q q } } f x f y f 9 s p q dy dx . Ap,q Ω | ¡ |sq d Ω Ω x y Trivially » £» p ¸ | p q ¡ p q|q q } }p Á f x f y f 9 s p q sq d dy dx. Ap,q Ω p q |x ¡ y| QPW Q Sh Q Next, we will use the seminorm in the duality form » » | p q ¡ p q| } } f x f y p q f 9 s p q sup d g x, y dy dx. (2.49) Ap,q Ω s }g} 1 1 ¤1 Ω Ω | ¡ | q Lp pLq pΩqq x y

¡ 1 } } 1 1 ¤ Let g 0 be an Lloc function with g Lp pLq pΩqq 1. Since the shadow of every cube Q contains 2Q, we just use Hölder’sinequality to find that £ ¢ p 1 » » » » p ¸ | p q ¡ p q| ¸ | p q ¡ p q|q q f x f y p q ¤ f x f y d g x, y dy dx sq d dy dx . (2.50) s q |x ¡ y| QPW Q 2Q |x ¡ y| QPW Q 2Q Therefore, we only need to prove the estimate ¤ £ p 1 » » » » p ¸ | p q ¡ p q| ¸ | p q ¡ p q|q q f x f y p q À ¥ f x f y d g x, y dy dx sq d dy dx . (2.51) z s q p q |x ¡ y| Q,S Q S 2Q |x ¡ y| QPW Q Sh Q If x P Q, y P Sz2Q, then |x ¡ y| DpQ, Sq, so we can write » » » » ¸ | p q ¡ p q| ¸ | p q ¡ p q| f x f y p q À f x f y p q d g x, y dy dx d g x, y dy dx. (2.52) z s q s q Q,S Q S 2Q |x ¡ y| Q,S Q S DpQ, Sq Since Ω is a uniform domain, for every pair of cubes Q and S in this sum, there exists an admissible r s chain Q, S joining them. Thus, writing fQ Q f dm for the mean of f in Q, the right-hand side of (2.52) can be split as follows: ffl » » » » ¸ |fpxq ¡ fpyq| ¸ |fpxq ¡ f | gpx, yq dy dx ¤ Q gpx, yq dy dx s d s d Q S DpQ, Sq q Q S DpQ, Sq q Q,S Q,S » » ¸ |f ¡ f | Q QS gpx, yq dy dx s d Q S DpQ, Sq q Q,S » » ¸ |f ¡ fpyq| QS gpx, yq dy dx s d Q,S Q S DpQ, Sq q : 1 2 3 (2.53) The first term can be immediately bounded by the Cauchy-Schwarz inequality. Namely, writing p q } p ¤q} 1 G x g x, Lq pΩq, by (1.29) we have that » £ » 1 £ 1 ¸ ¸ q1 ¸ d q 1 `pSq 1 ¤ |fpxq ¡ f | gpx, yqq dy dx Q p qsq d Q S D Q, S QPW ³ SPW SPW ¸ |fpxq ¡ fQ|Gpxq dx ¤ Q . `pQqs QPW 2.4. EQUIVALENT NORMS WITH REDUCTION OF THE INTEGRATION DOMAIN. 57

¡ ³ © 1 | p q ¡ | ¤ 1 | p q ¡ p q|q q p q Á | ¡ | By Jensen’s inequality, f x fQ `pQqd Q f x f y dy and thus, since ` Q d x y for x, y P Q, we have that £ » ¢» p 1 ¸ p |fpxq ¡ fpyq|q q 1 À dy dx }G} 1 . (2.54) |x ¡ y|sq d Lp QPW Q Q

} } 1 } } 1 1 ¤ Since G Lp g Lp pLq q 1, this ﬁnishes this part. For the second one, for all cubes Q and S we consider the subchain rQ, QSq rQ, Ss. Then » » ¸ gpx, yq ¸ 2 ¤ dy dx |fP ¡ f p q|. s d N P Q S p q q Q,S D Q, S P PrQ,QS q

Recall that all the cubes P P rQ, QSs contain Q in their shadow and the properties of the Whitney covering grant that N pP q 5P . Moreover, by (1.23) we have that DpQ, Sq DpP,Sq. Thus, » » ¸ ¸ ¸ gpx, yq 2 Àd |fpξq ¡ fpζq| dζ dξ dy dx s d p q q P P 5P QPSHpP q Q SPW S D P,S and, using H¨older’s inequality, and by (1.29), we have that

£ 1 » ¢» 1 ¸ ¸ 1 ¸ d q 1 q `pSq 2 À |fpξq ¡ fpζq| dζ dξ gpx, yqq dy dx p qsq d P 5P P p q Q Ω P D P,S P Q SH P » S W ¸ ¸ 1 À |fpξq ¡ fpζq| dζ dξ Gpxq dx . d,s,q `pP qs P P 5P QPSHpP q Q ³ p q À p q p qd By (1.28) we have that ShpP q G x dx d,ε infyPP MG y ` P , so » » ¸ `pP qd¡s 2 À |fpξq ¡ fpζq| dζMGpξq dξ `pP q2d P P 5P ¢ ¸ » » 1 q d q 1 1 À |fpξq ¡ fpζq| dζ `pP q q MGpξq dξ . d,p `pP qd s P P 5P

Note that for ξ, ζ P 5P , we have that |ξ ¡ ζ| Àd `pP q. Thus, using H¨older’sinequality again and } } 1 À } } 1 ¤ the fact that MG Lp p G Lp 1, we bound the second term by

£ 1 » ¢» 1 » ¢» p ¸ ¸ p |fpξq ¡ fpζq|q q |fpξq ¡ fpζq|q q 2 À dζ MGpξq dξ À dζ dξ . |ξ ¡ ζ|sq d |ξ ¡ ζ|sq d P P 5P P P 5P (2.55)

Now we face the boundedness of » » ¸ |f ¡ fpyq| 3 QS gpx, yq dy dx. s d Q,S Q S DpQ, Sq q 58 CHAPTER 2. T(P) THEOREMS

Given two cubes Q and S, we have that for every admissible chain rQ, Ss the cubes Q, S P SHpQSq by Deﬁnition 1.15 and DpQ, Sq `pQSq by (1.21). Thus, we can reorder the sum, writing » » ¸ ¸ ¸ |f ¡ fpyq| 3 À R gpx, yq dy dx (2.56) s d R P p q P p q Q S `pRq q »Q SH R S SH R » » ¸ ¸ ¸ |fpξq ¡ fpyq| ¤ gpx, yq dy dx dξ. s p1 1 qd R R QPSHpRq SPSHpRq Q S `pRq q

Using H¨older’sinequality, Lemma 1.18 and the fact that for S P SHpRq one has `pRq DpS, Rq, we get that

1 and, using the H¨olderinequality again and the boundedness of the maximal operator in Lp , we get ¤ £ p 1 » » p ¸ |fpξq ¡ fpyq|q q À ¥ } } 1 3 sq d dy dξ MG Lp p q |ξ ¡ y| R R Sh R ¤ £ p 1 » » p ¸ | p q ¡ p q|q q À ¥ f ξ f y sq d dy dξ . (2.57) p q |ξ ¡ y| R R Sh R

This fact, together with (2.52) proves (2.51) and thus, using (2.49) and (2.50), we get that } } À } } f s p q ε,s,p,q,d f rs p q. Ap,q Ω Ap,q Ω

Finally, by (2.47) we have that f P Lmaxtp,qupΩq. Remark 2.19. Note that we have proven that the homogeneous seminorms are equivalent, that is,

» £» p ¸ | p q ¡ p q|q q f x f y } }p sq d dy dx f 9 s p q, p q |x ¡ y| Ap,q Ω QPW Q Sh Q which improves (2.48). 2.4. EQUIVALENT NORMS WITH REDUCTION OF THE INTEGRATION DOMAIN. 59

In some situations we can reﬁne Lemma 2.18. Lemma 2.20. Let! Ω be a) uniform domain with an admissible Whitney covering W, let 1 q ¤ 8 d ¡ d P s p q p and max p q , 0 s 1. Then, f Fp,q Ω if and only if £ » ¢» p 1 ¸ p |fpxq ¡ fpyq|q q }f} dy dx 8. LppΩq |x ¡ y|sq d QPW Q 5Q

Furthermore, this quantity deﬁnes a norm which is equivalent to }f} s p q. Fp,q Ω

¡ } } 1 1 ¤ Proof. Arguing as before by duality, we consider a function g 0 with g Lp pLq pΩqq 1. Com- bining (2.54) and (2.55) we know that £ » » » ¢» p 1 ¸ ¸ p |fpxq ¡ f | |fpxq ¡ fpyq|q q QS p q À d g x, y dy dx sq d dy dx s q |x ¡ y| Q,S Q S DpQ, Sq QPW Q 5Q and, thus, we have £ ¢ p 1 » » » » p ¸ | p q ¡ p q| ¸ | p q ¡ p q|q q f x f y p q f x f y d g x, y dy dx sq d dy dx 3 . (2.58) s q |x ¡ y| Q,S Q S DpQ, Sq QPW Q 5Q where » » » » ¸ |f ¡ fpyq| ¸ ¸ |f ¡ fpyq| 3 : QS gpx, yq dy dx À R gpx, yq dy dx s d s d p q q p q q Q,S Q S D Q, S R Q,SPSHpRq Q S ` R by (2.56). Using H¨older’sinequality and Lemma 1.18 we get that

¤ 1 ¸ ¸ » q ¸ » 1 ¥ q 3 À |fR ¡ fpyq| dy Gpxq dx s d p q q R ` R SPSHpRq S QPSHpRq Q

¤ 1 » q ³ ¸ ¸ p q ¥ q R MG ξ dξ À |fR ¡ fpyq| dy s d p q q R SPSHpRq S ` R and, using the H¨olderinequality again, we get ¤ ¤ p 1 p » q ¸ ¸ p qd ¦ ¥ q ` R 3 À ¥ |fR ¡ fpyq| dy }MG} p1 . sp dp L R SPSHpRq S `pRq q

1 p } } 1 À By the boundedness of the maximal operator in L we have that MG Lp 1. Now, given R,S P W there exists an admissible chain rS, Rs, and we can decompose the previous expression as ¤ § § p q § s § q ¸ ¸ § ¸ ¨ p q q § p ` P d¡sp¡d p À ¥ § ¡ § p qd p q q 3 § fP fN pP q s § ` S ` R (2.59) § p q q § R SPSHpRq P PrS,Rq ` P ¤ p ¸ ¸ » q ¥ q d¡sp¡d p |fS ¡ fpyq| dy `pRq q : 3.1 3.2 , R SPSHpRq S 60 CHAPTER 2. T(P) THEOREMS where we wrote rS, Rq rS, RsztRu. Using H¨older’sinequality

¤ p ¤ q q q1 ¸ ¸ ¸ | ¡ |q ¸ 1 ¦ fP fN pP q ¥ sq d d¡sp¡d p 3.1 À ¥ `pP q q `pSq `pRq q . `pP qs R SPSHpRq P PrS,Rq P PrS,Rq

1 1 ° sq sq P p q p q q À p q q But for S SH R by Remark 1.17 we have that P PrS,Rq ` P ` R . Moreover, by P r s P 7p q p q (1.26) there exists a ratio ρ7 such that° for P S, R we have that S SH P : SHρ7 P and P 7p q p qd À p qd P SH R . We also know that SPSH7pP q ` S ` P , so writing UP for the union of the neighbors of P , we get

¡ © p ¤ q d q ¸ ¸ |fpξq ¡ fP | dξ `pP q ¥ UP d sp ¡sp¡ dp 3.1 À ﬄ `pRq q q . `pP qs R P PSH7pRq

Recall that p ¥ q and, therefore, by H¨older’sinequality and (1.24) we have that

To bound 3.2 we follow the same scheme. Since p ¥ q we have that

¤ p » q dp1¡ q q ¸ ¸ p q p ¥ q ` S d¡sp¡d p 3.2 |fS ¡ fpyq| dy `pRq q dp1¡ q q R SPSHpRq S `pSq p ¤ q ¤ p ¤ p ¡ q q p ³ ¨ p p q 1 p q ¸ ¸ | ¡ p q|q q ¸ ¥ S fS f y dy ¥ d d¡sp¡d p ¤ `pSq `pRq q , dp p ¡1q R SPSHpRq `pSq q SPSHpRq ° p qd p qd and, since SPSHpRq ` S ` R , reordering and using (1.24) we get that

³ ¨ p ¢³ p ¸ q q ¸ ¸ q q d |fS ¡ fpyq| dy |fS ¡ fpyq| dy `pSq 3.2 À S `pRq¡sp À S . dp p ¡1q `pSqd `pSqsp S `pSq q R:SPSHpRq S 2.4. EQUIVALENT NORMS WITH REDUCTION OF THE INTEGRATION DOMAIN. 61

s Remark 2.21. An analogous result to Lemma 2.20 for Besov spaces Bp,p can be found in [Dyd06, Proposition 5] where it is stated in the case of Lipschitz domains. Corollary 2.22. Let Ω be a uniform domain. Let δpxq : distpx, BΩq for every x P C. 8 ¡ d ¡ d s p q s p q Given 1 p q and 0 s 1 with s p q , we have that Ap,q Ω Fp,q Ω and, moreover, for ρ1 ¡ 1 big enough, we have that ¤ £ p 1 » » p | p q ¡ p q|q q } } } } ¥ f x f y P s p q f F s pΩq f LppΩq sq d dy dx for all f Fp,q Ω . p,q p qX |x ¡ y| Ω Bρ1δpxq x Ω ¤ 8 s p q s p q Given 1 q p and 0 s 1, we have that Ap,q Ω Fp,q Ω and, moreover, for 0 ρ0 1 we have that ¤ £ p 1 » » p | p q ¡ p q|q q } } } } ¥ f x f y P s p q f F s pΩq f LppΩq sq d dy dx for all f Fp,q Ω . p,q p q |x ¡ y| Ω Bρ0δpxq x Proof. This comes straight forward from Corollary 2.17, Lemma 2.18 and Lemma 2.20, taking smaller cubes in the Whitney covering if necessary when ρ0 1. 8 s p q s p q Remark 2.23. In particular, for every 1 p and 0 s 1 we have that Ap,p Ω Bp,p Ω , with £» » 1 | p q ¡ p q|p p } } } } f x f y P s p q f Bs pΩq f LppΩq sp d dy dx for all f Bp,p Ω . p,p p q |x ¡ y| Ω Bρ0δpxq x ¡ d ¡ d s p q s,pp q ¥ If in addition s p 2 , then Ap,2 Ω W Ω . If p 2 we have that ¤ £ p 1 » » p | p q ¡ p q|2 2 } } } } ¥ f x f y P s,pp q f W s,ppΩq f LppΩq 2s d dy dx for all f W Ω , p q |x ¡ y| Ω Bρ0δpxq x and, if 1 p 2, we have that ¤ £ p 1 » » p | p q ¡ p q|2 2 } } } } ¥ f x f y P s,pp q f W s,ppΩq f LppΩq 2s d dy dx for all f W Ω . p qX |x ¡ y| Ω Bρ1δpxq x Ω

Chapter 3

Characteristic functions of planar domains

In this chapter we prove that for p ¡ 2 the Beurling transform is bounded on W n,ppΩq when the n¡1{p boundary of the domain is regular enough. In particular, we will see that if N P Bp,p pBΩq, n,p then BχΩ P W pΩq, in the same spirit of [CT12]. Recall that this result is sharp for n 1 and Lipschitz constant small enough by [Tol13].

n¡1{p Theorem 3.1. Let p ¡ 2, let n P N and let Ω be a bounded Lipschitz domain with N P Bp,p pBΩq. Then, for every f P W n,ppΩq we have that

}BpχΩfq} n,pp q ¤ C}N} n¡1{p }f} n,pp q, W Ω Bp,p pBΩq W Ω where C depends on p, n, diampΩq and the Lipschitz character of the domain. The proof will be slightly more tricky since we will need to approximate the boundary of the domain by polynomials instead of straight lines: the derivative of the Beurling transform of the characteristic function of a half-plane is zero (see [CT12]), but the derivative of the Beurling transform of the characteristic function of a domain bounded by a polynomial of degree greater than one is not zero anymore. Using the T pP q-Theorem 2.1 this will suffice to see the boundedness of the Beurling transform in W n,ppΩq. Section 3.1 is devoted to present a family of convolution operators in the plane which include the Beurling transform as a particular case. The purpose of the notation that we will introduce at this point, which is not standard, is to simplify the proofs of Chapter 4. In Section 3.2 one finds the definition of some generalized β-coefficients related to Jones and David-Semmes’ celebrated betas and an equivalent norm for the Besov spaces introduced in Section 1.2 in terms of the generalized β-coefficients is presented using a result by Dorronsoro in [Dor85]. From that point, the proof of a quantitative version of Theorem 3.1 begins (see Theorem 3.28). The first step is to study the case of unbounded domains whose boundary can be expressed as the graph of a Lipschitz function. Section 3.3 contains the outline of the proof, reducing it to two lemmas. The first one studies the relation with the β-coefficients and is proven in Section 3.4. The second one, proven in Section 3.5, is about the case where the domain is bounded by the graph of a polynomial, and here one finds the exponential behavior of the bounds for the iterates of the Beurling transform, which entangles the more subtle details of the proof. Section 3.6 stops the flow for a while to relate the beta coefficients with the normal vector. Finally, in Sections 3.7 and 3.8 one finds the quantitative version (in the precise shape that we need in Chapter 4) of Theorem 3.1 for bounded Lipschitz domains using a localization principle and the T pP q-theorem.

63 64 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

3.1 A family of convolution operators in the plane

zt u Ñ P 1 Definition 3.2. Consider a function K : C 0 C. For any f Lloc we define » T K fpzq lim Kpz ¡ wqfpwq dmpwq εÑ0 z p q C Bε z as long as the limit exists, for instance, when K³ is bounded away from 0, f P L1 and y R supppfq P ¡ 1 ¡ or when f χU for an open set U with y U, p qz p q K dm 0 for every ε ε 0 and K Bε 0 Bε1 0 is integrable at infinity. We say that K is the kernel of T K . For any multiindex γ P Z2, we will consider Kγ pzq zγ zγ1 z¯γ2 and then we will put shortly γ Kγ T f : T f, that is, » T γ fpzq lim pz ¡ wqγ fpwq dmpwq (3.1) εÑ0 z p q C Bε z as long as the limit exists. For any operator T and any domain Ω, we can consider TΩf χΩ T pχΩ fq.

Example 3.3. As the reader may have observed, the Beurling and the Cauchy transforms are in p q ¡2 p¡ q ¡1 γ that family of operators. Namely, when K z z , that is, for γ 2, 0 , then π T is the 1 p¡1,0q Beurling transform. The operator π T coincides with the Cauchy transform. Consider the iterates of the Beurling transform Bm for m ¡ 0. For every f P Lp and z P C we have » p¡1qmm pz ¡ τqm¡1 p¡1qmm Bmfpzq lim fpτq dmpτq T p¡m¡1,m¡1qfpzq. (3.2) Ñ p ¡ qm 1 π ε 0 |z¡τ|¡ε z τ π

γ That is, for γ pγ1, γ2q with γ1 γ2 ¡2 and γ1 ¤ ¡2, the operator T is an iteration of the Beurling transform modulo constant (see [AIM09, Section 4.2]), and it maps LppUq to itself for γ every open set U. If γ2 ¤ ¡2, then T is an iterate of the conjugate Beurling transform and it is bounded on Lp as well. In both cases T γ is an admissible convolution Calder´on-Zygmund operator of order 8 (see Deﬁnition 1.26).

3.2 Besov norm and beta coeﬃcients

In [Dor85], Dorronsoro introduces a characterization of Besov spaces in terms of the mean oscillation of the functions on cubes, and he uses approximating polynomials to do so. If the polynomials are of degree one, that is straight lines, this deﬁnition can be written in terms of a certain sum of David-Semmes betas (see [CT12] for instance). Following the ideas of Dorronsoro in our case we will use higher degree polynomials to approximate the Besov function that we want to consider, giving rise to some generalized betas. The following proposition comes from [Dor85], where it is not explicitly proven. We give a short proof of it for the sake of completeness.

Proposition 3.4. Given a locally integrable function f : Rd Ñ R and a cube Q Rd, there exists n P n a unique polynomial RQf P which we will call approximating polynomial of f on Q, such that given any multiindex γ with |γ| ¤ n one has that » p n ¡ q γ RQf f x 0. (3.3) Q 3.2. BESOV NORM AND BETA COEFFICIENTS 65

Remark 3.5. In case of existence, the approximating polynomial veriﬁes » | n p q| ¤ 1 | | sup RQf x Cn,d | | f dm. xPQ Q Q Proof. By Remark 1.19, for any P P Pn we have that 2 2 1 2 1 2 }P } 8 P P }P } . L pQq L8pQq |Q| L1pQq |Q| L2pQq Using the linearity of the integral in (3.3), one has » » 1 | n |2 1 n ¤ | | RQf dm | | RQf f dm. Q Q Q Q Combining both facts one gets 2 1 Rn f À Rn f }f} . Q L8pQq |Q| Q L8pQq L1pQq

Proof of Proposition 3.4. By the Hilbert Projection Theorem, L2pQq Pn ` pPnqK. Thus, if P 2p q | n p | ¡ n q f L Q , we can write f Q RQf f Q RQf satisfying (3.3). 1 2 P t u P p q | | ¤ | | For general f L , we can deﬁne a sequence of functions fj j N L Q such that fj f ÝÝÑa.e. n and fj f. By Remark 3.5 we have that the approximating polynomials RQfj are uniformly bounded on Q by » » | n p q| À 1 | | ¤ 1 | | sup RQfj x | | fj dm | | f dm. xPQ Q Q Q Q t n u 1 Therefore there exists a convergent subsequence of RQfj j in L (and in any other norm). We n call RQf the limit of one such partial. By the Dominated Convergence Theorem we get (3.3). To see uniqueness, we observe that if we ﬁnd two polynomials P1 and P2 satisfying (3.3), then »

pP1 ¡ P2qP 0 Q P n ¡ } ¡ } for any P P . In particular, if we take P P1 P2 we get that P1 P2 L2pQq 0. Remark 3.6. Given P P Pn, a cube Q and 1 ¤ p ¤ 8 we have that f ¡ Rn f ¤ C }f ¡ P } , (3.4) Q LppQq d,n LppQq and given any cubes Q Q1, n n f ¡ R f ¤ C f ¡ R 1 f . (3.5) Q LppQq d,n Q LppQ1q Proof. By means of the triangle inequality and (3.3), we have that for any P P Pn f ¡ Rn f ¤ }f ¡ P } P ¡ Rn f }f ¡ P } Rn pP ¡ fq . Q LppQq LppQq Q LppQq LppQq Q LppQq Therefore, we use twice H¨older’sInequality and Remark 3.5 to get f ¡ Rn f ¤ }f ¡ P } |Q|1{pRn pP ¡ fq Q LppQq LppQq Q L8pQq |Q|1{p À }f ¡ P } }P ¡ f} ¤ 2}f ¡ P } . n,d LppQq |Q| L1pQq LppQq n The inequality (3.5) is just a consequence of (3.4) replacing P by RQ1 f. 66 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

Remark 3.7. In the one dimensional case, if f is continuous and I is an interval one can easily ¡ n see that f RI f has n 1 zeroes at least. Indeed, if it did not happen, one could ﬁnd a polynomial P n ¡ n P P with a simple zero at every point where f RI f changes its sign, and no more. Therefore, p ¡ n q¤ f RI f P would have constant sign and, thus, the integral in (3.3) would not vanish (see Figure 3.1).

³ ¡ 2 P 2 p ¡ 2 q ¡ Figure 3.1: If f RI f had only 2 zeroes, there would exist P P with f RI f P dm 0.

Now we can deﬁne the generalized betas.

Deﬁnition 3.8. Let f : Rd Ñ R be a locally integrable function and Q Rd a cube. Then we deﬁne » |fpxq ¡ Rn fpxq| p q 1 3Q p q βpnq f, Q | | p q dm x . Q 3Q ` Q

Remark 3.9. Taking into account (3.4), we can conclude that » 1 |fpxq ¡ P pxq| βpnqpf, Qq inf dmpxq. P n | | p q P P Q 3Q ` Q

This can be seen as a generalization of David and Semmes β1 coeﬃcient since βp1q and β1 are comparable as long as some Lipschitz condition is assumed on f.

9 s In [CT12] the authors point out that the seminorm of the homogeneous Besov space Bp,q for 0 s 1 can be deﬁned in terms of the approximating polynomials of degree 1 above. The same ¥ P s ¥ r s can be said for s 1. Indeed, [Dor85, Theorem 1] says that for f Bp,q and n s we have the equivalent seminorm

¤ q 1 ¡ ¡ © © q ³ p p » 8 | p q ¡ n p q| d sup Q | | d f y R f y dy dx ¦ R Q x: Q t Q Q dt } } 9 ¥ f Bs ﬄ sq p,q 0 t t

(see (1.17)). Note that, given x P Rd and t P R, every cube Q Q x with `pQq t satisfies that Q p q | p q¡ n p q| À p p qq Q x, t . Therefore, by (3.5) we have that supQQx:|Q|td Q f y RQf y dy d,n tβpnq f, Q x, t , p p qq À | p q ¡ ffln p q| and tβpnq f, Q x, t d,n supQQx:|Q|p6tqd Q f y RQf y dy. Thus, ffl £ £ { » 8 q 1 q βp qpf, Qp¤, tqq n Lp dt } } 9 f Bs s¡1 . p,q 0 t t 3.3. THE CASE OF UNBOUNDED DOMAINS 67

In the particular case when p q, which is in fact the one we are interested on, via Fubini’s Theorem and (3.5) one can conclude that for the canonical dyadic grid D, for instance, we have that

» » ¢ ¸8 2j 1 p βp qpf, Qpx, tqq dt }f}p n dx B9 s s¡1 p,p j d t t j¡8 2 R ¢ ¢ ¸8 ¸ p ¸ p βp qpf, Qq βp qpf, Qq n |Q| n |Q|. (3.6) `pQqs¡1 `pQqs¡1 j¡8 QPD:`pQq2j QPD

3.3 The case of unbounded domains Ω C

Deﬁnition 3.10. Given δ ¡ 0 and R ¡ 0, we say that Ω tx i y P C : y ¡ Apxqu is a pδ, R, n, pq-admissible domain with deﬁning function A if

n 1¡1{p n¡1,1 • The defining function A P Bp,p X C . • We have Ap0q 0 and, if n ¥ 2, A1p0q 0. pjq δ ¤ ¤ • We have Lipschitz bounds on the function and its derivatives A L8 Rj¡1 for 1 j n. We associate a Whitney covering W with appropriate constants to Ω. The constants will be fixed along this section, depending on n and δ. In this Section we will prove the next result for the operators T γ defined in (3.1). Theorem 3.11. Consider δ, R, ¡ 0, p ¡ 1 and a natural number n ¥ 1. There exists a radius p q P 2 ρ R such that for every δ, R, n, p -admissible domain Ω and every multiindex γ Z with γ p γ1 γ2 ¡n ¡ 2 and γ1 ¤ γ2 ¤ 0, we have that T χΩ P L pΩ X Bp0, ρqq. In particular, if A is the defining function of Ω, we have that ¡ © } γ }p ¤ } }p 2¡npp q|γ|p T χΩ LppΩXBp0,ρ qq C A 9 n¡1{p 1 ρ 1 , Bp,p where C depends on p, n and the Lipschitz character of Ω.

Figure 3.2: Disposition in Theorem 3.11. 68 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

Definition 3.12. Consider a given pδ, R, n, pq-admissible domain with defining function A. Then, n n for every interval I we have an approximating polynomial R3I : R3I A, and » | p q ¡ n p q| p q p q 1 A x R3I x βpnq I : βpnq A, I p q p q dx. ` I 3I ` I We call n t ¡ n p qu ΩI : x i y : y R3I x . Let π : C Ñ R be the vertical projection (to the real axis) and Q a cube in C. If πpQq I we will n n write ΩQ : ΩI . Remark 3.13. Note that π sends dyadic cubes of C to dyadic intervals of R and, in particular, any dyadic interval has a finite number of pre-images in the Whitney covering W of Ω uniformly bounded by a constant depending on δ and the Whitney constants of W. Proof of Theorem 3.11. By (3.6) we have that ¢ ¸ p q p βpnq I p `pIq }A} ¡ { , n¡1{p 9 n 1 p 1 `pIq Bp,p IPD so it is enough to prove that £ ¢ ¸ p βp qpIq } γ }p ¤ n p q 2¡npp q|γ|p T χΩ pp X p qq C ` I ρ 1 , (3.7) L Ω B 0,ρ `pIqn¡1{p IPD where D stands for tI P D : `pIq ¤ 2ρ and Dpt0u,Iq ¤ 3ρqu. P We begin the proof by some basic observations. Let j1, j2 Z such that j2 j1 1. Then, the line integral » » 2π p ¡ q wj1 wj2 dw i eiθ j1 j2 1 dθ 0 (3.8) B D 0 so, as long as j2 ¡ 0, given 0 ε 1 Green’s formula (1.12) says that » » ¡ i wj1 wj2 1 dmpwq wj1 wj2 dw 0. (3.9) z p q 2j B YB p q D B 0,ε 2 D B 0,ε P 2 ¡ ¡ ¥ ¥ Consider a given γ Z with γ1 γ2 n 2 and assume that γ2 0 (the case γ1 0 can be proven mutatis mutandis). Consider a Whitney cube Q and z P Bp0, ρq X Q. Then by (3.9) we have that § § §» § γ § γ § |T χΩpzq| § pw ¡ zq χΩpwq dmpwq§ (3.10) § | ¡ |¡ p q § § z w ` Q § §» § » |χΩn pwq ¡ χΩpwq| § γ § Q ¤ § pw ¡ zq χΩn pwq dmpwq§ dmpwq. § Q § | ¡ |n 2 |z¡w|¡`pQq |z¡w|¡`pQq w z

p q p B n q If we have taken appropriate Whitney constants, then we also have that ` Q dist Q, ΩQ (see Remark 3.5) and, thus, by (3.9) again, we have that » γ γ pw ¡ zq χ n pwq dmpwq T χ n pzq. (3.11) ΩQ ΩQ |z¡w|¡`pQq We will see in Section 3.5 that the following claim holds. 3.3. THE CASE OF UNBOUNDED DOMAINS 69

Claim 3.14. There exists a radius ρ (depending on δ, R, n and ) such that for every z P Bp0, ρq with z P Q P W, we have that p q|γ| γ 1 |T χ n pzq| À . (3.12) ΩQ n n ρ The last term in (3.10) will bring the beta coefficients into play. Recall that we defined the symmetric difference of two sets A1 and A2 as A1∆A2 : pA1 Y A2qzpA1 X A2q. For our choice of n X the Whitney constants we have that 2Q ΩQ Ω so » » |χΩn pwq ¡ χΩpwq| 1 Q p q p q n 2 dm w n 2 dm w . (3.13) | ¡ |¡ p q |w ¡ z| n |w ¡ z| z w ` Q ΩQ∆Ω t P Next we split the domain of integration in vertical strips. Namely, if we call Sj w C : j |Re pw ¡ zq| ¤ 2 `pQqu for j ¥ 0 and S¡1 H, we have that » » » ¸ p q p q 1 p q dm w dm w dm w n | ¡ |n 2 n | ¡ |n 2 | ¡ |n 2 Ω ∆Ω w z j pΩ ∆ΩqXSj zSj¡1 w z |w¡z|¡ρ{2 w z Q j¥0: 2 `pQq¤ρ Q ¸ § § 1 1 À §pΩn ∆Ωq X S § . (3.14) Q j p j¡1 p qqn 2 n j 2 ` Q ρ j¥0: 2 `pQq¤ρ We will see in Section 3.4 the following: Claim 3.15. We have that ¸ § § βp qpIq §pΩn ∆Ωq X S § À n p2j`pQqqn 1. (3.15) Q j n `pIqn¡1 IPD πpQqI2j 1πpQq

Summing up, plugging (3.11) and (3.12) in the ﬁrst term of the right-hand side of (3.10) and plugging (3.13), (3.14) and (3.15) in the other term, we get

¸ ¸ | | βp qpIq 1 p1 q γ |T γ χ pzq| À n p2j`pQqqn 1 . Ω n `pIqn¡1 p2j`pQqqn 2 ρn j¥0 IPD j p q j 1 p q 2 `pQq¤ρ π Q I 2 π Q

Note that the intervals I in the previous sum are in D tI P D : `pIq ¤ 2ρ and Dpt0u,Iq ¤ 3ρqu. Reordering and computing, ¸ ¸ | | ¸ | | βp qpIq 1 p1 q γ βp qpIq p1 q γ |T γ χ pzq| À n À n . Ω n p qn¡1 j p q n p qn n P ` I P 2 ` Q ρ P ` I ρ I D j N0 I D πpQqI I2j 1πpQq πpQqI

Raising to power p, integrating in Q and adding we get that for ρ small enough ¤ p ¸ ¸ | | ¦ βp qpIq p q γ } γ }p À | | ¦ n 1 T χΩ pp X p qq n Q L Ω B 0,ρ ¥ p qn n ` I ρ QPW IPD X p qH πpQqI Q B 0,ρ ¤ p ¸ ¸ ¦ βp qpIq À |Q| ¦ n ρ2¡npp1 q|γ|p. (3.16) p ¥ `pIqn QPW IPD QXBp0,ρqH πpQqI 70 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

Regarding the double sum, we use H¨olderInequality to ﬁnd that

¤ ¤ p p £ p1 ¸ ¦ ¸ p q ¸ ¸ p q p ¦ ¸ ¦ βpnq I βpnq I ¦ 1 |Q| ¤ |Q| 1 ¥ p qn n¡ 1 ¥ p ` I p q 2p p q 2p QPW IPD QPW IPD ` I IPD ` I X p qH πpQqI πpQqI πpQqI Q B 0,ρ £ ¸ ¸ p q p 2 βpnq I ¡1 À `pQq `pQq 2 (3.17) p n¡ 1 p q 2p QPW IPD ` I p q £ π Q I £ ¸ p ¸ ¸ p βpnqpIq 3 βpnqpIq ¤ `pQq 2 À `pIq, n¡ 1 W n¡ 1 p q 2p p q p IPD ` I QPW IPD ` I πpQqI where the constant in the last inequality depends on the maximum number of Whitney cubes that can be projected to a given interval, depending only on δ and n. Thus, by (3.16) and (3.17) we have proven (3.7) when γ2 ¥ 0. The case γ2 ¤ 0 can be proven analogously.

3.4 Beta-coeﬃcients step in

Figure 3.3: Disposition in the proof of Claim 3.15.

Proof of Claim 3.15. Consider N ¥ 0. Recall that we have a point z P Q P W, and a vertical strip t P | p ¡ q| ¤ N p qu p q SN w C : Re w z 2 ` Q . Let J0 π Q and let JN be the dyadic interval of length N 2 `pQq containing J0. Then it is enough to see that

§ § ¸ p qn¡1 § n § ` JN 2 pΩ ∆Ωq X S À βp qpIq `pJ q . (3.18) Q N n n `pIqn¡1 N IPD J0IJN 3.5. DOMAINS WHICH ARE BOUNDED BY THE GRAPH OF A POLYNOMIAL 71

First note that » § § Re pzq `pJN q §pΩn ∆Ωq X S § |A ¡ Rn | dm (3.19) Q N 3J0 1 p q¡ p q »Re z ` JN » ¤ |A ¡ Rn | dm |Rn ¡ Rn | dm 1 2 . 3JN 1 3JN 3J0 1 3JN 3JN Trivially, 2 1 βpnqpJN q`pJN q . (3.20) To deal with the second term, we consider the chain of dyadic intervals

J0 ¤ ¤ ¤ Jk Jk 1 ¤ ¤ ¤ JN ,

k with 0 k N and `pJkq 2 `pJ0q. We use the Triangle Inequality in the chain of intervals:

¡ » ¡ N¸1 N¸1 ¤ | n ¡ n | n ¡ n 2 R3J R3J dm1 R3J R3J . (3.21) k 1 k k 1 k L1p3J q k0 3JN k0 N

Fix 0 ¤ k N. By Remark 1.19 we have that `pJ qn 1 n ¡ n À n ¡ n N R3J R3J n R3J R3J , k 1 k 1 k 1 k 1 p qn 1 L p3JN q L p3Jkq ` Jk with constants depending only on n. Thus, by Remark 3.6 ¢ `pJ qn 1 n ¡ n À n ¡ ¡ n N R3J R3J n R3J A A R3J 1 k 1 k 1 k 1 1 k L p3Jkq p qn 1 L p3JN q L p3Jkq ` Jk ¨ p qn 1 ` JN 2 À p q p q p q n βpnq Jk 1 βpnq Jk n 1 ` Jk . (3.22) `pJkq

Thus, combining (3.19), (3.20), (3.21) and (3.22) we get (3.18).

3.5 Domains which are bounded by the graph of a polynomial

We will consider only very “ﬂat” polynomials. Let us see what we can say about their coeﬃcients. ¥ P n¡1,1p q p q 1p q pjq δ ¤ Lemma 3.16. Let n 2, A C R with A 0 0, A 0 0, A L8 Rj¡1 for j n and consider two intervals J and I with 3J I r¡R,Rs. Then we have the following bounds n for the derivatives of the approximating polynomial P RJ A in the interval I: n¡j pjq ¤ 3 δ ¤ P ¡ for j n. L8pIq Rj 1

Furthermore, if ρ ¡ 0 and 3J r¡ρ, ρs, then

n 2 n¡1 3 δρ 1 3 δρ }P } 8 ¤ and P ¤ . (3.23) L p¡ρ,ρq R L8p¡ρ,ρq R 72 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

0 ¤ ¤ ¤ 0 P Proof. By Remark 3.7 we know that there are at least n 1 common points τ0 , , τn 3J for p 0q p 0q A and P , that is, A τj P τj for every j. By the Mean Value Theorem, there are n common 1 ¤ ¤ ¤ 1 P k ¤ ¤ ¤ k P points τ0 , , τn¡1 3J for their derivatives. By induction we find points τ0 τn¡k 3J where ¤ ¤ ¡ pkqp kq pkqp kq ¤ ¤ ¡ the k-th derivatives coincide for 0 k n 1, that is, A τj P τj for every 0 j n k. Note that the polynomial derivative P pnq, which is in fact a constant, coincides with the differ- p ¡ q ential quotient of P n 1 evaluated at any pair of points. In particular given x P R, for the points n¡1 n¡1 τ0 and τ1 we have that § § § § § § § § §P pn¡1qpτ n¡1q ¡ P pn¡1qpτ n¡1q§ §Apn¡1qpτ n¡1q ¡ Apn¡1qpτ n¡1q§ δ §P pnqpxq§ § 0 1 § § 0 1 § ¤ . § n¡1 ¡ n¡1 § § n¡1 ¡ n¡1 § n¡1 τ0 τ1 τ0 τ1 R pj 1q ¤ n¡j¡1 { j Now we argue by induction again. Assume that P L8pIq 3 δ R for a certain j ¤ n ¡ 1. Consider x P I and, by the Mean Value Theorem, there exists a point ξ such that | pjqp q ¡ pjqp jq| | pj 1qp q|| ¡ j| pjqp jq pjqp jq P x P τ0 P ξ x τ0 . Thus, since P τ0 A τ0 we have that 3n¡j¡1δ δ 3n¡jδ |P pjqpxq| ¤ |P pj 1qpξq||x ¡ τ j| |Apjqpτ jq| ¤ 2R . 0 0 Rj Rj¡1 Rj¡1 We have not used yet the fact that A1p0q Ap0q 0. Let us fix ρ ¤ R and assume that 3J r¡ρ, ρs. Then for every x P r¡ρ, ρs, we can write A1pxq A1pxq ¡ A1p0q so δ |A1pxq| ¤ A2 |x| ¤ ρ, (3.24) L8pIq R 1p q 1p q ¡ 1p 1q 1p 1q ¡ 1p q and we can also write P x P x P τ0 A τ0 A 0 , so 3n¡2δ δ 3n¡1δρ |P 1pxq| ¤ P 2 |x ¡ τ 1| A2 |τ 1| ¤ 2ρ ρ ¤ . L8pIq 0 L8pIq 0 R R R By the same token, and using the estimate (3.24) on A1, we get

3n¡1δρ δρ 3nδρ2 |P pxq| ¤ P 1 |x ¡ τ 0| A1 |τ 0| ¤ 2ρ ρ ¤ . L8pr¡ρ,ρsq 0 L8pr¡ρ,ρsq 0 R R R

Now we can prove Claim 3.14. Recall that we want to ﬁnd a radius ρint R depending on p ρint q such that every point z contained in a Whitney cube Q B 0, 2 satisﬁes (3.12), that is, p q|γ| γ 1 |T χ n pzq| À , ΩQ n n ρint P tp¡ q P ¡ u ¥ where γ j1, j2 : j1, j2 N0 and j1 j2 n 2 (recall that we assumed that γ2 0). ¥ n According to the previous lemma, when° n 2 we are dealing with a domain ΩQ whose boundary p q n j is the graph of a polynomial P x j0 ajx such that

3nδρ2 |a | |P p0q| ¤ int , 0 R 3n¡1δρ |a | |P 1p0q| ¤ int and 1 R |P pjqp0q| 3n¡jδ |a | ¤ for 2 ¤ j n. (3.25) j j! j!Rj¡1 3.5. DOMAINS WHICH ARE BOUNDED BY THE GRAPH OF A POLYNOMIAL 73

We call ΩP : tx i y : y ¡ P pxqu to such a domain. Note that (3.23) implies that for ρint small | p q| ρint | | enough the polynomial P is “ﬂat”, namely P x 4 for x ρint. One can think of the “exterior” radius ρext below as a geometric version of , namely ρext 2 p{16q . Further, we can assume that ρext R.

Proposition 3.17. Consider two real numbers δ, R ¡ 0 and n ¥ 2. For ρext small enough, there P ¡ exists 0 ρint ρext depending also on n, δ and R such that for all j1, j2 N0 with j1 j2 n 2, n all P P P satisfying (3.25), all z P Qp0, ρintq X ΩP and 0 ε distpz, BΩP q we have §» § § j § ¡ © § p ¡ q 2 § j2 z w p q ¤ Cn 1{2 § dm w § 1 16ρext , (3.26) § p ¡ qj1 § n ΩP zBpz,εq z w ρint with Cn depending only on n. P ¡ P 1 If n 1 instead, then for all j1, j2 N0 with j1 j2 3 and all P P we have that » pz ¡ wqj2 dmpwq 0. (3.27) p ¡ qj1 ΩP zBpz,εq z w

ΩP P

ε z

ρint I

Qint

ρext

Qext

Figure 3.4: Disposition in Proposition 3.17.

Proof. First consider n 1. In that case, ΩP is a half plane. By rotation and dilation, we j 2 pz¡wq 2 can assume Ω : tw x i y : y ¡ 0u. Note that ¡ is infinitely many times P R pz¡wqj1 1 differentiable with respect to w in any ring centered in z. Then we can apply Green’s formula (1.12) and use the decay at infinity of the integrand and (3.8) to see that for ε ¡ 0 small enough

» ¡ » ¡ » ¡ p ¡ qj1 3 p ¡ qj1 3 p ¡ qj1 3 z w p q z w z w j dm w cj1 j ¡1 dw cj1 j ¡1 dw 2 z p q pz ¡ wq 1 pz ¡ wq 1 pz ¡ wq 1 R B z,ε R R » ¡ p ¡ qj1 4 z w p q cj1 j ¡1 dm w . 2 z p q pz ¡ wq 1 R B z,ε

When j1 3 the last constant is zero. By induction, all these integrals equal zero. Now we assume that n ¥ 2. Consider a given ρext ¡ 0. We deﬁne the interval I : r¡ρext, ρexts, the exterior window Qext : Qp0, ρextq, and the interior window Qint : Qp0, ρintq. Note that 74 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

(3.25) implies that for ρext small enough, the set tx i P pxq : x P Iu Qext, that is, the boundary BΩP , intersects the vertical sides of the window Qext but does not intersect the horizontal ones. The same can be said for the sides of Qint (see Figure 3.4). Fix z P Qint and ε distpz, BΩq. Splitting the domain of integration in two regions we get » » » pz ¡ wqj2 pz ¡ wqj2 pz ¡ wqj2 dmpwq dmpwq dmpwq. p ¡ qj1 p ¡ qj1 p ¡ qj1 ΩP zBpz,εq z w ΩP zQext z w ΩP XQextzBpz,εq z w (3.28) We bound the non-local part trivially by taking absolute values and using polar coordinates. Choosing ρint ρext{2, we have that » » 8 » 1 j1¡j2¡2 1 p q ¤ 1 2π 2 j ¡j dm w j ¡j dm1 2πr dr j ¡j ¡2 , (3.29) z |z ¡ w| 1 2 ρext r 1 2 j1 ¡ j2 ¡ 2 pρextq 1 2 ΩP Qext 2 0 where dm1 stands for the Lebesgue length measure. Note that j1 ¡ j2 ¡ 2 n. To bound the local part, we can apply Green’s Theorem again and we get » » p ¡ q p ¡ qj2 p ¡ qj2 2 j1 1 z w p q ¡ z w j dm w j ¡1 dw i X z p q pz ¡ wq 1 | ¡ | pz ¡ wq 1 ΩP Qext B z,ε » z w ε p ¡ qj2 ¡ z w j ¡1 dw XB pz ¡ wq 1 »ΩP Qext pz ¡ wqj2 dw. (3.30) p ¡ qj1¡1 BΩP XQext z w The ﬁrst term in the right-hand side of (3.30) is zero arguing as in (3.8). For the second term we note that z P Qint, and every w in the integration domain is in BQext, so |z ¡ w| ¡ ρext ¡ ρint. Thus, » 1 1 dw ¤ 6ρext. (3.31) | ¡ |j1¡j2¡1 | ¡ |j1¡j2¡1 ΩP XBQext z w ρext ρint ρext Summing up, by (3.28), (3.29), (3.30) and (3.31), since ρint , we get that § § 2 §» § §» § p ¡ qj2 § p ¡ qj2 § § z w § § z w § Cn § dmpwq§ ¤ § dw§ , (3.32) § p ¡ qj1 § p ¡ qj1¡1 n ΩP zBpz,εq z w BΩP XQext z w ρext with Cn depending only on n. It remains to bound the ﬁrst term in the right-hand side of (3.32). We begin by using the change of coordinates w x i P pxq to get a real variable integral: » » pz ¡ wqj2 pz¯ ¡ px ¡ i P pxqqqj2 dw p1 ¡ i P 1pxqq dx. (3.33) p ¡ qj1¡1 p ¡ p p qqqj1¡1 BΩP XQext z w I z x i P x

Note that the denominator on the right-hand side never vanishes because z RBΩP . Now we take a closer look to the fraction in order to take as much advantage of cancellation as we can, namely ¡ © j p ¡ p qq p ¡ p p qqq 2 p ¡ p ¡ p qqqj2 z¯ z 2i P x z x i P x z¯ x i P x ¡ ¡ pz ¡ px i P pxqqqj1 1 pz ¡ px i P pxqqqj1 1 ¢ ¸j2 j2 ¡ ¡ pz¯ ¡ z 2i P pxqqjpz ¡ px i P pxqqqj2 j j1 1 j j0 ¢ ¸j2 j p¡2i Impzq 2i P pxqqj 2 . (3.34) j pz ¡ px i P pxqqqn 1 j j0 3.5. DOMAINS WHICH ARE BOUNDED BY THE GRAPH OF A POLYNOMIAL 75

Next, we complexify the right-hand side of (3.34) so that we have a holomorphic function in a certain neighborhood of I to be able to? change the integration path. To do this change we need a r n key observation. If τ P Qext, then |τ| 2ρext and by (3.25) writing δ 3 δ we have that ¢ ρ 2 3 |P 1pτq| ¤ |a | 2|a ||τ| ¤ ¤ ¤ ¤ δr int 2ρ p2ρ q2 ¤ ¤ ¤ 1{2 (3.35) 1 2 R R ext R2 ext p 1p qq ¡ 1 if ρext is small enough. Thus, we have that Re 1 i P τ 2 in Qext and, by the Complex Rolle Theorem 1.10, we can conclude that τ ÞÑ τ i P pτq is injective in Qext. In particular, z ¡ pτ i P pτqq has one zero at most in Qext, and this zero is not real because z RBΩP . Therefore, since the real line divides Qext in two congruent open rectangles, there is one of them whose closure has a neighborhood containing no zeros of this function. We call this open rectangle R. Now, for ¥ ÞÑ pP pτq¡Impzqqj p ¡ 1p qq any j 0 we have that τ pz¡pτ i P pτqqqn 1 j 1 i P τ is holomorphic in R, so we can change the path of integration and get » » 2jpP pxq ¡ Impzqqj 2jpP pτq ¡ Impzqqj p1 ¡ i P 1pxqq dx ¡ p1 ¡ i P 1pτqq dτ. p ¡ p p qqqn 1 j p ¡ p p qqqn 1 j I z x i P x BRzI z τ i P τ ? (3.36) On the other hand, if |τ| 2ρext, then we have that |P pτq| ¤ |a | |a ||τ| |a ||τ|2 |a ||τ|3 ¤ ¤ ¤ (3.37) ¢0 1 2 3 2 ρ ρ 1 1 { ¤ δr int int 2ρ p2ρ q2 p2ρ q3 ¤ ¤ ¤ ¤ ρ3 2 R R ext R ext R2 ext ext for ρext small enough. Then, taking absolute values inside the last integral in (3.36) and using (3.35) and (3.37) we get » » { 2j|P pτq ¡ Impzq|j 3 2jpρ3 2 ρ qj |1 ¡ i P 1pτq| |dτ| ¤ ext int |dτ|. (3.38) | ¡ p p qq|n 1 j | ¡ p p qq|n 1 j BRzI z τ i P τ 2 BRzI z τ i P τ

Finally, we have that for any τ PBRzI BQext, ? 3 ρ |z ¡ pτ i P pτqq| ¥ |τ| ¡ |z| ¡ |P pτq| ¥ ρ ¡ 2ρ ¡ ρ 2 ¥ ext ¡ 2ρ ext int ext 2 int for ρext small enough. Using this fact we rewrite (3.38) as » { » 2j|P pτq ¡ Impzq|j 3 2jpρ3 2 ρ qj |1 ¡ i P 1pτq| |dτ| ¤ ext int |dτ|. (3.39) | ¡ p p qq|n 1 j p { ¡ qn 1 j BRzI z τ i P τ 2 ρext 2 2ρint BRzI Putting together (3.33), (3.34), (3.36) and (3.39) we can write § § £ ¢ » j 3{2 j § p ¡ qj2 § ¸2 § z w § ¤ 3 ¤ ρext ρint j2 § j ¡1 dw§ n 1 2 4ρext B X pz ¡ wq 1 2 pρ {2 ¡ 2ρ q ρ {2 ¡ 2ρ j ΩP Qext ext int j0 ext int £ j 3{2 2 6ρext ¤ ρext ρint n 1 1 2 , pρext{2 ¡ 2ρintq ρext{2 ¡ 2ρint

t { 3{2u and, choosing ρint min ρext 8, ρext , §» § ¡ © § p ¡ qj2 § j2 § z w § ¤ Cn 1{2 § dw§ 1 16ρext , (3.40) p ¡ qj1¡1 n BΩP XQext z w ρext where the constant Cn depends only on n. Now, (3.32) together with (3.40) prove (3.26). 76 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

Remark 3.18. Note that we have assumed γ2 ¥ 0 in the proof Theorem 3.11. When proving the ¤ P tp ¡ q P ¡ case γ2 0, we would have to prove Proposition 3.17 with γ j1, j2 : j1, j2 N0 and j2 j1 n 2u. The proof is analogous to the one shown above with slight modiﬁcations, and it is left to the reader to complete the details.

3.6 The geometric condition

In this rather technical section we give equivalent expressions of the sufficient geometric condition n¡1,1 n,p for a pδ, Rq ¡ C domain Ω to satisfy that BχΩ P W pΩq (see Theorem 3.27). Consider a window Q (see Definition 1.4) of Ω, and its associated parameterization A. In particular, we assume that there is a rigid transformation F which sends the center of Q to the origin and, in case n ¡ 1 we also assume that the tangent to the curve BΩ at 0 is sent to the real axis so that F pQ XBΩq tpx, Apxqq : x P IRu, t¡ R R u where IR 2 x 2 . In case n 1, this assumption is too restrictive since the existence of a tangent to BΩ in the center of Q is not granted and, therefore, }A} 8 can reach δR{2 even L pIRq t¡ R R u if we take smaller radius R. Thus, we must use instead IR 2δ x 2δ . We will use an IR c auxiliary bump function ϕR such that ϕR 1 in , ϕR 0 in I and 3 R pjq ¤ { j ¤ ϕR C R for every j n. L8 ¨ Q XB r Thus, we have that F 3 Ω is parameterized by A ϕRA (with obvious modifications for n 1). p q ¡ n¡1,1 t uM Theorem 3.19. Let! Ω be a bounded) δ, R C domain and let Qk k1 be a collection of R- M 1 t u windows such that maxt20,20δu Qk cover the boundary of Ω. Let Ak k be the parameterizations k1 of the boundary associated to each window. Consider N : BΩ Ñ R2 to be the unitary outward normal vector. Then, for any 1 p 8

¸M ¸ p ¸M βp qpA ,Iq n k À } }p À} }p n p¡2 ϕRAk 9 n 1¡1{p N n¡1{ppB q. (3.41) `pIq Bp,p Bp,p Ω k1 P IR k1 I D:I 6 with constants depending on n, p, δ, the length of the boundary H1pBΩq and M. Moreover, ¸M } }p } }p N n¡1{p R,H1pBΩq ϕRAk 9 n 1¡1{p 1. (3.42) Bp,p pBΩq Bp,p k1 The proof of this lemma for n 1 can be found in [CT12, Section 3]. We proceed to prove the general case using the same tools. The expert reader may skip this H1pBΩq part. Note that M can be chosen to be M R . Proof. We only need to prove the case n ¥ 2. By (3.6) the ﬁrst estimate in (3.41) is immediate, while the second one is a consequence of Proposition 3.21 below and (3.42) is a consequence of all this facts and Corollary 3.22.

Lemma 3.20. Let n P N with n ¥ 2 and s n ¡ 1 tsu with 0 ¤ tsu 1. With the notation introduced above, for 1 ¤ k ¤ M we have that » » p q p q |A n pxq ¡ A n pyq|p } }p À k k ϕRAk 9 s 1 t u dy dx 1. Bp,p | ¡ | s p 1 IR IR x y 3.6. THE GEOMETRIC CONDITION 77

r pn¡1q Proof. Let us write ﬁx a window Qk, and take Ak ϕRAk. First of all, note that, since Ak is Lipschitz, for almost every x P IR ¢ ¸n p q n p q p ¡ q Ar n pxq A j pxqϕ n j pxq. k j k R j0

Thus, using the lifting property for homogeneous Besov spaces (see [Tri83, Theorem 5.2.3/1]), since s n ¡ 1 tsu and 0 ¤ tsu 1, we have that

¸n p p qp p q p ¡ qp Ar Ar n À A j ϕ n j k 9 s 1 k 9 tsu k R 9 tsu Bp,p Bp,p Bp,p § j 0 § p § pjq pjq § n » § p q pn¡jqp q ¡ p q pn¡jqp q§ ¸ Ak y ϕ y Ak x ϕ x dy dx. tsup 1 ¢ |y ¡ x| j0 R R

c But ϕR 0 in IR, so we can reduce the integration domain and, using the symmetry between x and y, it is enough to consider the case x P IR, that is, § ¡ © § p » » § pjq p ¡ q § ¸n § n j p q§ p ∆h Ak ϕ x dh r À Ak t u dx B9 s 1 |h| s p |h| p,p j0 IR R and, since |∆hpfgqpxq| ¤ |gpxq||∆hfpxq| |fpx hq||∆hgpxq|, (3.43) we get » » » » ¸n pjq p pjq p p p ¡ q |∆ ϕ pxq| dh |∆ A pxq| dh r À | n j p q| h | pn¡jqp q| h k Ak Ak x t u dx ϕ x h t u dx B9 s 1 |h| s p |h| |h| s p |h| p,p j0 IR R IR R » » » » p q p q ¸n 1 |∆ ϕpjqpxq|p dh 1 |A j pxq ¡ A j pyq|p À h dx k k dy dx Rpn¡j¡1qp |h|tsup |h| Rpn¡jqp |x ¡ y|tsup 1 j0 IR R IR IR ¢ ¸n À I.j II.j . j0

We take a closer look to the summands I.j and we separate the integral by the size of h,

£» » » » 1 |∆ ϕpjqpxq|p |∆ ϕpjqpxq|p I.j h dhdx h dhdx . pn¡j¡1qp | |tsup 1 | |tsup 1 R IR |h| 3R h IR |h|¡3R h

Now we apply the Mean Value Theorem to the local part and we bound by the supremum in the non-local one to get £ » » » » 1 p |ϕpjqpxq ¡ 0|p ¤ pj 1q | |pp1¡tsuq¡1 I.j p ¡ ¡ q ϕ h dh dx t u dhdx n j 1 p L8 | | s p 1 R |h| 3R IR IR |h|¡3R h ¢ 1 p p 1 À pj 1q p¡tsup pjq À ¡jp¡tsup 1¡pn¡j¡1qp 1¡sp p ¡ ¡ q ϕ R R ϕ R t u R R , R n j 1 p L8 L8 R s p 78 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS that is, I.j are just error terms for 0 ¤ j ¤ n. When j n, using the Lipschitz character of Ω, we have that

» » p q p q » » 1 |A j pxq ¡ A j pyq|p 1 δp |x ¡ y|p II.j : k k dy dx ¤ dy dx Rpn¡jqp |x ¡ y|tsup 1 Rpn¡jqp Rjp |x ¡ y|tsup 1 IR »IR IR IR À 1 pp1¡tsuq 1¡sp np R dx R , R IR that is, II.j are also error terms for 0 ¤ j n. Summing up, p Ar À II.n R1¡sp, k 9 s 1 Bp,p and the lemma follows.

1 Recall that we deﬁned the arc parameter of the curve, z : R ÑBΩ with |z ptq| 1 and p 1pB qq p q Ñ | z t H Ω z t for every t and the auxiliary bump function ϕΩ : R R such that ϕΩ 2I 1 | p¡ 1pB q{ 1pB q{ q } }p 1pB q and ϕΩ p4Iqc 0 where I H Ω 2, H Ω 2 . Then, since N LppBΩq H Ω , using (1.18), (1.16) and the lifting property, we get p } }p }p ¥ q }p }p ¥ q }p 1pB q rp ¥ q spn¡1q N s pB q N z ϕΩ p N z ϕΩ 9 s H Ω N z ϕΩ t u Bp,p Ω L B 9 s p,p Bp,p and, by the Leibniz rule, Proposition 1.11 and (3.43), the reader can check with some eﬀort that

¡ n¸1 p p p pjq pn¡1¡jq pn¡1q }N} 1pB q 1 pN ¥ zq ϕ 1pB q 1 pN ¥ zq ϕ Bs pBΩq H Ω Ω tsu H Ω Ω tsu p,p B9 B9 j0 p,p p,p » » ¨ |∆ pN ¥ zqpn¡1qϕ ptq|p dh 1 h Ω dt . |h|tsup |h| R R Moreover, using (3.43) again, Fubini and the periodicity of z, we get » » » » | p q|p | p ¥ qpn¡1qp q|p } }p À |p ¥ qpn¡1qp q|p ∆hϕΩ t dh ∆h N z t dh N s pB q 1 N z t dt dt Bp,p Ω tsup | | tsup | | |h| h 4I¡t |h| h »R » R R | p ¥ qpn¡1qp q|p p À ∆h N z t dh p ¥ qpn¡1q À} }p 1 dt N z N s pB q. (3.44) tsup | | p Bp,p Ω I 2I |h| h L pIq Proposition 3.21. With the notation introduced above,

¸M } }p À} }p ϕRAk 9 s 1 N s pB q. Bp,p Bp,p Ω k1

Proof. Let us write ﬁx a window Qk. By Lemma 3.20, it only remains to bound » » p q p q |A n pxq ¡ A n pyq|p II.n k k dy dx. | ¡ |tsup 1 IR IR x y pnqp q pn¡1qp q p p qq To do so, we need to relate ∆hAk x and ∆hNk x , where Nk is the unit vector in x, Ak x normal to the graph of Ak deﬁned as p q p p q p qq p qp 1 p q ¡ q Nk x Nk,1 x ,Nk,2 x gk x Ak x , 1 , 3.6. THE GEOMETRIC CONDITION 79 with 1 g pxq a and, thus, k 1 p q2 1 Ak x A2pxqA1 pxq 1 p q ¡ k k ¡ 2p q 1 p q p q3 gk x a 3 Ak x Ak x gk x , 1 p q2 1 Ak x ¤ ¤ ¤ .

First we note the trivial pointwise bounds of the derivatives of gk. The ﬁrst two bounds are obvious and the rest of them can be deduced by induction, § § § § § 1 § |g pxq| §a § ¤ 1, k § 1 p q2 § 1 Ak x § § δ2 |g1 pxq| §A2pxqA1 pxqg pxq3§ ¤ , k k k k R ¤ ¤ ¤ , p q C |g j pxq| ¤ δ for all j n. k Rj 1 Analogously, we have similar bounds for the multiplicative inverse of gk, grk , gk a 2 |grkpxq| ¤ 1 δ , § § δ2 §gr1 pxq§ |g pxqA1 pxqA2pxq| ¤ , k k k k R ¤ ¤ ¤ , § § § p q § C §gr j pxq§ ¤ δ for every j n. k Rj Thus, for the k-th window normal vector § § p q § p q § C |N j pxq| §g j pxq§ ¤ δ for all j n and k,2 k Rj § ¢ § §¸j § ¸j p q § j p q p ¡ q § 1 1 1 |N j pxq| § A i 1 pxqg j i pxq§ À for all j n. k,1 § i k k § δ,j Ri Rj¡i Rj i0 i0 Summing up, we have that p q p q p q p q 1 j 1 j r j j À Ak , gk , gk , Nk δ,n for j n. (3.45) L8 L8 L8 L8 Rj Therefore, using the Mean Value Theorem one gets

pnqp q Thus, solving for Ak x we get

p ¡ q ° ¡ ¨ p q p ¡ ¡ q N n 1 pxq ¡ n 2 n¡1 A i 1 pxqg n 1 i pxq pnqp q k,1 i0 i k k Ak x , gkpxq and taking diﬀerences

§ ¡ © § ¡ § ¡ © § § § n¸2 § § | pnqp q| À § pn¡1qr p q§ § pi 1q pn¡1¡iqr p q§ ∆hAk x ∆h Nk,1 gk x ∆h Ak gk gk x . (3.47) i0

On one hand, using (3.45) and (3.46) we have that § ¡ © § § § § pn¡1q § § pn¡1q § pn¡1q §∆ N gr pxq§ ¤ }gr } 8 §∆ N pxq§ N |∆ gr pxq| h k,1 k k L h k,1 k,1 8 h k § § L § p ¡ q § 1 |h| À §∆ N n 1 pxq§ . h k,1 Rn¡1 R On the other hand, if we consider 0 i ¤ n ¡ 2, we obtain analogously § ¡ © § § § § § § p q p ¡ ¡ q § 1 § p q § 1 § p ¡ ¡ q § 1 §∆ A i 1 g n 1 i gr pxq§ À §∆ A i 1 pxq§ §∆ g n 1 i pxq§ |∆ gr pxq| h k k k Rn¡1¡i h k Ri h k Rn¡1 h k 1 |h| 1 |h| 1 |h| À . Rn¡1¡i Ri 1 Ri Rpn¡iq Rn¡1 R pn¡1qp q ¡ pn¡1qp q When i 0, instead, using that Nk,2 x gk x , we obtain that § ¡ © § § § § p ¡ q § 1 § § § p ¡ q § 1 § 1 n 1 r p q§ À § 1 p q§ § n 1 p q§ | r p q| ∆h Akgk gk x n¡1 ∆hAk x ∆hgk x n¡1 ∆hgk x R § § R 1 |h| § p ¡ q § 1 |h| À §∆ N n 1 pxq§ . Rn¡1 R h k,2 Rn¡1 R Back to (3.47), we have deduced that § § p q § p ¡ q § |h| |∆ A n pxq| À §∆ N n 1 pxq§ . h k h k Rn Applying this result, we obtain that § § p » » § pn¡1q pn¡1q § » » § p q ¡ p q§ R p Nk x Nk y 1 |h| II.n À dy dx dh dx tsup 1 np tsup 1 |x ¡ y| R ¡ |h| IR IR § § R IR § §p » » § pn¡1qp q ¡ pn¡1qp q§ Nk x Nk y À dy dx R1¡sp. (3.48) | ¡ |tsup 1 IR IR x y ³ p q x r Finally, note that t τk x 0 gk is the arc parameter of the curve, since dx 1 1 a . r p q 1 p q2 dt gk x 1 Ak x

r p q p ¡1p qq Thus, we have that Nk t : Nk τk t is the normal vector parameterized by the arc: according r ¡1 ¡1 to our deﬁnitions, for t P τkpIRq we have that Nkptq N ¥ zpt z pzkqq where z pzkq is assumed 3.6. THE GEOMETRIC CONDITION 81

r 1 to be chosen in I. That is, Nk is a translation of N : BΩ Ñ S parametrized by the arc z : 2I ÑBΩ ¡1 r for values close to z pzkq. Of course, we have that Nkpxq Nkpτkpxqq. Therefore, 1 p q r 1 p p qq 1 p q r 1 p p qqr p q Nk x Nk τk x τk x Nk τk x gk x and, by induction, for j ¤ n ¡ 1 we get

¸j ¸ ¹i p q p q p q j p q r i p p qq r αl p q Nk x Nk τk x Cα gk x . (3.49) i1 P i l1 α N |α|j¡i

r pjq ¤ ¡ Solving this equation for Nk and using (3.45), for j n 1 we have that p q 1 Nr j ¤ . (3.50) k 8 j L pτkpIRqq R r In consequence, taking t τkpxq and h τkpyq ¡ τkpxq, and applying (3.49), we get | pn¡1qp q ¡ pn¡1qp q| ¤ | r pn¡1qp q|}r }n¡1 Nk y Nk x ∆hr Nk t gk L8 n¸¡2 ¸ ¹j p q p q | r j p q| r αi ∆hr Nk t Cα gk L8 j1 P j i1 α N |α|n¡1¡j n¸¡1 ¸ ¸j ¹ § § p q § p q p q § p q r j §r αi p q ¡ r αi p q§ r αl Nk Cα gk x gk y gk . L8 L8 j1 P j i1 li α N |α|n¡1¡j Using (3.45), (3.46) and (3.50) we get | r pn¡1qp q|}r }n¡1 À | r pn¡1qp q| ∆hr Nk t gk L8 ∆hr Nk t , for all j ¤ n ¡ 2 and |α| n ¡ 1 ¡ j we get

¸j ¹ r p q p q p q p q 1 |x ¡ y| 1 |h| r j |r αi p q ¡ r αi p q|r αl À Nk gk x gk y gk | |¡ . L8 L8 Rj Rαi 1 R α αi Rn i1 li Thus, |r| pn¡1q pn¡1q r pn¡1q h |N pxq ¡ N pyq| À |∆r N ptq| . k k h k Rn r Therefore, using the bilipschitz change of variables t τkpxq and h τkpyq ¡ τkpxq in (3.48), we have that § § § §p » » § pn¡1qp q ¡ pn¡1qp q§ Nk x Nk y II.n À dy dx R1¡sp tsup 1 I I |x ¡ y| » R R» £ r pn¡1q p r |∆r N ptq| |h|p À h k drh dt R1¡sp. (3.51) r tsup 1 np r tsup 1 τkpIRq t τkpIRq |h| R |h| 82 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

Taking sums on 1 ¤ k ¤ M and using Lemma 3.20, (3.48) and (3.51) we get £» » ¸M ¸M r pn¡1q p |∆r N ptq| } }p À h k r 1¡sp 1¡sp ϕRAk 9 s 1 dh dt R R . Bp,p p q p q |r|tsup 1 k1 k1 τk IR t τk IR h

r p q r pn¡1qp q Recall that Nk t is a translation of the vector N parameterized by the arc. Namely, Nk t pn¡1q ¡1 pN ¥ zq pt z pzkqq and » » ¸M ¸M pn¡1q ¡1 p |∆r pN ¥ zq pt z pz qq| } }p À h k r 1¡sp ϕRAk 9 s 1 dh dt MR , Bp,p p q p q |r|tsup 1 k1 k1 τk IR t τk IR h and changing variables, » » ¸M | p ¥ qpn¡1qp q|p } }p À ∆h N z t dh À} }p ϕRAk 9 s 1 1 t u dt N Bs pBΩq. Bp,p |h| s p |h| p,p k1 I 2I

Corollary 3.22. We have that

¸N } }p À } }p N s pB q ϕRAk 9 s 1 1. Bp,p Ω Bp,p k1

1 Sketch of the proof. Use (3.44), the overlapping of the windows 20 Qk (with the obvious modiﬁca- tion for n 1) and then argue as before for the small values of h. For the remaining part, use the Lp norm of ∇pn¡1qpN ¥ zq and Proposition 1.11.

By the same reasoning, one can also prove the following corollary.

Corollary 3.23. Let Q1, Q2 be two windows of the same size of a Lipschitz domain and consider Ñ Ñ Ñ parameterizations A1 : I1 R, A2 : I2 R such that there exist rigid transformations F1 : Q1 Ñ C and F2 : Q2 C such that ¡1 ptp p qq P uq ¡1 ptp p qq P uq F1 x, A1 x : x I1 F2 x, A2 x : x I2 .

p q p q 1 ¤ Assume further that ` I1 ` I2 R and take ϕ1, ϕ2 to be bump functions such that χ Ij ¤ 3 ϕj χIj . For s n 1 we have that } }p } }p ϕ1A1 9 s 1 ϕ2A2 9 s 1. Bp,p Bp,p

3.7 A localization principle: bounded smooth domains

We are going to follow a standard localization argument, so we will give a sketch, leaving some details for the reader. Let us start with some remarks. First we make some general observations on admissible domains. In these ﬁrst two remarks we assume n ¥ 2 since the case n 1 is simpler (there is no need for rotations). 3.7. A LOCALIZATION PRINCIPLE: BOUNDED SMOOTH DOMAINS 83

Remark 3.24. If Ω is a pδ, R, n, pq-admissible domain with defining function A, then for every τ PBΩ one can perform a translation of the domain that sends τ to the origin and a rotation in the r ¡ same spirit of Definition 1.4, so that BΩ coincides with the graph of a new function A P Cn 1,1pRq r r r r1 in a certain ball Bp0, Rq with fixed radius R (depending on δ and R) with Ap0q 0, A p0q 0, r1 r r r r r A ¤ δ and supppAq r¡2R, 2Rs. By Corollary 3.23, we have that A À}A} s 1 for 8 9 s Bp,p L Bp,p s n 1. Therefore Ar determines a pδ,r R,r n, pq-admissible domain Ωr with compactly supported defining function (see Figure 3.5). P 2 ¡ ¡ ¤ ¤ p q p q P p rq Consider γ Z with¡ γ1© γ2 n 2 and γ1 γ2 0. Note that χΩ z χΩr z for z B 0, R . P X Rr γ p q γ p q γ p ¡ qp q For every z Ω B 0, 2 we use the decomposition T χΩ z T χΩr z T χΩ χΩr z : » | p q ¡ p q| γ γ χΩ w χΩr w γ 1 |T χ pzq| ¤ |T χr pzq| dmpwq À |T χr pzq| . (3.52) Ω Ω | ¡ |n 2 Ω r |w|¡Rr{2 w z Rn

Figure 3.5: Disposition in Remark 3.24 before the rotation and the translation.

Next we take a look at admissible domains with compact support.

Remark 3.25. Let Ω be a pδ, R, n, pq-admissible domain with deﬁning function A supported in I r¡2R, 2Rs. For a given ¡ 0 small enough, take ρ to be the radius ρ from Theorem 3.11 associated to the parameters δ,r R,r n, p of the previous remark. We assume ρ Rr{2. Since A is supported in I, we can cover the area close to the graph G tx i Apxq : x P Iu by a ﬁnite number of balls of radius ρ (see Figure 3.6). In each ball we can apply (3.52) for the r P 2 ¡ ¡ ¤ ¤ corresponding domain ¨ Ω. Thus, given γ Z with γ1 γ2 n 2 and γ1 γ2 0, writing ρ UρG zPG B z, 2 and using Theorem 3.11 we have that ¡ © } γ }p ¤ } }p p q|γ|p T χΩ LppΩXU Gq C A n 1¡1{p 1 , ρ Bp,p pBΩq with C depending on n, p, δ, R and (but not on |γ|). r Finally, for z R UρG we can use the same argument of (3.52) replacing the domain Ω by the 2 half plane R . Namely, § § » γ § γ § 1 |T χ pzq| ¤ §T χ 2 pzq§ dmpwq. Ω R n 2 2 |w ¡ z| Ω∆R

In that case, the ﬁrst term is zero just by (3.27). Since A is compactly supported in r¡2R, 2Rs and it is Lipschitz with constant δ, the domain of integration of the second term is contained in γ Qp0, 2p1 δqRq. Thus, when z P ΩzQp0, 4p1 δqRq then |T χΩpzq| is bounded by a constant times R2 P X p p q qz | γ p q| C |z|n 2 . When z Ω Q 0, 4 1 δ R UρG then T χΩ z is bounded by ρn . Summing up, we 84 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

Figure 3.6: Decomposition of a pδ, R, n, pq-admissible domain Ω with defining function A supported in I r¡2R, 2Rs considered in Remark 3.25. have a global bound ¢ 2 } γ }p ¤ } }p p q|γ|p R À} }p p q|γ|p T χΩ LppΩq C A n 1¡1{p 1 np A n 1¡1{p 1 , Bp,p ρ Bp,p with constants depending on n, p, δ, R and . Now we turn to the case of bounded domains. First we note how differentiation works for γ T χΩ. p ¡ q Remark 3.26. Consider a bounded pδ, Rq ¡ C n 1,1 domain Ω and let us fix γ P Z2 with either ¥ ¥ P 2 | | P γ1 0 or γ2 0, and α N0 with modulus α n. Then for z Ω we have $ 'CnχΩpzq if γ pn ¡ 1, ¡1q and α pn, 0q &' or γ p¡1, n ¡ 1q and α p0, nq, DαT γ 1pzq Ω '0 if α ¡ γ ¥ 0 or α ¡ γ ¥ 0 except in the previous case, %' 1 1 2 2 γ¡α p q Cγ,αTΩ 1 z otherwise,

α n where D stands for the weak derivative in Ω. The constants satisfy |Cγ,α| À p|γ| nq .

Proof. Let us assume that γ2 ¥ 0. If γ1 ¥ 0 as well, diﬀerentiating a polynomial under the integral γ γ1 γ2 sign makes the proof trivial, so we assume γ1 ¤ ¡1. Recall that we write w w w . For every z P Ω choose εz : distpz, BΩq{2. By (3.9), Green’s formula and (3.8) we get that » » γ p q p ¡ qγ p q i p ¡ qγ p0,1q T χΩ z z w dm w p q z w dw, (3.53) ΩzBpz,εz q 2 γ2 1 BΩ and we can diﬀerentiate under the integral sign. If γ2 ¥ α2, then we have » i pγ2 1q! p¡γ1 α1 ¡ 1q! ¡ p q α γ p q p¡ qα1 p ¡ qγ α 0,1 D T χΩ z p q 1 p ¡ q p¡ ¡ q z w dw. 2 γ2 1 γ2 α2 1 ! γ1 1 ! BΩ 3.8. BOUNDED SMOOTH DOMAINS, SUPERCRITICAL CASE 85

Since γ2 ¡ α2 ¥ 0 and γ1 ¡ α1 0, we can apply (3.53) to γ ¡ α instead of γ and, thus, pγ q! p¡γ α ¡ 1q! α γ α1 2 1 1 γ¡α D T χΩpzq p¡1q T χΩpzq. pγ2 ¡ α2q! p¡γ1 ¡ 1q!

If γ2 1 α2 we must pay special attention. In that case diﬀerentiating under the integral sign in (3.53) we get » i pγ2q! p¡γ1 α1 ¡ 1q! ¡ p q α γ p q p¡ qα1 p ¡ qγ α 0,1 D T χΩ z 1 p ¡ q p¡ ¡ q z w dw 2 » γ2 α2 1 ! γ1 1 ! BΩ 1 Cγ,α dw, p ¡ q¡γ1 α1 BΩ z w

n where |Cγ,α| À p|γ| nq . If, moreover, γ1 ¡ α1 ¤ ¡2, we can use (3.8) to write » » α γ 1 D T χΩpzq Cγ,α dw Cγ,α 0 dmpwq 0. (3.54) p ¡ q¡γ1 α1 BΩYBBp0,εz q z w ΩzBBp0,εz q

Otherwise, that is, if γ2 1 α2 and γ1 ¡ α1 ¡1, then α p0, nq and γ p¡1, n ¡ 1q. This implies that » α γ p q 1 p q D T χΩ z Cn p ¡ q dw CnχΩ z , (3.55) BΩ z w with |Cn| À pn ¡ 1q!. Let us remark the fact that γ p¡1, 0q together with α p0, 1q is the case of the B¯-derivative of the Cauchy transform, which is the identity. Finally, if γ2 α2 ¡ 1, then diﬀerentiating (3.54) or (3.55) we get

α γ D T χΩpzq 0.

One can argue analogously if γ1 ¥ 0. By the preceding remarks, Theorem 3.19 and other standard arguments, one gets the following theorem.

n¡1,1 n 1¡1{p Theorem 3.27. Let Ω be a bounded pδ, Rq-C domain with parameterizations in Bp,p . P 2ztp¡ ¡ qu ¡ γ P n,pp q Then, for any γ Z 1, 1 with γ1 γ2 2, we have that T χΩ W Ω and, in particular, for any ¡ 0, we have that ¡ © } n γ }p À | |np } }p p q|γ|p ∇ T χΩ LppΩq C γ N n¡1{p 1 , Bp,p pBΩq

1 where C depends on p, n, δ, R, the length of the boundary H pBΩq and but not on |γ|.

3.8 Bounded smooth domains, supercritical case

Using Theorems 3.27 and 2.1, we will prove the following theorem. Theorem 3.28. Consider p ¡ 2, n ¥ 1 and let Ω be a bounded Lipschitz domain with param- n 1¡1{p eterizations in Bp,p . Then, for every ¡ 0 there exists a constant C such that for every P 2ztp¡ ¡ qu ¥ ¡ multiindex γ Z 1, 1 with γ1 γ2 2, one has ¡ © | | } γ } ¤ | |n γ1 γ2 2 } } p q γ p qγ1 γ2 2 T n,pp qÑ n γ γ 2,pp q C γ N n¡1{p 1 diam Ω . Ω W Ω W 1 2 Ω Bp,p pBΩq (3.56) 86 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

P p mq In particular, for every m N we have that the iteration of the Beurling transform B Ω is bounded on W n,ppΩq, with norm ¡ © m n 1 m }pB qΩ} n,pp qÑ n,pp q ¤ Cm }N} n¡1{p p1 q . (3.57) W Ω W Ω Bp,p pBΩq

n 1¡1{p n 1¡2{p Proof. Note that by (1.15), we have that Bp,p B8,8 and, since 1 ¡ 2{p ¡ 0, we also n 1¡2{p n,1¡2{p n¡1,1 have that B8,8 C (see Remark 1.12) so Ω is in fact a pδ, Rq-C -domain, where δ and R depend on the size of the local parameterizations of the boundary and on }N} n¡1{p , Bp,p pBΩq and we can use Theorem 3.27. P 2ztp¡ ¡ qu ¡ First we study the case γ1 γ2 2 0. Consider a given γ Z 1, 1 with γ1 γ2 2. m m p¡1q m p¡m¡1,m¡1q p Recall that for m 0, B π T by (3.2). The proof of the L boundedness of 2 these operators with norm smaller than Cpm can be found in [AIM09, Corollary 4.5.1]. Thus,

π |γ| }T γ } }Bm} À m . (3.58) LpÑLp m LpÑLp 2

On the other hand, a short computation shows that the constant CK in (1.41) for these kernels is | n p q|| |j 2 À | |n CKγ sup ∇ Kγ z z γ , (3.59) z0 with constant depending on n. P 2 p q λ1 λ2 Following the notation of Theorem 2.1, given a multiindex λ N0, we write Pλ z z z¯ , λ that is, Pλpzq z . In order to use this theorem, it only remains to check the bounds for }DαT γ P } for all multiindices α, λ P 2 with |α| n and |λ| n. Using the binomial Ω λ LppΩq ° ¨ N0 λ p¡ q|ν| λ p ¡ qν λ¡ν expansion w ν¤λ 1 ν z w z , we can write » ¢ ¸ λ T γ P pzq lim pz ¡ wqγ wλ dmpwq p¡1q|ν| zλ¡ν T γ ν χ pzq. Ω λ Ñ Ω ε 0 zB pzq ν C ε ~0¤ν¤λ

Diﬀerentiating (and assuming that 0 P Ω) we ﬁnd that

¸ ¸n | n γ p q| À n p p qqn| j γ ν p q| ∇ TΩPλ z 2 1 diam Ω ∇ T χΩ z ~0¤ν¤λ j0 and, thus, by the equivalence of norms in the Sobolev space (1.8), we have that ¢ ¸ p } n γ }p À n |ν| γ ν γ ν p ∇ TΩPλ LppΩq Ω ∇ T χΩ T χΩ pp q , LppΩq L Ω ~0¤ν¤λ with constants depending on n, p and the diameter and the Sobolev embedding constant of Ω. By Remark 3.26, Theorem 3.27 and (3.58), we have that ¸ ¡ © ¸ } n γ }p À | |np } }p p q|ν|p } ν }p ∇ TΩPλ LppΩq ν N n¡1{p 1 T χΩ LppΩq. (3.60) Bp,p pBΩq γ¤ν¤γ λ γ¤ν¤γ λ

P p P 1 } ¦ } ¤ } } } } The Young Inequality (1.10) says that for all functions f L and g L , f g Lp f Lp g L1 . Thus, for γ ν ¤ γ λ we have that

} ν } ¤ p qν1 ν2 2} } TΩf Lp diam Ω f Lp , (3.61) 3.8. BOUNDED SMOOTH DOMAINS, SUPERCRITICAL CASE 87

} ν }p À p qpn¡1qp 2 and taking f χΩ, T χΩ Lp 1 diam Ω . For ν γ, the same holds by the boundedness of the iterates of the Beurling transform with a slightly worse constant. Namely,

} γ } ¤ | |} } TΩf Lp Cp γ f Lp . (3.62) Since p ¡ 2, putting (2.1), (3.58), (3.59), (3.60), (3.61) and (3.62) together, we get ¸ } n γ } À } γ } } np q} ∇ TΩ W n,ppΩqÑLppΩq CK T LpÑLp ∇ TΩPλ LppΩq | | ¡ λ n © À | |n | | | |n } } p q|γ| γ γ γ N n¡1{ppB q 1 ¡ Bp,p Ω © n |γ| À |γ| }N} n¡1{p p1 q , (3.63) Bp,p pBΩq with constants depending on n, p, δ, the diameter of Ω, its Sobolev embedding constant and , but not on γ. This proves (3.56) when γ1 γ2 ¡2 and (3.57) for every m ¡ 0. It remains to study the operators of homogeneity greater than ¡2. In that case we will see that we can diﬀerentiate under the integral sign to recover the previous situation. Fix γ P Z2 such that γ ¡ } } ¤ p qγ1 γ2 2} } γ1 γ2 2 0. By (3.61) we have that TΩf Lp diam Ω f Lp . Thus, to prove (3.56), it suﬃces to see that for f P W n,ppΩq we have ¡ © p q p | | n γ1 γ2 2 γ ¤ | | n γ1 γ2 2 p } } p q γ p } } ∇ TΩf pp q C γ N n¡1{p 1 f W n,ppΩq. L Ω Bp,p pBΩq

P 2 | | By (3.63) it is enough to check that for any ν N0 with ν γ1 γ2 2, we have $ 'CnχΩfpzq if γ p|ν| ¡ 1, ¡1q and ν p|ν|, 0q &' or γ p¡1, |ν| ¡ 1q and ν p0, |ν|q, Dν T γ fpzq (3.64) Ω '0 if ν ¡ γ ¥ 0 or ν ¡ γ ¥ 0 except in the previous case, %' 1 1 2 2 γ¡ν p q Cν,γ TΩ f z otherwise,

|ν| with |Cn|, |Cν,γ | À p|ν| |γ|q . To prove this statement, take α ¤ ν ¡ p1, 0q, and note that the partial derivative is

γ¡α γ¡α ¡ B T fpzq ¡ iB T fpzq BT γ αfpzq x Ω y Ω Ω 2 ¡ ¡ T γ αpf ¡ fpzqqpz hq ¡ T γ αpf ¡ fpzqqpzq lim Ω Ω hÑ0 2h ¡ ¡ T γ αpf ¡ fpzqqpz i hq ¡ T γ αpf ¡ fpzqqpzq Ω Ω B γ¡α p q p q lim TΩ χΩ z f z hÑ0 2ih : I II III , where h is assumed to be real. Now, the principal value is not needed because γ1 ¡ α1 γ2 ¡ α2 ¡ ¡2, so » ppz h ¡ wqγ¡α ¡ pz ¡ wqγ¡αq rfpwq ¡ fpzqs I lim dmpwq. Ñ h 0 Ω 2h Moreover, since f P C0,σ for a certain σ ¡ 0 by the Sobolev Embedding Theorem, we get » p|z h ¡ w|γ¡α |z ¡ w|γ¡αq |fpwq ¡ fpzq| lim dmpwq 0. Ñ h 0 Bpz,2|h|q 2h 88 CHAPTER 3. CHARACTERISTIC FUNCTIONS OF PLANAR DOMAINS

On the other hand, using the Taylor expansion of order two of pz ¡w ¤qγ¡α around 0, there exists ε εph, w, zq with |ε| h such that » ¢ B pz ¡ w ¤qγ¡αp0q B2pz ¡ w ¤qγ¡αpεqh I lim x x pfpwq ¡ fpzqq dmpwq. Ñ h 0 ΩzBpz,2|h|q 2 2

Arguing analogously for II , we get that » γ¡α¡p1¡0q I II lim pγ1 ¡ α1qpz ¡ wq fpwq dmpwq hÑ0 Ω»zBpz,2|h|q γ¡α¡p1¡0q ¡ lim pγ1 ¡ α1qpz ¡ wq dmpwqfpzq Ñ h 0 ΩzBpz,2|h|q

(when taking limits, the Taylor remainder vanishes by the H¨oldercontinuity of f). If γ1 ¡ α1 0 then this part is null and III will be also null unless γ2 ¡ α2 ¡1 by Remark 3.26. Otherwise, the last term coincides with III and they cancel out. By induction, we get (3.64). Chapter 4

An application to quasiconformal mappings

P 8 } } 1 k Let µ L supported in a certain ball B C with k : µ L8 1 and consider K : 1¡k . We say that f is a K-quasiregular solution to the Beltrami equation

B¯f µ Bf (4.1)

P 1,2 with Beltrami coefficient µ if f Wloc , that is, if f and ∇f are square integrable functions in any compact subset of C, and B¯fpzq µpzqBfpzq for almost every z P C. Such a function f is said to be a K-quasiconformal mapping if it is a homeomorphism of the complex plane. If, moreover, p q p 1 q Ñ 8 f z z O z as z , then we say that f is the principal solution to (4.1). Given a compactly supported Beltrami coefficient µ, the existence and uniqueness of the principal solution is granted by the measurable Riemann mapping Theorem (see [AIM09, Theorem 5.1.2], for instance). The principal solution can be given by means of the Cauchy and the Beurling transforms. If we call h : pI ¡ µBq¡1µ, then fpzq Chpzq z is the principal solution of (4.1) because B¯f h and Bf Bh 1. In this chapter we use the results of the previous one to improve the known results on the relation between the smoothness of a quasiconformal mapping and its Beltrami coefficient.

Theorem 4.1. Let n P N, let Ω be a bounded Lipschitz domain with outward unit normal vector n¡1{ppB q 8 P n,pp q } } p q N in Bp,p Ω for some 2 p and let µ W Ω with µ L8 1 and supp µ Ω. Then, the principal solution f to (4.1) is in the Sobolev space W n 1,ppΩq.

Section 4.1 is devoted to collecting some results that we will use in this chapter. Then in Section 4.2 we perform the main argument to present the outline of the proof of Theorem 4.1, reducing it to two lemmas, one on the compactness of a commutator which is proven in Section 4.3 and one on m the compactness of χΩB pχΩc B pχΩ¤qq which is studied in Section 4.5. Finally, Subsection 4.4 is devoted to establishing a generalization of some results in [MOV09] to be used in the last section.

89 90 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

4.1 Some tools

Let us sum up some properties of the Cauchy transform which will be useful in this section (see P 1 [AIM09, Theorems 4.3.10, 4.3.12 and 4.3.14]). We write IΩg : χΩ g for every g Lloc. Theorem 4.2. Let 1 p 8. Then

• For every f P Lp, we have that BCf Bf and B¯Cf f.

• For every function f P L1 with compact support, if p ¡ 2 or f dm 0, then we have that ﬄ } } À p p qq} } Cf Lp p diam supp f f Lp . (4.2)

• Let Ω be a bounded open subset of C. Then, we have that ¥ pp q Ñ 1,pp q IΩ C : L C W Ω (4.3)

is bounded.

We will use some results from [RS96, Section 4.6.4]. P d 8 Theorem 4.3. Let n N and n p . If Ω is a Lipschitz domain, then for every pair f, g P W n,ppΩq we have that

} } ¤ } } } } f g W n,ppΩq Cd,n,p,Ω f W n,ppΩq g W n,ppΩq.

Moreover, if the boundary of Ω has parameterizations in C1 and p ¡ d, then for m P N we have that } m} ¤ n } }m¡n } }n f W n,ppΩq Cd,n,p,Ω m f L8pΩq f W n,ppΩq.

Proof. We have that W n,ppRdq is a multiplicative algebra (see [RS96, Section 4.6.4]), that is, if f, g P W n,ppRdq, then } } ¤ } } } } f g W n,p Cn,p f W n,p g W n,p . Since Ω is an extension domain (see [Eva98, Section 5.4]), we have a bounded operator E : n,pp q Ñ n,pp q | | P n,pp q W Ω W C such that Ef Ω f Ω for every f W Ω . The ﬁrst property is a consequence of this fact. 8 To prove the second property,¡ assume that f ©P C pΩq. By (1.8) we only need to prove that }Bnp mq} ¤ n } }m¡n } }n ¤ ¤ k f LppΩq Cd,n,p,Ωm f L8pΩq f W n,ppΩq for 1 k d. By the Leibniz’ rule, it is an exercise to check that ¸ ¹n Bnp mq m¡n Bji k f f c~j,m k f, (4.4) ~P n i1 j N0 ji¥ji 1 for 1¤i n |~j|n ° ¡ n ~ p ¤ ¤ ¤ q with c~j,m 0 and ~j c~j,m m . Consider j n, 0, , 0 . Then, by (1.14), that is, the Sobolev embedding Theorem, we get ¹n ji n n¡1 n n¡1 n B f B f f ¤ }B f} }f} 8 À }f} . (4.5) k k LppΩq k LppΩq L pΩq Ω,p W n,ppΩq i1 LppΩq 4.2. A FREDHOLM THEORY ARGUMENT 91

Otherwise, the indices ji n for 1 ¤ i ¤ n and, since p ¡ d, we use (1.14) again to state that ¹n ¹n ¹n j j 1 j n B i ¤ B i | | p À B i ¤ } } k f k f Ω Ω,p k f f W n,ppΩq. (4.6) L8pΩq W 1,ppΩq i1 LppΩq i1 i1

By (4.4), (4.5), (4.6) and the triangle inequality, this implies that ¸ ¹n n m m¡n ji n m¡n n }B pf q} ¤ f c B f À m }f} 8 }f} . k LppΩq L8pΩq ~j,m k L pΩq W n,ppΩq ~P n i1 pp q j N0 L Ω ji¥ji 1 for 1¤i n |~j|n

By an approximation procedure this property applies to every f P W n,ppΩq.

Finally, let us recall some results on compact operators and Fredholm theory (see [Sch02, Chapters 4 and 5], for instance). In the following deﬁnitions and results, X,Y,Z are Banach spaces, and all the operators are assumed to be bounded and linear.

Deﬁnition 4.4. An operator K : X Ñ Y is called compact if every bounded sequence txkuk X t u t p qu has a subsequence xkj j such that K xkj j converges in Y . Proposition 4.5. If A : X Ñ Y , B : Z Ñ X are operators and K : Y Ñ Z is compact, then K ¥ A and B ¥ K are compact.

Theorem 4.6. If L : X Ñ Y and there is a sequence of compact operators Kj : X Ñ Y such that jÑ8 }L ¡ Kj} ÝÝÝÑ 0, then L is compact. Deﬁnition 4.7. An operator A : X Ñ Y is said to be a Fredholm operator if it has closed range and the dimensions of the kernels NpAq and NpA1q are ﬁnite. The Fredholm index of A is ipAq : dim NpAq ¡ dim NpA1q.

Theorem 4.8. Let A : X Ñ Y , A1,A2 : Y Ñ X, let K1,B1 : X Ñ X and K2,B2 : Y Ñ Y with Bj invertible and Kj compact and such that A1 ¥ A B1 ¡ K1 and A ¥ A2 B2 ¡ K2. Then A is Fredholm.

Theorem 4.9. The index is continuous with respect to the operator norm.

4.2 A Fredholm theory argument

P p mq mp q P 1 Consider m N. Recall that B Ωg χΩB χΩg for g Lloc (see Deﬁnition 3.2) and IΩg n,p 2 χΩ g. Note that IΩ is the identity in W pΩq. Let us deﬁne Pm : IΩ µBΩ pµBΩq m¡1 n,p ¤ ¤ ¤ pµBΩq . Since W pΩq is a multiplicative algebra (by Theorem 4.3), we have that Pm is bounded on W n,ppΩq. It follows that

m Pm ¥ pIΩ ¡ µBΩq pIΩ ¡ µBΩq ¥ Pm IΩ ¡ pµBΩq , (4.7) and

m m m m m m m m m IΩ ¡ pµBΩq pIΩ ¡ µ pB qΩq µ ppB qΩ ¡ pBΩq q pµ pBΩq ¡ pµBΩq q p1q m p2q p3q Am µ Am Am . (4.8) 92 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

m m m Note the diﬀerence between pBΩq g χΩBp. . . χΩBpχΩBpχΩgqqq and pB qΩg χΩB pχΩgq. m Next we will see that for m large enough, the operator IΩ ¡ pµBΩq is the sum of an invertible operator and a compact one. p3q m m m First we will study the compactness of Am µ pBΩq ¡ pµBΩq . To start, writing rµ, BΩsp¤q for the commutator µBΩp¤q ¡ BΩpµ¤q we have the telescopic sum

m¸¡1 ¨ p3q m¡jr s j¡1p qm¡1 p qp m¡1p qm¡1 ¡ p qm¡1q Am µ µ, BΩ µ BΩ µBΩ µ BΩ µBΩ j1 m¸¡1 ¨ m¡jr s j¡1p qm¡1 p q p3q µ µ, BΩ µ BΩ µBΩ Am¡1. j1

p3q Arguing by induction we can see that Am can be expressed as a sum of operators bounded on n,p W pΩq which have rµ, BΩs as a factor. It is well-known that the compactness of a factor implies the compactness of the operator (see Proposition 4.5). Thus, the following lemma, which we prove p3q in Section 4.3 implies the compactness of Am .

n,p Lemma 4.10. The commutator rµ, BΩs is compact in W pΩq. ¨ p2q m m m¡1 Consider now Am pB qΩ ¡ pBΩq . We deﬁne the operator Rmg : χΩB χΩc B pχΩ gq whenever it makes sense. This operator can be understood as a (regularizing) double reﬂection with respect to the boundary of Ω. For every g P W n,ppΩq we have that ¨ ¨¨¨ p2q m¡1 m¡1 A g χΩ B pχΩ χΩc qB pχΩ gq ¡ B χΩ pBΩq g m ¨ ¨ ¨ m¡1p q m¡1p ¤q ¡ p p¤qqm¡1 ¥ p2q χΩB χΩc B χΩg χΩB χΩ B χΩ BΩ g Rmg BΩ Am¡1g.

Note that by deﬁnition ¡ © p2q ¡ ¥ p2q Rm Am BΩ Am¡1 (4.9)

n,p is bounded on W pΩq. In Section 4.5 we will prove the compactness of Rm, which, by induction, p2q will prove the compactness of Am .

n,p Lemma 4.11. For every m, the operator Rm is compact in W pΩq. Now, the following claim is the remaining ingredient for the proof of Theorem 4.1.

p1q Claim 4.12. For m large enough, Am is invertible. Proof. Since p ¡ 2 we can use Theorem 4.3 to conclude that for every g P W n,ppΩq } mp mq } À} m} }p mq } µ B Ωg W n,ppΩq µ W n,ppΩq B Ωg W n,ppΩq À n} }m¡n} }n }p mq } } } m µ L8 µ W n,ppΩq B Ω W n,ppΩqÑW n,ppΩq g W n,ppΩq. By Theorem 3.28, for any ¡ 0 there are constants depending on the Lipschitz character of Ω (and other parameters) but not on m, such that ¡ © }p mq }p À pn 1qp p qmp } }p B Ω W n,ppΩqÑW n,ppΩq m 1 N n¡1{p . Bp,p pBΩq

In particular, if we choose 1 1 , we get that for m large enough, the operator norm }µ}8 } mp mq } p1q µ B Ω W n,ppΩqÑW n,ppΩq 1 and, thus, Am in (4.8) is invertible. 4.3. COMPACTNESS OF THE COMMUTATOR 93

Proof of Theorem 4.1. Putting together Lemmas 4.10 and 4.11, Claim 4.12, and (4.8), we get that m IΩ ¡ pµBΩq can be expressed as the sum of an invertible operator and a compact one for m big enough and, by (4.7), we can deduce that IΩ ¡ µBΩ is a Fredholm operator (see Theorem 4.8). The same argument works with any other operator IΩ ¡ tµBΩ for 0 t 1{}µ}8. It is well known that the Fredholm index is continuous with respect to the operator norm on Fredholm operators (see Theorem 4.9), so the index of IΩ ¡ µBΩ must be the same index of IΩ, that is, 0. It only remains to see that this operator is injective to prove that it is invertible. Since µ is continuous, we know from [Iwa92, Section 1] that the operator I ¡ µB is injective in Lp. Thus, if n,p g P W pΩq, and pIΩ ¡ µBΩqg 0, we deﬁne Gpzq gpzq if z P Ω and Gpzq 0 otherwise, and then we have that

pI ¡ µBqG pI ¡ µχΩBqpχΩGq pIΩ ¡ µBΩqg 0.

By the injectivity of the former, we get that G 0 and, thus, g 0 as a function of W n,ppΩq. Now, remember that the principal solution of (4.1) is fpzq Chpzq z where

h pI ¡ µBq¡1µ, that is, h µBphq µ, so suppphq supppµq Ω and, thus, for almost every z P Ω we have that χΩpzqhpzq µpzqBΩphqpzq hpzq µpzqBphqpzq µpzq, so

¡1 h|Ω pIΩ ¡ µBΩq µ, proving that h P W n,ppΩq. By Theorem 4.2 we have that Ch P LppCq. Since the derivatives of ¯ n,p the principal solution, Bf h and Bf Bh 1 BΩh χΩc Bh 1, are in W pΩq, we have f P W n 1,ppΩq.

4.3 Compactness of the commutator

n,p 8 Proof of Lemma 4.10. We want to see that for any µ P W pΩq X L , the commutator rµ, BΩs is compact. The idea is to show that it has a regularizing kernel. In particular, we will prove that assuming some extra condition on the regularity of µ, then the commutator maps W n,ppΩq to W n 1,ppΩq. This will imply the compactness of the commutator as a self-map of W n,ppΩq and, by a classical argument on approximation of operators, this will be extended to any given µ. 8p q First we will see that we can assume µ to be Cc C without loss of generality by an approximation procedure. Indeed, since Ω is an extension domain, for every µ P W n,ppΩq, there is a function Eµ with }Eµ} n,pp q ¤ C}µ} n,pp q such that Eµ|Ω µχΩ. Now, Eµ can be approxi- W C W Ω 8 n,p t u P p q p q mated by a sequence of functions µj j N Cc C in W C and one can deﬁne the operator n,p n,p n,p rµj, BΩs : W pΩq Ñ W pΩq. Since W pΩq is a multiplicative algebra, one can check that tr su P r s µj, BΩ j N is a sequence of operators converging to µ, BΩ in the operator norm. Thus, it is n,p enough to prove that the operators rµj, BΩs are compact in W pΩq for all j by Theorem 4.6. 8p q r s Let µ be a Cc C function. We will prove that the commutator µ, BΩ is a smoothing operator, n,p n 1,p n,p mapping W pΩq into W pΩq. Consider f P W pΩq, a Whitney covering W with appropriate P ¤ ¤ j À Cj constants and, for every Q W, choose a bump function χ 3 ϕQ χ2Q with ∇ ϕQ 8 j . 2 Q L `pQq n¡1 Recall that in Section 1.5 we deﬁned P3Q f to be the approximating polynomial of f around 3Q. 94 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

Then, we break the norm in three terms, ¡¡ © © ¸ p n 1r s p À n 1r s ¡ n¡1 ∇ µ, BΩ f p p ∇ µ, BΩ f P f ϕQ (4.10) L pΩq 3Q pp q P L Q Q W ¡¡ © © ¸ p n 1 n¡1 ∇ rµ, BΩs f ¡ P f pχΩ ¡ ϕQq 3Q pp q P L Q Q W ¡ © ¸ p n 1r s n¡1 ∇ µ, BΩ P3Q f : 1 2 3 . LppQq QPW

First we study 1 . In this case, we can use the following classical trick for compactly supported P 8p q P p P 1,pp q functions. Given a ball B C, ϕ Cc B and g L , then Cg W 2B by (4.3). Therefore, we can use Leibniz’ rule (1.11) for the ﬁrst order derivatives of ϕ ¤ Cg, and by Theorem 4.2 we get

ϕ ¤ Bpgq ¡ Bpϕ ¤ gq ϕ ¤ BCg ¡ Bpϕ ¤ B¯Cgq ¡Bϕ ¤ Cg Bpϕ ¤ Cgq ¡ B¯Bpϕ ¤ Cgq BpB¯ϕ ¤ Cgq BpB¯ϕ ¤ Cgq ¡ Bϕ ¤ Cg. (4.11) P 8p q Thus, for a ﬁxed cube Q, since we assumed that µ Cc C , we have that ¡¡ © © ¡ ¡¡ © ©© ¡¡ © © r s ¡ n¡1 B¯ ¤ ¡ n¡1 ¡B ¤ ¡ n¡1 µ, B f P3Q f ϕQ B µ C f P3Q f ϕQ µ C f P3Q f ϕQ .

Therefore, using the boundedness of the Beurling transform and the fact that it commutes with derivatives, we have that ¡¡ © © ¸ p n 1r s ¡ n¡1 1 ∇ µ, B f P3Q f ϕQ LppQq Q ¡ ¡¡ © ©© ¡ ¡¡ © ©© ¸ p ¸ p À n 1 B¯ ¤ ¡ n¡1 n 1 B ¤ ¡ n¡1 p ∇ µ C f P3Q f ϕQ ∇ µ C f P3Q f ϕQ Lp Lp Q Q ¡¡ © © ¸ n¸1 p ¤ } }p j ¡ n¡1 µ W n 2,8 ∇ C f P3Q f ϕQ Lp Q j0 and, using the identities BC B, B¯C Id (when j ¡ 0 in the previous sum) together with (4.2) from Theorem 4.2 (when j 0) we can estimate £ ¡¡ © © ¸ n¸1 p p À } }p j¡1 ¡ n¡1 p qp ¡ n¡1 1 p µ W n 2,8 ∇ f P3Q f ϕQ ` Q f P3Q f Lpp2Qq Lpp2Qq Q j1 and, by the Poincar´einequality (1.33) we get

¸ n¸ 1 À } }p p qpn 1¡jqp} n }p À } }p } n }p 1 n,p µ W n 2,8 ` Q ∇ f Lpp2Qq n,Ω µ W n 2,8 ∇ f LppΩq. Q j0 Second, we bound using duality ¡¡ © © ¸ p n 1 n¡1 2 ∇ rµ, BΩs f ¡ P f pχΩ ¡ ϕQq 3Q pp q P L Q ¤Q W p ¸ » § ¡¡ © © § ¥ § n 1 n¡1 § sup §∇ rµ, BΩs f ¡ P f pχΩ ¡ ϕQq pzq gpzq§ dmpzq . 1 3Q gPLp :}g} 1 ¤1 P Q Lp Q W 4.3. COMPACTNESS OF THE COMMUTATOR 95

Let Q be a Whitney cube, let z P Q and let α P N2 with |α| n 1. Then, if we call µpzq ¡ µpwq K pz, wq , µ pz ¡ wq2 ¡ © ¡ n¡1 p ¡ q then, since z is not in the support of f P3Q f χΩ ϕQ , we have that ¡¡ © © » ¡ © αr s ¡ n¡1 p ¡ q p q α p q p q ¡ n¡1 p q p ¡ p qq D µ, BΩ f P3Q f χΩ ϕQ z Dz Kµ z, w f w P3Q f w 1 ϕQ w dm. Ω Note that ¢ 1 ¸ α 1 DαK pz, wq pµpzq ¡ µpwqqDα Dα¡γ µpzqDγ , z µ z pz ¡ wq2 γ z pz ¡ wq2 γ α | p q ¡ p q| ¤ } } | ¡ | so using µ z µ w ∇µ L8 z w we get

α 1 |D K pz, wq| ¤ C p q}µ} 8 . z µ n,diam Ω W n 1, |z ¡ w|n 2 Note the similitude between this estimate and the size condition (1.41) (take smoothness n, di- } } mension 2 and Calder´on-Zygmund constant Cn,diampΩq µ W n 1,8 ). Using (1.34) and Lemma 2.3 we get ³ § § § n¡1 § » 1 ¸ §fpwq ¡ P fpwq§ dmpwq p S 3Q 2 À}µ} 8 sup |gpzq| dmpzq W n 1, p qn 2 }g} 1 ¤1 D Q, S Q Lp Q,S ¸ ¸ } } } n } p qn¡1 p q2 g L1pQq ∇ f L1p3P qD P,S ` S À}µ} 8 sup W n 1, p q p qn 2 }g} 1 ¤1 ` P D Q, S Lp Q,S P PrQ,Ss À } } } n } n,Ω µ W n 1,8 ∇ f LppΩq.

Next we use a T p1q argument reducing 3 to the boundedness of rµ, BΩsp1q. Consider the p q p ¡ qγ p ¡ qγ1 p ¡ qγ2 P 2 monomials PQ,γ z : z zQ z zQ z zQ for γ N0 where zQ stands for the center n¡1 of Q (see complex notation° in Section 1.2). The Taylor expansion (1.31) of P3Q f around zQ can n¡1 p q p q be written as P3Q f z |γ| n mQ,γ PQ,γ z . Thus, we have that ¸ ¡ r s n¡1 p q p¡2,0q n¡1 p q p¡2,0q p q π µ, BΩ P3Q f z µ, TΩ P3Q f z mQ,γ µ, TΩ PQ,γ z , |γ| n ° ¨ p ¡ qγ p¡ qλ γ p ¡ qλp ¡ qγ¡λ and using the binomial expansion w zQ λ¤γ 1 λ z w z zQ we have ¸ ¸ ¢ n¡1 λ γ p¡2,0q λ ¡πrµ, B sP fpzq m p¡1q µ, T p1qpzqP ¡ pzq, (4.12) Ω 3Q Q,γ λ Ω Q,γ λ |γ| n ~0¤λ¤γ that is, ¡ © ¸ p n 1 n¡1 3 ∇ rµ, BΩs P f 3Q pp q P L Q Q W ¡ © ¸ ¸ ¸ p À | |p n 1 p¡2,0q λ p q ¤ mQ,γ ∇ µ, TΩ 1 PQ,γ¡λ . LppQq |γ| n ~0¤λ¤γ QPW 96 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

| | } } But every coeﬃcient mQ,γ is bounded by C f W n¡1,8pQq by (1.32) and all the derivatives of PQ,γ are uniformly bounded on Ω. Therefore, we have that ¸ ¸ p À} }p p¡2,0q λ 3 f W n¡1,8pΩq µ, TΩ 1 . W n 1,ppQq QPW 0¤|λ| n

Using the Sobolev Embedding Theorem, we get ¤ ¸ p ¸ p À} }p ¥ p¡2,0q λ p¡2,0q 3 f W n,ppΩq µ, TΩ 1 µ, TΩ 1 . W n 1,ppΩq W n 1,ppQq 0 |λ| n QPW

¡ ~ p¡2,0q λ ¡ ¡¡ Note that if λ 0, then the operator TΩ has homogeneity 2 λ1 λ2 2 and, therefore, p¡2,0q λ n,pp q Ñ n 1,pp q ¡ n 1,pp q by Theorem 3.28, TΩ : W Ω W Ω is bounded and, since p 2 and W Ω is a multiplicative algebra, we have that p p p¡2,0q λ p¡2,0q λ À } }p µTΩ 1 TΩ µ n,p,Ω µ W n 1,ppΩq. W n 1,ppΩq W n 1,ppΩq Therefore, ¡ © À } }p }r sp q}p } }p 3 µ W n 1,ppΩq µ, BΩ 1 W n 1,ppΩq f W n,ppΩq, so we have reduced the proof of Lemma 4.10 to the following claim.

Claim 4.13. Let 2 p 8, n P N. Given a bounded Lipschitz domain Ω with parameterizations n 1¡1{p P 8p q r sp q P n 1,pp q in Bp,p and a function µ Cc C , then µ, BΩ 1 W Ω . n,p n 1 We know that rµ, BΩsp1q µBΩp1q ¡ BΩpµq P W pΩq. We want to prove that ∇ rµ, BΩs1 P Lp. To do so, we split the norm in the same spirit of (4.10), but chopping µ instead of f: ¡ © ¸ p n 1r sp qp À n 1 ¡ n 2 p q ∇ µ, BΩ 1 p p ∇ µ P µ ϕQ, BΩ 1 L pΩq 3Q pp q P L Q Q W ¡ © ¸ p n 1 n 2 ∇ µ ¡ P µ pχΩ ¡ ϕQq, BΩ p1q 3Q pp q P L Q Q W ¸ p n 1 n 2 p q ∇ P3Q µ, BΩ 1 : 4 5 6 . LppQq QPW ¡ © ¡ n 2 P 8 First we consider 4 . Since µ P3Q µ ϕQ Cc , by (4.11) we have that ¡ © ¡ ¡¡ © © © ¸ p ¸ p n 1 n 2 n 1 n 2 ∇ µ ¡ P µ ϕQ, B χΩ Àp ∇ B µ ¡ P µ ϕQ ¤ CχΩ 3Q pp q 3Q pp q L C L 2Q Q Q ¡ ¡¡ © © © ¸ p n 1 B¯ ¡ n 2 ¤ ∇ µ P3Q µ ϕQ CχΩ Lpp2Qq Q and, using Leibniz’ rule (1.11), H¨older’sinequality, and the ﬁnite overlapping of double Whitney cubes,

¢ ¡¡ © © n¸1 p À j 1 ¡ n 2 ¤ n 1¡j p 4 p sup ∇ µ P3Q µ ϕQ ∇ CχΩ pp q. (4.13) P L8p2Qq L Ω j0 Q W 4.3. COMPACTNESS OF THE COMMUTATOR 97 ¡¡ © © p j 1 ¡ n 2 8 To estimate 4 it remains to see that supQPW ∇ µ P3Q µ ϕQ . But L8p2Qq this is an immediate consequence of the Poincar´einequality (1.33), which shows that ¡¡ © © j 1 ¡ n 2 À p qn 2¡j n 3 ∇ µ P3Q µ ϕQ ` Q ∇ µ 8p q. (4.14) L8p2Qq L 2Q

Thus, the bounds (4.13) and (4.14) yield ¤ n 3 p } }p 4 Cp,n,diamΩ ∇ µ L8pΩq CχΩ W n 1,ppΩq, which is ﬁnite by Theorem 3.28.

Next we face 5 . Note that for a given Whitney cube Q, if z P Q, then χΩpzq ¡ ϕQpzq 0, so ¡¡ © © ¸ p n 1 ¡ n 2 p ¡ q 5 ∇ B µ P3Q µ χΩ ϕQ . LppQq QPW

Moreover, for z P Q P W, we have ¡ © ¡¡ © © » n 2 µpwq ¡ P µpwq p1 ¡ ϕQpwqq Bn 1 ¡ n 2 p ¡ q p q 3Q p q B µ P3Q µ χΩ ϕQ z cn 3 n dm w . z 3 pz ¡ wq Ω 2 Q ¡¡ © © B¯ ¡ n 2 p ¡ q p q Bn 1 p q Since B µ P3Q µ χΩ ϕQ z 0, only is non zero in the n 1 -th gradient, so § ¡¡ © © § § § ¸ 1 § n 1 ¡ n 2 p ¡ q p q§ À ¡ n 2 ∇ B µ P3Q µ χΩ ϕQ z µ P3Q µ . DpQ, Sq3 n L1pSq SPW

For every pair of Whitney cubes Q, S, consider an admissible chain rS, Qs. By (1.34) we have that ¸ `pSq2DpP,Sqn 2 ¡ n 2 À n 3 µ P3Q µ ∇ µ 1p q. L1pSq `pP q L 3P P PrS,Qs

Combining all these facts with the expression of the norm by duality and Lemma 2.3, we get » 1 ¸ ¸ 1 ¸ `pSq2DpP,Sqn 2 p À n 3 5 sup g dm ∇ µ 1p q 1 DpQ, Sq3 n `pP q L 3P gPLp pΩq:}g} 1 ¤1 Q P Pr s p Q S W P S,Q ¸ ¸ ¸ p q2 n 3 } } ` S ∇ µ 1 g 1p q À p q2 L p3P q L Q À n 3 diam Ω sup ∇ µ pp q. 1 p q p q3 L Ω P p p q } } ¤ ` P D Q, S g L Ω : g p1 1 Q S P PrS,Qs

Finally we focus on ¸ p n 1 n 2 p q 6 ∇ P3Q µ, BΩ 1 . LppQq QPW p q p ¡ qγ P 2 Consider ﬁrst a monomial PQ,γ z z z°Q for a multiindex ¨ γ N . Then, as we did in (4.12), p q p¡ q|λ| γ p ¡ qλp ¡ qγ¡λ we use the binomial expression PQ,γ w λ¤γ 1 λ z w z zQ to deduce that ¸ ¢ p¡ q γ p¡ q ¡πB P pzq T 2,0 P pzq p¡1q|λ| T 2,0 λp1qpzqpz ¡ z qγ¡λ. Ω Q,γ Ω Q,γ λ Ω Q ~0¤λ¤γ 98 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

~ p¡2,0qp qp q p q Note that the term for λ 0 in the right-hand side of this expression is TΩ 1 z PQ,γ z , so it cancels out in the commutator: ¸ ¢ |λ| γ p¡2,0q λ ¡ πrP , B sp1qpzq p¡1q T p1qpzqP ¡ pzq. (4.15) Q,γ Ω λ Ω Q,γ λ ~0 λ¤γ ° n 2 p q p q Now, writting P3Q µ z |γ|¤n 2 mQ,γ PQ,γ z we have that ¸ p ¸ ¸ n 1r n 2 sp q ¤ | |p n 1r sp qp 6 ∇ P3Q µ, BΩ 1 mQ,γ ∇ PQ,γ , BΩ 1 pp q, LppQq L Q QPW QPW γ¤n 2 so using (1.32) and (4.15) together with Leibniz’ rule (1.11), we get ¸ ¸ ¸ n¸1 p À} }p j p¡2,0q λp q n 1¡j p 6 µ W n 2,8 ∇ TΩ 1 ∇ PQ,γ¡λ 8p q LppQq L Q QPW γ¤n 2 ~ ¤ j0 0 λ γ ¸ p ¤ } }p p¡2,0q λp q Cn,p,Ω µ W n 2,8 TΩ 1 . (4.16) W n 1,ppΩq ~0 λ:|λ|¤n 2

p¡2,0q λp q P n 1,pp q ¡ ~ In the last sum we have that TΩ 1 W Ω for all λ 0 by Theorem 3.28 because the operators T p¡2,0q λ have homogeneity bigger than ¡2. Thus, the right-hand side of (4.16) is ﬁnite.

4.4 Some technical details

p q P 3 Given ~m m1, m2, m3 N , let us define the line integral » pw ¡ ξqm3 K ~mpz, ξq : dw (4.17) p ¡ qm1 p ¡ qm2 BΩ z w w ξ for all z, ξ P Ω, where the path integral is oriented counterclockwise. Given a j times differentiable function f, we will write ¸ D~ifpzq P jpfqpξq pξ ¡ zq~i z ~i! |~i|¤j for its j-th degree Taylor polynomial centered in the point z. p q P Mateu, Orobitg and Verdera study the kernel Kp2,m 1,mq z, ξ for m N in [MOV09, Lemma 6] assuming the boundary of the domain Ω to be in C1,ε for ε 1. They prove the size inequality 1 |Kp qpz, ξq| À 2,m 1,m |z ¡ ξ|2¡ε and a smoothness inequality in the same spirit.¨ In [CMO13], when dealing with the compactness m¡1 s,p of the operator Rmf χΩB χΩc B pχΩfq on W pΩq for 0 s 1, this is used to prove that the Beltrami coefficient µ P W s,ppΩq implies the principal solution of B¯f µBf being in W s 1,ppΩq only for s ε. This bounds are not enough for us in this form and, moreover, we will consider m1 ¡ 2 (this comes from differenciating the kernel of Rm, something that we have to do in order to study the classical Sobolev spaces). Nevertheless, their argument can be adapted to the case of n 1¡1{p n,1¡2{p the boundary being in the space Bp,p C to get Proposition 4.15 below, which will be used to prove Lemma 4.11. The proof follows the same pattern but it is more sophisticated and some combinatorial lemma will be handy. We will use some auxiliary functions. 4.4. SOME TECHNICAL DETAILS 99

Deﬁnition 4.14. Let us deﬁne » 1 pτ ¡ ξqm3 H pwq : dτ for every w, ξ RBΩ, m3,ξ ¡ 2πi BΩ τ w and » pτ ¡ zqm3 h pzq : dτ 2πiH pzq for every z P Ω. (4.18) m3 ¡ m3,z BΩ τ z p q P 3 ¥ Proposition 4.15. Let Ω be a Lipschitz domain, and let ~m m1, m2, m3 N with m1 3, ¥ ¤ ¡ m2, m3 1 and m2 m1 m3 2. Then, the weak derivatives of order m3 of hm3 are

¡ BjB¯m3 j j ¤ ¤ hm3 cm3,jB χΩ, for 0 j m3. (4.19)

Moreover, for every pair z, ξ P Ω with z ξ, we have that

m m3¡1 ¸ 3 p q ¡ pξ ¡ zq c ~m,jRm m ¡3,j z, ξ p q Bm1 2 p q 1 3 K ~m z, ξ c ~m BχΩ z ¡ ¡ , (4.20) pξ ¡ zqm2 pξ ¡ zqm2 m1 1 j j¤m2¡1 where ¡ m3 p q Bj p q ¡ M jpBj qp q RM,j z, ξ : hm3 ξ Pz hm3 ξ (4.21) ¡ Bj is the Taylor error term of order M j for the function hm3 .

We begin by noting some remarkable properties of these functions.

Remark 4.16. Given ξ RBΩ and w PBΩ, if we write H¡ pwq for the interior non-tangential m3,ξ limit of H pζq when ζ Ñ w and H pwq for the exterior one, we have the Plemelj formula m3,ξ m3,ξ

¡ pw ¡ ξqm3 H pwq ¡ H pwq (4.22) m3,ξ m3,ξ

(see [Ver01, p. 143] for instance).

Remark 4.17. Given ~j pj1, j2q with j2 ¤ m3, taking partial derivatives in (4.18) we get » m3¡j2 ~ m3!j1! pτ ¡ zq j j1 ¯j2 j2 D hm pzq B B hm pzq p¡1q dτ for every z P Ω 3 3 p ¡ q p ¡ q1 j1 m3 j2 ! BΩ τ z

and, in particular, hm3 is inﬁnitely many times diﬀerentiable in Ω. Therefore, by Green’s formula (1.12) and the cancellation of the integrand (3.8), for j ¡ 0 we have » » j j¡1 p ¡ q pτ ¡ zq pw ¡ zq D j,m3 j h pzq c dτ c dmpwq c Bjχ pzq m3 m3,j p ¡ q1 j m3,j p ¡ qj 1 m3,j Ω BΩ τ z ΩzBpz,εq w z for ε distpz, BΩq and, in case j 0, by the Residue Theorem (1.13) » 1 B¯m3 h pzq c dτ c 2πiχ pzq, m3 m3 ¡ m3 Ω BΩ τ z proving (4.19). 100 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

p q p q P Remark 4.18. We can also relate the derivatives of both hm3 z and Hm3,ξ z for any pair z, ξ Ω. By Deﬁnition 4.14 and the previous remark, we have that ¢ m » ¡ ¸3 p ¡ qm3 lp ¡ ql p q m3 τ z z ξ 2πiHm3,ξ z dτ B l τ ¡ z l0 Ω ¸m3 m ! pm ¡ lq! 3 B¯lh pzq 3 p¡1qlpξ ¡ zqlp¡1ql, pm ¡ lq!l! m3 m ! l0 3 3 that is,

¸m3 1 2πiBjH pzq Dpj,lqh pzqpξ ¡ zql. (4.23) m3,ξ l! m3 l0

p q Hm3,ξ w 1 Proof of Proposition 4.15. Consider z, ξ P Ω. Then decays at 8 as pz¡wqm1 pw¡ξqm2 |w|m1 m2 1 and it is holomorphic in Ωc and, thus, by Green’s Theorem we have that

» » m p ¡ qm3 pw ¡ ξq 3 H pwq w ξ m3,ξ K ~mpz, ξq dw dw, p ¡ qm1 p ¡ qm2 p ¡ qm1 p ¡ qm2 BΩ z w w ξ BΩ z w w ξ and using (4.22), » H¡ pwq m1 m3,ξ K ~mpz, ξq p¡1q dw. p ¡ qm1 p ¡ qm2 BΩ w z w ξ p q Note that Hm3,ξ w is holomorphic in Ω, implying that the integrand above is meromorphic in Ω with poles in z and ξ. Moreover, H¡ P L2pBΩq by the boundedness of the Cauchy transform m3,ξ in L2pΓq on a Lipschitz graph Γ (see [Ver01], for instance). Thus, combining then Dominated Convergence Theorem and the Residue Theorem, we get " * 1 ¡ Hm ,ξp¤q 1 ¡ Hm ,ξp¤q p¡ qm1 p q Bm1 1 3 p q Bm2 1 3 p q 1 K ~m z, ξ 2πi m z m ξ . pm1 ¡ 1q! p¤ ¡ ξq 2 pm2 ¡ 1q! p¤ ¡ zq 1 Therefore,

m1 ¸ j2 p¡1q 1 pm1 ¡ 1q! B Hm ,ξpzq pm2 j1 ¡ 1q! p q 3 p¡ qj1 K ~m z, ξ m j 1 2πi pm1 ¡ 1q! j1!j2! pz ¡ ξq 2 1 pm2 ¡ 1q! j1,j2¥0 j1 j2m1¡1

¸ j2 1 pm2 ¡ 1q! B Hm ,ξpξq pm1 j1 ¡ 1q! 3 p¡ qj1 m j 1 . pm2 ¡ 1q! j1!j2! pξ ¡ zq 1 1 pm1 ¡ 1q! j1,j2¥0 j1 j2m2¡1

Simplifying and using (4.23) on the ﬁrst sum of the right-hand side and (4.18) on the second one, we get ¢ ¸ ¸m3 m2 j1 ¡ 1 1 1 1 p q p¡ qm1 m2 p q j2,l p qp ¡ ql 1 K ~m z, ξ m j D hm3 z ξ z m2 ¡ 1 j2! pξ ¡ zq 2 1 l! j1,j2¥0 l0 ¡ j1 j2 m1 1 ¢ ¸ j2 m1 j1 ¡ 1 1 B hm pξq 3 p¡ qj2 1 m j 1 . (4.24) m1 ¡ 1 j2! pξ ¡ zq 1 1 j1,j2¥0 j1 j2m2¡1 4.4. SOME TECHNICAL DETAILS 101

The key idea for the rest of the proof is that the ﬁrst term in the right-hand side of (4.24) contains the Taylor expansion of the functions in the second one. Let m2 ¡ 1 ¤ M ¤ m1 m3 ¡ 2 (we will consider M m1 m3 ¡ 3). Using the Taylor ¡ Bj2 p ¡ qm1 m2 1 approximating polynomial (4.21) of each hm3 and multiplying by ξ z we get ¢ m¸1¡1 ¸m3 ¡ m2 m1 ¡ 2 ¡ j 1 1 p q p q ¡K pz, ξqpz ¡ ξqm1 m2 1 D j,l h pzqpξ ¡ zq j,l ~m m ¡ 1 j! l! m3 j0 2 l0 ¡ ¢ m¸2 1 m m ¡ 2 ¡ j p¡1qj ¡ 1 2 pξ ¡ zqjRm3 pz, ξq m ¡ 1 j! M,j j0 1 ¡ ¢ m¸2 1 m m ¡ 2 ¡ j p¡1qj ¸ D~iBjh pzq ¡ 1 2 m3 pξ ¡ zq~i pj,0q. m1 ¡ 1 j! ~i j0 |~i|¤M¡j

To simplify notation, let us deﬁne the error

¡ ¢ m¸2 1 j ¡ m1 m2 ¡ 2 ¡ j p¡1q E : ¡K pz, ξqpz ¡ ξqm1 m2 1 pξ ¡ zqjRm3 pz, ξq. (4.25) M ~m m ¡ 1 j! M,j j0 1

Then, ¢ ¸ ¡ ¡ α p q m1 m2 2 α1 D hm3 z α EM pξ ¡ zq m2 ¡ 1 α! α¥~0 ¤p ¡ q α m1 1,m3 ¢ ¸ ¸ m m ¡ 2 ¡ j p¡1qj Dαh pzq ¡ 1 2 m3 pξ ¡ zqα. m1 ¡ 1 j! pα1 ¡ jq!α2! α¥~0 0¤j¤mintm2¡1,α1u |α|¤M

¡ α p q Note that if α2 m3, we have that D hm3 z 0 (apply (4.19) with j 0). The same happens¨ m1 m2¡2¡α1 for the case α pα1, m3q with α1 ¡ 0. On the other hand, if α1 ¡ m1 ¡1, then 0. ¨ m2¡1 m1 m2¡2¡j By the same token, if j ¡ m2 ¡ 1, 0. Thus, we can write m1¡1

¸ Dαh pzq E m3 pξ ¡ zqα M α! |α|¤m1 m3¡2 ¢ ¢ ¢ ¡ ¡ ¸ ¡ ¡ m1 m2 2 α1 j m1 m2 2 j α1 ¤ ¡ χ|α|¤M p¡1q . m2 ¡ 1 m1 ¡ 1 j j¤α1

Note that we have added many null terms in the previous expression, but now the proof of the proposition is reduced to Claim 4.19 below which implies that ¢ ¸ ¡ ¡ α p q m1 m2 2 α1 D hm3 z α EM pξ ¡ zq . m2 ¡ 1 α! M |α|¤m1 m3¡2

Taking M m1 m3 ¡ 3 in this expression, only the¨ terms with |α| m1 m3 ¡ 2 remain and, m1 m2¡2¡α1 arguing as before, if α1 ¡ m1 ¡ 1 then 0 and m2¡1 ¥ α p q if α2 m3 then D hm3 z 0 (4.26) 102 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

(|α| ¡ m3 because we assume that m1 ¥ 3). Summing up, by (4.19) we have that

pm1¡1,m3¡1q D hm pzq p ¡ ¡ q ¡ p ¡ ¡ q 3 p ¡ q m1 1,m3 1 Bm1 2 p qp ¡ q m1 1,m3 1 Em1 m3¡3 ξ z c ~m BχΩ z ξ z . pm1 ¡ 1q!pm3 ¡ 1q! By (4.25) this implies (4.20).

Claim 4.19. For any natural numbers m1, m2 and α1 we have that ¢ ¢ ¢ m m ¡ 2 ¡ α ¸α1 α m m ¡ 2 ¡ j 1 2 1 p¡1qj 1 2 1 . m ¡ 1 j m ¡ 1 2 j0 1 Proof. We have the trivial identity ¢ ¢ ¢ ¢ m m ¡ 2 ¡ α m m ¡ 2 ¡ α ¸0 0 m m ¡ 2 ¡ α ¡ i 1 2 1 1 2 1 p¡1qi 1 2 1 . m ¡ 1 m ¡ 1 ¡ α i m ¡ 1 ¡ α 2 1 1 i0 1 1 P ¥ Let κ1, κ2, κ3 Z with κ1 0. We have that ¢ ¢ ¢ ¢ ¢ ¢ ¸κ1 κ κ ¡ i ¸κ1 κ κ 1 ¡ i κ κ ¡ i p¡1qi 1 3 p¡1qi 1 3 ¡ 1 3 i κ i κ 1 i κ 1 i0 2 i0 2 2 ¢ ¢ ¢ ¢ κ¸1 1 κ κ 1 ¡ j κ κ 1 ¡ j p¡1qj 1 3 1 3 j κ 1 j ¡ 1 κ 1 j0 2 2 ¢ ¢ κ¸1 1 κ 1 κ 1 ¡ j p¡1qj 1 3 . j κ 1 j0 2 Arguing by induction we get that ¢ ¢ ¢ ¢ ¸0 0 m m ¡ 2 ¡ α ¡ i ¸α1 α m m ¡ 2 ¡ j p¡1qi 1 2 1 ¤ ¤ ¤ p¡1qj 1 2 1 . i m ¡ 1 ¡ α j m ¡ 1 i0 1 1 j0 1

Remark 4.20. In Proposition 4.15, if we assume that m1 2, then for every pair z, ξ P Ω with z ξ we have that

m ¡1 m ¸ m3 pξ ¡ zq 3 pξ ¡ zq 3 c ~m,jR ¡ pz, ξq p q p q p q m3 1,j K ~m z, ξ c ~mBχΩ z c ~mχΩ z ¡ . (4.27) pξ ¡ zqm2 pξ ¡ zqm2 1 pξ ¡ zqm2 1 j j¤m2¡1

The proof is exactly the same, but (4.26) is only valid for α2 ¡ m3, leading to

p1,m3¡1q p0,m3q D hm pzq p ¡ q m2D hm pzq p q 3 p ¡ q 1,m3 1 3 p ¡ q 0,m3 Em3¡1 ξ z ξ z , pm3 ¡ 1q! m3! leading to (4.27) by (4.19).

The thoughtful reader may wonder why we do not use M m1 m3 ¡ 2 in (4.25) to get an analogous result to [MOV09, (27)], namely Em1 m3¡2 0. This formula is still valid in our context (with ~m p2, m 1, mq), and taking ~m pn 2, m 1, mq we could get a generalization with no extra terms, only Taylor remainders up to degree M. In the present dissertation, however, m n,p we deal with the situation hm P W pΩq with p ¡ 2 and, therefore, we will only use Taylor polynomials up to degree m n ¡ 1 to use the H¨olderestimates for Taylor remainders of Lemma 4.22, which is a consequence of the following lemma. 4.4. SOME TECHNICAL DETAILS 103

Lemma 4.21. Let z, ξ be two points in an extension domain Ω Rd (open and connected), M ¥ 1 ¤ ¡ { P M s,pp q ¡ d a natural number, 0 s 1, p d s and f W Ω . Then, writing σ σd,s,p s p , the Taylor error term satisﬁes the estimate § § § p q ¡ M p q§ ¤ } } | ¡ |M σ f ξ Pz f ξ C f W M s,ppΩq z ξ .

P M s,pp q Ñ M s,pp p p qq Proof. Let us assume that 0 Ω. Using the extension E : W Ω W0 B 0, 2 diam Ω and the Sobolev Embedding Theorem, it suﬃces to prove the estimate for f P CM,σpRdq. We will prove only the case d 1 leaving to the reader the generalization. In that case, we deﬁne

fptq ¡ P M fptq F puq : u t pt ¡ uqM P | p q| ¤ } } | ¡ |σ for any u t R. We want to see that Ft u C f CM,σ u t for t u. Note that the p q M p q M¡1 p q M-diﬀerentiability of f implies that limτÑt Ft τ 0. Thus, decomposing Pu f t Pu f t 1 pMqp qp ¡ qM M! f u t u , we have that ¨ ¨ fptq ¡ P M¡1fptq ¡ fptq ¡ P M¡1fptq F puq lim F puq ¡ F pτq lim u τ t Ñ t t Ñ p ¡ qM τ t τ t t u¢ ¨ p q ¡ M¡1 p q 1 ¡ 1 lim f t Pτ f t τÑt pt ¡ uqM pt ¡ τqM ¡ © 1 lim ¡f pMqpuq f pMqpτq I II III . (4.28) τÑt M! The ﬁrst term in (4.28) is ¨ fptq ¡ P M¡1fptq I u pt ¡ uqM and, using the mean value form of the remainder term of the Taylor polynomial, there exists a point c1 P pu, tq such that f pMqpc q I 1 . M! The second term in (4.28) is ¢ ¨ p ¡ qM ¡ p ¡ qM p q ¡ M¡1 p q t τ t u II lim f t Pτ f t τÑt pt ¡ uqM pt ¡ τqM £ ¨ ¸M p q ¡ M¡1 p q p ¡ q 1 lim f t Pτ f t u τ τÑt p ¡ qjp ¡ qM 1¡j t u t τ £ j 1 u ¡ τ ¸M fptq ¡ P M¡1fptq ¸M fptq ¡ P M¡1fptq lim τ ¡ lim τ . τÑt t ¡ u pt ¡ uqj¡1pt ¡ τqM 1¡j τÑt pt ¡ uqj¡1pt ¡ τqM 1¡j j1 j1 Applying the Taylor Theorem, only the term j 1 has a non-null limit in the last sum, with

f pMqptq II ¡ , M! so § § § § § pMqp q pMqp q§ § § | p q| ¤ §f c1 ¡ f t § 1 pMqp q ¡ pMqp q ¤ 2 } } | ¡ |σ Ft u § § lim §f u f τ § f M,σ u t . M! M! M! τÑt M! C 104 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

Recall that in (4.21) we deﬁned the Taylor error terms

¡ m3 p q Bj p q ¡ M jpBj qp q RM,j z, ξ : hm3 ξ Pz hm3 ξ P P for M, j, m3 N and z, ξ Ω. Next we give bounds on the size and the smoothness of this terms. Lemma 4.22. Consider a real number p ¡ 2 and naturals n, m P N and let Ω C be a Lipschitz n 1¡1{p ¡ 2 ¤ domain with parameterizations of the boundary in Bp,p . Writing σp : 1 p , for j m we have that ¡ | m 1 p q| ¤ | ¡ |m n j σp Rm n,j z, ξ CΩ,n,m z ξ (4.29) P | ¡ | ¡ 3 | ¡ | and, if z1, z2, ξ Ω with z1 ξ 2 z1 z2 , then ¡ ¡ | m p q ¡ m p q| ¤ | ¡ |σp | ¡ |m n j 1 Rm n¡1,j z1, ξ Rm n¡1,j z2, ξ CΩ,n,m z1 z2 z1 ξ . (4.30)

k n,p Proof. Recall that B χΩ P W pΩq for every k by Theorem 3.28. Thus, by (4.19) we have that m 1 n,p ∇ hm 1 P W pΩq and, since hm 1 is continuous and bounded in Ω as well (see (4.18)), we j n m 1¡j,p have that B hm 1 P W pΩq for 0 ¤ j ¤ m n. By Lemma 4.21, it follows that m 1 j m n¡j σ | p q| ¤ B | ¡ | p Rm n,j z, ξ C hm 1 W m n¡j 1,ppΩq z ξ .

The second inequality is obtained by the same procedure as [MOV09, Lemma 7]. We quote it P | ¡ | ¡ 3 | ¡ | here for the sake of completeness. Assume that z1, z2, ξ Ω with z1 ξ 2 z1 z2 . Then Rm pz , ξq ¡ Rm pz , ξq P m n¡1¡jBjh pξq ¡ P m n¡1¡jBjh pξq. m n¡1,j 1 m n¡1,j 2 z1 m z2 m

But for a natural number M and a function f P CM,σp pΩq one has that

¸ ~i p q ¸ ~j p q M M D f z1 ~i D f z2 ~j P fpξq ¡ P fpξq pξ ¡ z1q ¡ pξ ¡ z2q . z1 z2 ~i! ~j! |~i|¤M |~j|¤M

° ~¨ Since pξ ¡ z q~j j pz ¡ z q~j¡~ipξ ¡ z q~i, one can write 2 ~i¤~j ~i 1 2 1 ¢ ¸ ~i p q ¸ ~j p q ¸ ~ M M D f z1 ~i D f z2 j ~j¡~i ~i P fpξq ¡ P fpξq pξ ¡ z1q ¡ pz1 ¡ z2q pξ ¡ z1q z1 z2 ~ ~ ~i |~|¤ i! |~|¤ j! ~¤~ i M ¤ j M i j ¦ ¸ p ¡ q~i ¸ ~j p q ξ z1 ¦ ~i D f z2 ~j¡~i ¦D fpz1q ¡ pz1 ¡ z2q ~i! ¥ p~j ¡~iq |~i|¤M |~j|¤M ~i¤~j ¸ p ¡ q~i ¡ © ξ z1 ~i M¡|~i| ~i D fpz1q ¡ P D fpz1q . ~i! z2 |~i|¤M

Therefore, arguing as before, ¸ M M i M¡i σp |P fpξq ¡ P fpξq| À |ξ ¡ z | }f} M,σ |z ¡ z | z1 z2 1 C p pΩq 1 2 i¤M À | ¡ |M | ¡ |σp } } ξ z1 z1 z2 f CM,σp pΩq. 4.5. COMPACTNESS OF THE DOUBLE REFLECTION 105

4.5 Compactness of the double reflection Rm ¨ m¡1 Proof of Lemma 4.11. Recall that we want to prove that Rm : f ÞÑ χΩB χΩc B pχΩfq is a compact operator in W n,ppΩq. n n,p p Since Rmf is analytic in Ω, it is enough to see that Tm : B Rm : W pΩq Ñ L pΩq is a compact operator. n,p Indeed, we have that Rm is bounded on W pΩq by (4.9) and, thus, since the inclusion W n,ppΩq ãÑ W n¡1,ppΩq is compact for any extension domain (see [Tri83, 4.3.2/Remark 1]), n,p n¡1,p we have that Rm : W pΩq Ñ W pΩq is compact. That is, given a bounded sequence t u n,pp q t u P n¡1,pp q fj j W Ω , there exists a subsequence fjk k and a function g W Ω such that Ñ n¡1,pp q n,pp q Ñ pp q Rmfjk g in W Ω . If Tm : W Ω L Ω was a compact operator, then there would be a subsubsequence tf u and a function g such that T f Ñ g in LppΩq. It is immediate jki i n m jki n n to see that gn is the weak derivative B g in Ω. Therefore, if Tm is compact then Rm is compact as well. n,p We will prove that Tm is compact using an approximation argument. Let f P W pΩq. For every cube Q, let fQ be the mean of f in Q. Consider a partition of the unity tψQuQPW such that 11 | j | À p q¡j supp ψQ 10 Q and ∇ ψQ ` Q for every Whitney cube Q. P t i u For every i N we can define a finite partition of the unity ψQ QPW such that p q ¡ ¡i i • If ` Q 2 then ψQ ψQ.

p q ¡i i p q | j i | À p q¡j • If ` Q 2 then supp ψQ Sh Q (see Deﬁnition 1.16) and ∇ ψQ ` Q .

p q ¡i i • If ` Q 2 then ψQ 0. p p ¡ qq p ¡ q Then, writing fQ Q f dm for the mean of f in Q and Tm f fQ Q Q Tm f fQ dm, we can define ffl ffl ¸ ¸ i p q p qp q p q p p ¡ qq i p q Tmf z Tm f z ψQ z Tm f fQ Q ψQ z . QPW:`pQq¡2¡i QPW:`pQq2¡i

We will prove the following two claims.

P i n,pp q Ñ pp q Claim 4.23. For every i N, the operator Tm : W Ω L Ω is compact. i ¡ i n,pp q Ñ pp q Claim 4.24. The norm of the error operator E : Tm Tm : W Ω L Ω tends to zero as i tends to inﬁnity.

Then the compactness of Tm is a well-known consequence of the previous two claims (see Theorem 4.6). By all the exposed above, this proves Lemma 4.11.

i n,pp q Ñ 1,pp q Proof of Claim 4.23. We will prove that the operator Tm : W Ω W Ω is bounded. As before, since Ω is an extension domain, the embedding W 1,ppΩq ãÑ LppΩq is compact. Therefore i n,pp q Ñ pp q we will deduce the compactness of Tm : W Ω L Ω . Note that the specific value of the i operator norm Tm W n,ppΩqÑW 1,ppΩq is not important for our argument, since we only care about compactness. Consider a fixed i P N and f P W n,ppΩq. For every z P Ω and every first order derivative D, since Tmf is analytic in Ω, we can use the Leibniz rule (1.11) to get ¸ ¸ ¸ i p q p q p p ¡ qq i DTmf DTm f ψQ Tm f DψQ Tm f fQ Q DψQ. Q:`pQq¡2¡i Q:`pQq¡2¡i Q:`pQq2¡i 106 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

| p ¡ q| ¤ } p ¡ q} p q¡2{p By Jensen’s inequality Tm f fQ Q Tm f fQ LppQq` Q , so ¸ ¸ i |∇T fpzq| ¤ χ 11 pzq|∇Tmfpzq| |∇ψQpzq||Tmfpzq| m 10 Q p q¡ ¡i p q¡ ¡i Q:` Q ¸2 Q:` Q 2 | i p q|} p ¡ q} p ¡iq¡2{p ∇ψQ z Tm f fQ LppQq 2 . (4.31) Q:`pQq2¡i

| i p q| À i Using the ﬁnite overlapping of the double Whitney cubes and the fact that ∇ψQ z 2 for every p q Whitney cube Q, writing Ωi for Q:`pQq¡2¡i supp ψQ we can conclude that ¸ ¡ © i p p p p p p ∇T f À }∇T f} p }T f} p }T f} p |f | }T 1} p . m LppΩq i,p m L pΩiq m L pΩiq m L pQq Q m L pQq Q:`pQq2¡i

By the Sobolev Embedding Theorem

| | ¤ } } À } } fQ f L8pΩq Ω,p f W 1,ppΩq. (4.32)

n,p p Thus, since Tm : W pΩq Ñ L pΩq is bounded, we have that i ∇T f À }∇T f} p }f} n,p . (4.33) m LppΩq p,i,Ω m L pΩiq W pΩq ¨ n m¡1 To see that }∇T f} p À }f} n,p , note that ∇T f ∇B B χ c B pχ fq . We have m L pΩiq i W pΩq m Ω Ω m¡1 p p c p c that B : L pΩq Ñ L pΩ q is bounded trivially, and for z P Ωi and g P L supported in Ω we have that » 1 |∇BnBgpzq| À gpwq dmpwq. | ¡ |n 3 |z¡w|¡2¡i z w This is the convolution of g with an L1 kernel, so Young’s inequality (1.10) tells us that

n }∇B Bg} p ¤ C }g} p , L pΩiq i L proving that m¡1 }∇T f} p À B pχ fq À}f} p À}f} n,p . (4.34) m L pΩiq i Ω LppΩcq L pΩq W pΩq Combining (4.33) and (4.34), we have seen that ∇T i f À}f} . The reader can use m LppΩq W n,ppΩq i À} } Jensen’s inequality as in (4.31) to check that Tmf LppΩq f W n,ppΩq as well. This, proves that i n,pp q Ñ 1,pp q the operator Tm : W Ω W Ω is bounded and, therefore, composing with the compact i n,pp q Ñ pp q inclusion, the operator Tm : W Ω L Ω is compact.

Proof of Claim 4.24. We want to see that the error operator

i ¡ i E Tm Tm satisfies that Ei tends to zero as i tends to infinity. W n,ppΩqÑLppΩq p q i Consider the set Ωi Q:`pQq¡2¡i supp ψQ . We define the modified error operator E0 acting in f P W n,ppΩq as § § ¸ ¸ § § i p q p q § p ¡ qp q ¡ p p ¡ qq § p q E0f z : χΩzΩi¡1 z Tm f fS z Tm f fQ Q χ2S z Q:`pQq2¡i S:`pSq¤2¡i SShpQq 4.5. COMPACTNESS OF THE DOUBLE REFLECTION 107 for every z P Ω. The first step will be proving that i À i } } E f LppΩq E0f LppΩq Ci f W 1,ppΩq, (4.35)

iÑ8 with Ci ÝÝÝÑ 0. ¨ n m¡1 Note that Tm1 TmχΩ because Tm¤ B χΩB χΩc B pχΩ¤q . Let us write ¸ ¸ Tmfpzq TmpfqpzqψSpzq pfSTmp1qpzq Tmpf ¡ fSqpzqq ψSpzq SPW:`pSq¡2¡i SPW:`pSq¤2¡i for z P Ω. Recall that ¸ ¸ i p q p qp q p q p p ¡ qq i p q Tmf z Tm f z ψQ z Tm f fQ Q ψQ z . QPW:`pQq¡2¡i QPW:`pQq2¡i

Thus, for the error operator Ei we have the expression ¸ i p q p q ¡ i p q p qp q p q E f z Tmf z Tmf z fSTm 1 z ψS z p q¤ ¡i ¤ S:` S 2 ¸ ¸ ¥ p ¡ qp q p q ¡ p p ¡ qq i p q Tm f fS z ψS z Tm f fQ Q ψQ z S:`pSq¤2¡i Q:`pQq2¡i i p q i p q E1f z E2f z . (4.36)

The ﬁrst part is easy to bound using again (4.32). Indeed, we have that ¸ i p p p p p E f À |f | }T p1q} p À }f} 1,p }T p1q} p , (4.37) 1 LppΩq p S m L p11{10Sq Ω W pΩq m L pΩzΩi¡1q S:`pSq¤2¡i

p iÑ8 where }T p1q} p ÝÝÝÑ 0. m L pΩzΩiq i P To control E2f in (4.36), note that every z Ω satisﬁes ¸ ¸ p q i p q ¤ ψS z ψQ z 1, (4.38) S:`pSq¤2¡i Q:`pQq2¡i R p q P z with equality when z `pQq¡2¡i supp ψQ , that is, when z Ω Ωi. Recall that ¸ ¸ i p q p ¡ qp q p q ¡ p p ¡ qq i p q E2f z Tm f fS z ψS z Tm f fQ Q ψQ z . S:`pSq¤2¡i Q:`pQq2¡i ° ° P z p q i p q If z Ω Ωi, we have equality in (4.38), i.e., S:`pSq¤2¡i ψS z Q:`pQq2¡i ψQ z 1. Thus ¸ ¸ i p q p ¡ qp q p q i p q E2f z Tm f fS z ψS z ψQ z p q¤ ¡i p q ¡i S:` S ¸2 Q:` Q 2¸ ¡ p p ¡ qq i p q p q Tm f fQ Q ψQ z ψS z Q:`pQq2¡i S:`pSq¤2¡i ¸ ¸ ¡ © p ¡ qp q ¡ p p ¡ qq p q i p q Tm f fS z Tm f fQ Q ψS z ψQ z . (4.39) Q:`pQq2¡i S:`pSq¤2¡i 108 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS P p q P p q If, instead, z Ωi Q:`pQq¡2¡i supp ψQ then there is a cube S0 with z supp ψS0 and ¡i 1 ¡i `pS0q ¥ 2 . Therefore, any other cube S with ψSpzq 0 must have side-length `pSq ¥ 2 1 p q because any neighbor cube of S0 has side-length at least 2 ` S0 (see Deﬁnition 1.5). Therefore, ¸ ¸ i p q p ¡ qp q p q ¡ p p ¡ qq i p q E2f z Tm f fS z ψS z Tm f fQ Q ψQ z S:`pSq2¡i Q:`pQq2¡i ¸ ¡ © p ¡ qp q p q ¡ p p ¡ qq i p q Tm f fQ z ψQ z Tm f fQ Q ψQ z . Q:`pQq2¡i

p ¡ qp q i p q Adding and subtracting Tm f fQ z ψQ z at each term of this sum, we get ¸ ¨ i p q p ¡ qp q p q ¡ i p q E2f z Tm f fQ z ψQ z ψQ z Q:`pQq2¡i ¸ ¡ © p ¡ qp q ¡ p p ¡ qq i p q Tm f fQ z Tm f fQ Q ψQ z . (4.40) Q:`pQq2¡i

Summing up, by (4.39) and (4.40) we have that ¸ ¸ ¡ © i p q p q p ¡ qp q ¡ p p ¡ qq p q i p q E2f z χΩzΩi z Tm f fS z Tm f fQ Q ψS z ψQ z Q:`pQq2¡i S:`pSq¤2¡i ¸ ¡ © p q p ¡ qp q ¡ p p ¡ qq i p q χΩizΩi¡1 z Tm f fQ z Tm f fQ Q ψQ z Q:`pQq2¡i ¸ ¨ p q p ¡ qp q p q ¡ i p q χΩizΩi¡1 z Tm f fQ z ψQ z ψQ z . Q:`pQq2¡i

p q ¡i i p q i Every cube Q with ` Q 2 satisﬁes that supp ψQ Sh Q , and, choosing conveniently ψQ we i X can assume that supp ψQ Ωi 2Q. Therefore, we get that § § ¸ ¸ § § | i p q| À p q § p ¡ qp q ¡ p p ¡ qq § p q E2f z χΩzΩi¡1 z Tm f fS z Tm f fQ Q χ2S z (4.41) Q:`pQq2¡i S:`pSq¤2¡i p q § S Sh Q § § § § ¸ ¨§ p q § p ¡ qp q p q ¡ i p q § χΩ zΩ ¡ z § Tm f fQ z ψQ z ψ z § . i i 1 § Q § Q:`pQq2¡i

For the last term, just note that for z P ΩizΩi¡1, using the ﬁrst equality in (4.38) we have that ¤ ¸ ¨ ¸ ¸ p qp q i p q ¡ p q p qp q ¥ i p q ¡ p q Tm f z ψQ z ψQ z Tm f z ψQ z ψQ z 0. Q:`pQq2¡i Q:`pQq2¡i Q:`pQq2¡i

Thus, ¸ ¨ ¸ ¨ p ¡ qp q i p q ¡ p q ¡ p qp q i p q ¡ p q Tm f fQ z ψQ z ψQ z Tm fQ z ψQ z ψQ z , Q:`pQq2¡i Q:`pQq2¡i

i which can be bounded as E1 in (4.37). This fact, together with (4.36), (4.37) and (4.41) settles (4.35), that is, i À i } } E f LppΩq E0f LppΩq Ci,Ω,n,p f W 1,ppΩq, 4.5. COMPACTNESS OF THE DOUBLE REFLECTION 109

iÑ8 with Ci,Ω,n,p ÝÝÝÑ 0. Next we prove that for the modiﬁed error term, § § ¸ ¸ § § i p q p q § p ¡ qp q ¡ p p ¡ qq § p q E0f z χΩzΩi¡1 z Tm f fS z Tm f fQ Q χ2S z , Q:`pQq2¡i S:`pSq¤2¡i SShpQq i À } } ÝÝÝÑiÑ8 we have that E0f LppΩq Ci f W 1,ppΩq with Ci 0. Arguing by duality, we have that » § § ¸ § § i § p ¡ qp q ¡ p p ¡ qq § p q | p q| p q E0f p sup Tm f fS z Tm f fQ χ2S z g z dm z . (4.42) L 1 Q p ΩzΩ ¡ ¡ gPL i 1 Q:`pQq2 i } } ¡ g p1 1 S:`pSq¤2 i SShpQq First note for every pair of Whitney cubes Q and S with S ShpQq and every point z, using an admissible chain rS, Qq rS, QsztQu we get that

Tmpf ¡ fSqpzq ¡ pTmpf ¡ fQqq Tmpf ¡ fSqpzq ¡ pTmpf ¡ fSqq Q ¸ S ¨ pT pf ¡ f qq ¡ T pf ¡ f p qq , m P P m N P N pP q P PrS,Qq where N pP q stands for the “next” cube in the chain rS, Qs (see Definition 1.13). Note that the shadows of cubes of fixed side-length have finite overlapping since |ShpQq| |Q| and, therefore, every Whitney cube S appears less than C times in the right-hand side of (4.42). Thus, ¢ » ¸ i À | p ¡ qp q ¡ p p ¡ qq | | p q| p q E0f Lp sup Tm f fS z Tm f fS S g z dm z (4.43) g:}g} 1 1 ¡ p S:`pSq¤2 i 2S § § » ¸ ¸ § ¨ § §pT pf ¡ f qq ¡ T pf ¡ f p qq § |gpzq| dmpzq . m P P m N P N pP q ¡ 2S Q:`pQq2 i P PrS,Qq S:`pSq¤2¡i SShpQq All the cubes P P rS, Qs with S P ShpQq, satisfy that `pP q À DpQ, Sq `pQq by Definition 1.13. If we assume that `pQq 2¡i this implies that `pP q ¤ C2¡i. Moreover, we have that § § § ¨ § ¸ §pT pf ¡ f qq ¡ T pf ¡ f p qq § ¤ |T pf ¡ f qpzq ¡ pT pf ¡ f qq | dmpzq. m P P m N P N pP q m P m L L LX2P H P (4.44) Finally, we observe that P P rS, Qs with S ShpQq imply that DpP,Sq ¤ C`pP q (see (1.26)). 1 Thus, for a fixed P with `pP q ¤ C2¡i and g P Lp , we have that ¸ » ¸ » |gpzq| dmpzq À C |gpzq| dmpzq À `pP qd inf Mg. (4.45) ¡ 2S 2S P Q:`pQq2 i S:DpP,Sq¤C`pP q S:SShpQq P PrS,Qs Note that in the first step, as we did in (4.43), we have used that every cube S appears less than C times in the left-hand side. By (4.43), (4.44) and applying (4.45) after reordering, we get that » ¸ i À |p p ¡ qp q ¡ p p ¡ qq q p| p q| p qq| p q E0f Lp sup Tm f fS z Tm f fL L g z Mg z dm z . }g} 1 1 ¡ 2S p S:`pSq¤C2 i LX2SH 110 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

Since }Mg} p1 À}g} p1 ¤ 1, we have that L L » ¸ i À | p ¡ qp q ¡ p p ¡ qq | | p q| p q E0f Lp sup Tm f fS z Tm f fL L g z dm z , }g} 1 1 2S p pS,LqPW0

¡i where W0 tpS, Lq : `pSq ¤ C2 and 2S X L Hu. For every cube Q, let ϕQ be a radial bump function with χ10Q ¤ ϕQ ¤ χ20Q and the usual bounds in their derivatives. Now we use these bump functions to separate the local and the non- local parts. In the local part we can neglect the cancellation and use the triangle inequality (and | | À the fact that 2S g dm inf7S Mg), but in the non-local part the smoothness of a certain kernel will be crucial,ffl so we write » ¸ i À | rp ¡ q sp q| | p q| p q E0f Lp sup Tm f fS ϕS z g z dm z }g} 1 1 ¡ p S:`pSq¤C2 i 2S ¸ » sup |Tmrpf ¡ fLqϕSspξq| Mgpξq dmpξq } } g p1 1 pS,LqPW 2L 0 » § § ¸ § § sup §Tmrpf ¡ fSqp1 ¡ ϕSqspzq ¡ pTmrpf ¡ fLqp1 ¡ ϕSqsqL§|gpzq|dmpzq }g} 1 1 2S p pS,LqPW0 7’ 7 8 . (4.46) | | ¤ 1 Note that the inequality g Mg (which is valid almost everywhere for g in Lloc) imply that 7’ ¤ 7 . First we take a look at 7 . For any pair of neighbor Whitney cubes S and L and z P 2L, using the definition of weak derivative and Fubini’s Theorem we find that » » p ¡ qm¡1 rp ¡ q sp q 1 w ξ p p q ¡ q p q p q p q Tm f fL ϕS z cn n 2 m 1 f ξ fL ϕS ξ dm ξ dm w c pz ¡ wq pw ¡ ξq Ω» 20»S p ¡ qm 1 w ξ Brp¯ ¡ q sp q p q p q cn,m n 2 m 1 f fL ϕS ξ dm ξ dm w c pz ¡ wq pw ¡ ξq »Ω ¢» 20S pw ¡ ξqm c dmpwq Brp¯ f ¡ f qϕ spξq dmpξq. n,m p ¡ qm 1p ¡ qn 2 L S 20S Ωc w ξ z w In the right-hand side above, we have that ξ, z P Ω. Therefore, we can use Green’s Theorem in the integral on Ωc and then (4.17) to get » ¢» pw ¡ ξqm 1 T rpf ¡ f qϕ spzq c dw Brp¯ f ¡ f qϕ spξq dmpξq m L S n,m p ¡ qm 1p ¡ qn 2 L S »20S BΩ w ξ z w p qBrp¯ ¡ q sp q p q cn,m K ~m0 z, ξ f fL ϕS ξ dm ξ , 20S where ~m0 : p2 n, m 1, m 1q. Using Proposition 4.15 we have that

m ¸ m 1 pξ ¡ zq cm,n,jR pz, ξq K pz, ξq c BnBχ pzq m n,j . ~m0 m,n Ω pξ ¡ zqm 1 pξ ¡ zqm n 2¡j j¤m

n p¡m¡1,mq The ﬁrst part is B BχΩpzq times the kernel of the operator T . For the second part, we have that by Lemma 4.22 |Rm 1 pz, ξq| m n,j À 1 ¡ ¡ , |ξ ¡ z|m n 2 j |ξ ¡ z|2 σp 4.5. COMPACTNESS OF THE DOUBLE REFLECTION 111

¡ 2 where σp 1 p . Thus, ¸ » 7 sup |Tmrpf ¡ fLqϕSspzq| Mgpzq dmpzq (4.47) } } g p1 1 pS,LqPW 2L 0 » § § ¸ § ¨ § n p¡m¡1,mq ¯ À sup §B BχΩpzqT Brpf ¡ fLqϕSs pzq§ Mgpzq dmpzq } } g p1 1 pS,LqPW 2L 0 » » § § ¸ §Brp¯ f ¡ f qϕ spξq§ L S p q p q p q sup ¡ dm ξ Mg z dm z 7.1 7.2 . | ¡ |2 σp }g} 1 1 2L 20S ξ z p pS,LqPW0

p¡ ¡ q p¡ ¡ q In the ﬁrst sum we use that in W 1,ppCq we have the identity T m 1,m ¥ B¯ B¯ ¥ T m 1,m m p¡m¡1,mq¯ m 1,p 8 cmB and, therefore, T Brpf ¡ fLqϕSs cmB rpf ¡ fLqϕSs P W L , so ¸ » À |Bn p q| p q p q} mrp ¡ q s} 7.1 sup BχΩ z Mg z dm z B f fL ϕS L8 }g} 1 1 p qP 2L p S,L¸W0 n m À sup }B BχΩ} pp q}Mg} p1 p q}B rpf ¡ fLqϕSs} 1,pp q. L 2L L 2L W C }g} 1 1 p pS,LqPW0

By the boundedness of Bm in W 1,ppCq we have that m }B rpf ¡ fLqϕSs} 1,pp q À }pf ¡ fLqϕS} 1,pp q. W C W 20S Moreover, the Poincar´einequality (1.33) allows us to deduce that }p ¡ q } À} } f fL ϕS W 1,pp20Sq ∇f Lpp20Sq.

¡i On the other hand, there is a certain i0 such that for `pSq ¤ C2 and L X 2S H, we have z that S, 2L Ω Ωi¡i0 , and n n }B BχΩ} pp q ¤ }B BχΩ} pp z q. L 2L L Ω Ωi¡i0

1 Thus, by the H¨olderinequality and the boundedness of the maximal operator in Lp we have that ¸ n 7.1 À }B BχΩ} pp z q sup }Mg} p1 p q}∇f} pp q L Ω Ωi¡i0 L 2L L 20S }g} 1 1 p pS,LqPW0

¤ } } } } 1 À } } CΩ,i ∇f LppΩq sup Mg Lp p CΩ,i ∇f LppΩq, (4.48) } } g p1 1

iÑ8 with CΩ,i ÝÝÝÑ 0. To bound the term 7.2 in (4.47), note that given two neighbor cubes S and L and a point z P 2L, by (1.28) we have that » § § §¯ § ¨ Brpf ¡ fLqϕSspξq p q À Brp¯ ¡ q s p q p qσp ¡ dm ξ M f fL ϕS z ` S . | ¡ |2 σp 20S ξ z Thus, » ¸ ¨ ¯ σp 7.2 À sup M Brpf ¡ fLqϕSs pzq`pSq Mgpzq dmpzq } } g p1 1 pS,LqPW 2L 0 ¸ ¨ ¡iσ À p Brp¯ ¡ q s } } 1 2 sup M f fL ϕS LppΩq Mg Lp p2Lq }g} 1 1 p pS,LqPW0 112 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS and, by the boundedness of the maximal operator, (1.33) and the H¨olderinequality, we get ¸ ¡iσ ¡iσ À p Brp¯ ¡ q s } } 1 À p } } 7.2 2 sup f fL ϕS Lpp20Sq Mg Lp p2Lq 2 ∇f LppΩq. (4.49) }g} 1 1 p pS,LqPW0 By (4.47), (4.48) and (4.49), we have that À } } 7 CΩ,i ∇f LppΩq, (4.50)

iÑ8 with CΩ,i ÝÝÝÑ 0. Back to (4.46) it remains to bound ¸ » | rp ¡ qp ¡ qsp q ¡ p rp ¡ qp ¡ qsq | | p q| p q 8 sup Tm f fS 1 ϕS z Tm f fL 1 ϕS L g z dm z . }g} 1 1 2S p pS,LqPW0 ¥ } } Fix g 0 such that g p1 1. Then we will prove that ¤ } } 8g CΩ,i f W 1,ppΩq,

iÑ8 with CΩ,i ÝÝÝÑ 0, where ¸ » 8g : |Tmrpf ¡ fSqp1 ¡ ϕSqspzq ¡ Tmrpf ¡ fLqp1 ¡ ϕSqspζq| dmpζqgpzq dmpzq. 2S L pS,LqPW0

First, we add and subtract Tmrpf ¡ fLqp1 ¡ ϕSqspzq in each term of the last sum to get ¸ » 8g ¤ |TmrpfL ¡ fSqp1 ¡ ϕSqspzq| dmpζqgpzq dmpzq 2S L pS,LqPW0 ¸ » |Tmrpf ¡ fLqp1 ¡ ϕSqspzq ¡ Tmrpf ¡ fLqp1 ¡ ϕSqspζq| dmpζqgpzq dmpzq. 2S L pS,LqPW0 For a given z P Ω, » » » |fpξq ¡ fL| | logpdistpw, Ωqq| | logpdiampΩqq| dmpξq dmpwq À }f} 8 dmpwq, | ¡ |n 2| ¡ |2 L | ¡ |n 2 Ωc Ω z w w ξ Ωc z w which is ﬁnite since Ω is a Lipschitz domain³ (hint: compare the last integral above with the length 1pB q 1 | p q| of the boundary H Ω times the integral 0 log t dt). Thus, we can use Fubini’s Theorem and then Green’s Theorem to state that » » p ¡ qm¡1 rp ¡ qp ¡ qsp q 1 w ξ p p q ¡ qp ¡ p qq p q p q Tm f fL 1 ϕS z cn n 2 m 1 f ξ fL 1 ϕS ξ dm ξ dm w c pz ¡ wq pw ¡ ξq Ω» ¢» Ω pw ¡ ξqm c dw rpf ¡ f qp1 ¡ ϕ qspξq dmpξq n,m p ¡ qm 1p ¡ qn 2 L S »Ω BΩ w ξ z w p qrp ¡ qp ¡ qsp q p q cn,m K ~m1 z, ξ f fL 1 ϕS ξ dm ξ , Ω where ~m1 : p2 n, m 1, mq. Arguing analogously, » rp ¡ qp ¡ qsp q p ¡ q p qrp ¡ qsp q p q Tm fL fS 1 ϕS z cn,m fL fS K ~m1 z, ξ 1 ϕS ξ dm ξ . Ωz10S 4.5. COMPACTNESS OF THE DOUBLE REFLECTION 113

Thus, we get that § § » §» § ¸ § § 8g À |f ¡ f | § K pz, ξqrp1 ¡ ϕ qspξq dmpξq§ gpzq dmpzq L S § ~m1 S § p qP 2S Ωz10S S,L W0 » §» § ¸ § § § p p q ¡ p qqrp ¡ qp ¡ qsp q p q§ p q p q p q § K ~m1 z, ξ K ~m1 ζ, ξ f fL 1 ϕS ξ dm ξ § dm ζ g z dm z 2S L Ω pS,LqPW0 8.1 8.2 . (4.51)

Recall that Proposition 4.15 states that for z P 2S and ξ P Ω,

m¡1 ¸ m pξ ¡ zq cm,n,jR ¡ pz, ξq K pz, ξq c BnBχ pzq m n 1,j (4.52) ~m1 m,n Ω pξ ¡ zqm 1 pξ ¡ zqm n 2¡j j¤m and, for any z, ξ P Ω, by (4.29) we have that § § § Rm pz, ξq § § m n¡1,j § ¤ 1 § ¡ § CΩ,n,m ¡ , (4.53) pz ¡ ξqm n 2 j |z ¡ ξ|3 σp

° ¡ pb¡aq j 1 aibj¡1¡i ¡ 2 1 ¡ 1 i0 P where σp 1 p . Thus, by (4.30) and the identity aj bj aj bj , when z, ζ 5S and ξ P Ωz20S we have that § § § ¢ § § m p q m p q § § § § Rm n¡1,j z, ξ Rm n¡1,j ζ, ξ § § m 1 1 § § ¡ § ¤ §R ¡ pz, ξq ¡ § pξ ¡ zqm n 2¡j pξ ¡ ζqm n 2¡j m n 1,j pξ ¡ zqm n 2¡j pξ ¡ ζqm n 2¡j § § m m §R pz, ξq ¡ R pζ, ξq§ | ¡ | | ¡ |σp | ¡ |σp § m n¡1,j m n¡1,j § À z ζ z ζ À z ζ § ¡ § Ω,n,m ¡ . (4.54) pξ ¡ ζqm n 2 j |z ¡ ξ|4 σp |z ¡ ξ|3 |z ¡ ξ|3

³ ¡ p p ¡ qq ¡ pξ¡zqm 1 rp ¡ qsp q p q Then, using that dist 2S, supp 1 ϕS 0, we have that Ω pξ¡zqm 1 1 ϕS ξ dm ξ mrp ¡ qsp q P cmBΩ 1 ϕS z for z 2S and, by (4.51), (4.52) and (4.53) we get that ¸ » À | ¡ | |Bn p q mrp ¡ qsp q| p q p q 8.1 fL fS BχΩ z BΩ 1 ϕS z g z dm z 2S pS,LqPW0 ¸ » » | ¡ | 1 p q p q p q fL fS ¡ dm ξ g z dm z | ¡ |3 σp 2S Ωz10S z ξ pS,LqPW0 8.1.1 8.1.2 . (4.55)

We begin by the ﬁrst term in the right-hand side of (4.55), that is, ¸ » | ¡ | |Bn p q mrp ¡ qsp q| p q p q 8.1.1 fL fS BχΩ z BΩ 1 ϕS z g z dm z . 2S pS,LqPW0

By the Poincar´eand the Jensen inequalities, we have that » p q 1 ` L 1¡ 2 |f ¡ f | ¤ |fpξq ¡ f | dmpξq À }∇f} À `pSq p }∇f} . (4.57) L S p q2 S p q2 L1p5Sq Lpp5Sq ` L L ` L

m On the other hand, by Lemma 4.25 below we have that B ϕSpzq 0 for z P 2S. Therefore, using (4.57) we have that ¸ » 1¡ 2 À} } p q p |Bn p q m p q| p q p q 8.1.1 ∇f LppΩq ` S BχΩ z BΩ χΩ z g z dm z (4.58) S:`pSq¤C2¡i 2S ¡ip1¡ 2 q n m ¡ip1¡ 2 q À p } } } } 1 }B } } } À p } } 2 ∇f LppΩq g Lp pΩq BχΩ LppΩq BΩ χΩ L8pΩq Ω 2 ∇f LppΩq.

Let us recall that the second term in the right-hand side of (4.55) is ¸ » » | ¡ | 1 p q p q p q 8.1.2 fL fS ¡ dm ξ g z dm z | ¡ |3 σp 2S Ωz10S z ξ pS,LqPW0 and, by (4.57), ¸ 1¡ 2 1 À p q p } } } } 8.1.2 ` S ∇f pp q ¡ g 1p q L 5S `pSq1 σp L 2S p q¤ ¡i S:` S¸C2 σ ¡ 2 2 À p q p p p } } } } 1 ` S ∇f Lpp5Sq g Lp p2Sq. S:`pSq¤C2¡i

By H¨older’sinequality,

¡iσ ¡iσ À p } } } } 1 p } } 8.1.2 2 ∇f LppΩq g Lp pΩq 2 ∇f LppΩq.

Using this fact together with (4.55) and (4.58), we have that À } } 8.1 CΩ,i ∇f LppΩq, (4.59)

iÑ8 with CΩ,i ÝÝÝÑ 0. Let us focus now on the ﬁrst term in the right-hand side of (4.56), that is, ¸ » |Bn p q| | mrp ¡ qp ¡ qsp q| p q p q 8.2.1 BχΩ z BΩ f fL 1 ϕS z g z dm z (4.60) p qP 2S S,L¸W0 n m À } } 1 }B } } rp ¡ qp ¡ qs} g Lp p2Sq BχΩ Lpp2Sq BΩ f fL 1 ϕS L8p2Sq. pS,LqPW0

m 1,pp q By the Sobolev Embedding Theorem and the boundedness of BΩ in W Ω (granted by Theorem 3.28) we have that

} mrp ¡ qp ¡ qs} ¤ } mrp ¡ qp ¡ qs} À }p ¡ qp ¡ q} BΩ f fL 1 ϕS L8pΩq BΩ f fL 1 ϕS W 1,ppΩq f fL 1 ϕS W 1,ppΩq 4.5. COMPACTNESS OF THE DOUBLE REFLECTION 115 and, using Leibniz’ rule, Poincar´e’sinequality and the Sobolev embedding Theorem, we get

m 1 }B rpf ¡ f qp1 ¡ ϕ qs} 8 ¤ }∇f} }f ¡ f } }f ¡ f } Ω L S L pΩq LppΩq `pSq L Lpp20Sq L LppΩq À } } } } } } } } À} } Ω ∇f LppΩq ∇f Lpp20Sq f LppΩq f L8 f W 1,ppΩq.

Thus, by H¨older’sinequality we have that

n n 8.2.1 À }f} 1,pp q}g} p1 p q}B BχΩ} pp z q }f} 1,pp q}B BχΩ} pp z q. (4.61) W Ω L Ω L Ω Ωi¡i0 W Ω L Ω Ωi¡i0

n iÑ0 Note that }B BχΩ} pp z q ÝÝÑ 0. L Ω Ωi¡i0 The second term in (4.56), that is, » » ¸ 1 8.2.2 |BnBχ pζqBmrpf ¡ f qp1 ¡ ϕ qspζq| dmpζq gpzq dmpzq, p q2 Ω Ω L S ` L L 2S pS,LqPW0 follows the same pattern. Since S and ³L in the sum above are neighbors, they have comparable P p q p q À p q2 p q side-length, and for ζ L we have that 2S g z dm z ` L Mg ζ . Therefore, ¸ » À |Bn p q mrp ¡ qp ¡ qsp q| p q p q 8.2.2 BχΩ ζ BΩ f fL 1 ϕS ζ Mg ζ dm ζ p qP L S,L ¸W0 n m À } } 1 }B } } rp ¡ qp ¡ qs} Mg Lp p5Sq BχΩ Lpp5Sq BΩ f fL 1 ϕS L8p5Sq. S:`pSq¤C2¡i

The last expression coincides with the right-hand side of (4.60) changing g by Mg and 2S by 5S. Arguing analogously to that case, we get that

n n 8.2.2 À}f} 1,pp q}Mg} p1 p q}B BχΩ} pp z q À}f} 1,pp q}B BχΩ} pp z q. (4.62) W Ω L Ω L Ω Ωi¡i0 W Ω L Ω Ωi¡i0 Finally, we consider ¸ » » |z ¡ ζ|σp 8.2.3 |fpξq ¡ f | dmpξq dmpζqgpzq dmpzq. | ¡ |3 L 2S L Ωz10S z ξ pS,LqPW0

Note that for z, ζ P 5S we have that |z ¡ ζ| À `pSq. Separating Ωz10S in Whitney cubes we get ¸ » ¸ `pSqσp 8.2.3 À gpzqdmpzq }f ¡ f } . p q3 L L1pP q 2S D S, P pS,LqPW0 P PW

Using the chain connecting two cubes P and L, by (1.34) we get that

¸ `pP q2 }f ¡ f } À }∇f} . L L1pP q L1p3Qq `pQq QPrP,Ls

Thus,

¸ ¸ p q2} } } } ¡ ` P ∇f L1p3Qq g L1p7Lq 8.2.3 À 2 iσp . `pQqDpL, P q3 L,P PW QPrP,Ls 116 CHAPTER 4. AN APPLICATION TO QUASICONFORMAL MAPPINGS

By Lemma 2.3 (with ρ 1), we get

¡ À iσp } } 8.2.3 2 ∇f LppΩq, (4.63) and Claim 4.24 is proven. Indeed, by (4.56), (4.61), (4.62) and (4.63), we have that

À } } 8.2 CΩ,i ∇f LppΩq.

This fact combined with (4.51) and (4.59) prove that

¤ À } } 8 sup 8g CΩ,i ∇f LppΩq } } g p1 1 and, together with (4.35), (4.46) and (4.50), gives i À } } E f LppΩq CΩ,i f W 1,ppΩq,

iÑ8 with CΩ,i ÝÝÝÑ 0. It remains to prove the following result. 2 | P Lemma 4.25. Let ϕ be a radial function in L such that ϕ D 0. Then, for every m N,

m B ϕpzq 0 for z P D. Proof. Since Bϕ is in L2 and it is radial by linearity, by induction, it is enough to prove that

Bϕpzq 0 for z P D. ¡ } ¡ } P Let ε 0 and consider a simple radial function s such that ϕ s L2 ε. Let z D. Recall p q that BχD z 0 (see [AIM09, (4.24)]). Since s is a ﬁnite combination of characteristic functions t uM P i p q of concentric disks Di i1 with z D for all i, then, Bs z 0. p ¡ q } } ¤ } p ¡ q} Therefore χDBϕ χDB ϕ s and, thus, we get χDBϕ L2 B ϕ s L2 ε. Since ε can be chosen as small as desired, χDBϕ 0. Chapter 5

Carleson measures on Lipschitz domains

Theorem 2.1 provides us with a useful tool to check if an operator is bounded on W n,ppΩq as long as p ¡ d. Our concern for this chapter is to find a sufficient condition valid even if p ¤ d. We want this condition to be related to some test functions (the polynomials of degree smaller than n seem the right choice) but somewhat more specific than the condition in the Key Lemma. In particular we seek for some Carleson condition in the spirit of the celebrated article [ARS02] by N. Arcozzi, R. Rochberg and E. Sawyer. In Section 5.2 we define the vertical shadows Shvpxq and Shvpxq for every point x in a Lipschitz domain Ω close enough to BΩ (see Definition 5.14 and Figure 5.2). This definition is similar to the shadow introduced in Section 1.4, but requires a local orientation. Therefore, in this chapter we will restrict ourselves to the study of Lipschitz domains, although there is hope that some of the techniques used here can be extrapolated to uniform domains (note that in the proof of the necessity of the Carleson condition, that is, Theorem 5.3 below, we use Lemma 5.8 which requires some restrictions in the dimension of the boundary). Those shadows, as before, can be understood as Carleson boxes of the domain. We say that a positive and finite Borel measure µ is an s, p-Carleson measure if for every a P Ω and close enough to the boundary, » pd¡spqp1¡p1q p1 dx distpx, BΩq pµpShvpxq X Shvpaqqq ¤ CµpShvpaqq. p B qd Shvpaq dist x, Ω In this chapter we study the relation between Carleson measures and Calderón-Zygmund operators. The first result we obtain is the following: Theorem 5.1. Let T be an admissible convolution Calderón-Zygmundoperator of order n, and n p consider a bounded Lipschitz domain Ω and 1 p ¤ d. If the measure |∇ TΩP pxq| dx is a 1, p- Carleson measure for every polynomial P of degree at most n ¡ 1, then TΩ is a bounded operator on W n,ppΩq. We also have a counterpart for Triebel-Lizorkin spaces with smoothness 0 s 1: 8 8 ¡ d ¡ d Theorem 5.2. Let 1 p , 1 q and 0 s 1 with s p q , let T be a p, q-admissible convolution Calderón-Zygmundoperator of order s, and consider a bounded Lipschitz domain Ω. p q | s p q|p If the measure µ x ∇qTΩ1 x dx is an s, p-Carleson measure, then TΩ is a bounded operator s p q on Fp,q Ω . The Carleson condition of Theorem 5.1 above is in fact necessary for n 1:

117 118 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

Theorem 5.3. Let T be an admissible convolution Calder´on-Zygmundoperator of order 1, and consider a bounded Lipschitz domain Ω and 1 p 8. The following statements are equivalent:

1,p 1. TΩ is a bounded operator on W pΩq.

p 2. The measure |∇T χΩpxq| dx is a p-Carleson measure for Ω. Section 5.1 is devoted to modifying the definitions of Shadow and admissible chain introduced in Section 1.4. In Section 5.2 some results from [ARS02] which will be used in the subsequent sections are collected. Section 5.3 provides an adaptation of the Key Lemma 2.2 to proper orientations of the Whitney coverings and the proof of Theorem 5.1, that is, the sufficient condition in terms of Carleson measures for an operator to be bounded on W n,ppΩq. After that, a counterpart to both s p q d ¡ d results for Fp,q Ω with p q s 1 (including Theorem 5.2 above) is given in Section 5.4. In Section 5.5 it is shown that when only the first derivatives are considered (that is, for W 1,ppΩq) this sufficient condition is in fact necessary, proving Theorem 5.3. Finally, Section 5.6 contains a different approach for the complex plane when the domain is more regular.

5.1 Oriented Whitney coverings

Along this section we consider Ω to be a ﬁxed bounded pδ, Rq-Lipschitz domain. Recall that we say that Q is an R-window of Ω if it is a cube centered in BΩ, with side-length R inducing a Lipschitz parameterization of the boundary (see Deﬁnition 1.4). We can choose a d¡1pB q{ d¡1 t uN number N H Ω R and a collection of windows Qk k1 such that

¤N BΩ δ1Qk, k1 where δ1 ¡ 0 is a value to ﬁx later (in Remark 5.5). Each window Qk is associated to a parameterization Ak in the sense that, after a rotation, X tp 1 q P p d¡1 ¢ q X ¡ p 1qu Ω 2Qk y , yd R R 2Qk : yd Ak y .

Thus, each Qk induces a vertical direction, given by the eventually rotated yd axis. The number of Whitney cubes in Qk with the same side-length intersecting a given vertical line is bounded by a constant depending only on the Lipschitz character of Ω. In the subsequent sections we will make use of a tree structure on the Whitney cubes compatible with admissible chains in a tubular neighborhood of the boundary. Therefore, we must modify the notions introduced in Section 1.4. We distinguish the cubes in the central region from those which are close to the boundary of the domain.

Definition 5.4. We say that Q is central if distpQ, BΩq ¡ δ2R, where 0 δ2 1 is a constant to fix in Remark 5.5. We denote this subcollection of cubes by W0. We say that Q is peripheral if it is not central. Given Whitney cubes Q, S Qk, we will say that S is above Q if the “vertical” projection of the open cube Q intersects the vertical projection of S and the center of S has vertical coordinate greater or equal than the center of Q (the vertical direction is the one induced by the window again). Consider δ1 δ0 1 to be fixed. We call δ0Qk XΩ the canvas of the window Qk, and we divide the peripheral cubes in collections Wk tQ P WzW0 : Q δ0Qk X Ωu.

Remark 5.5. For Whitney constants big enough and for δ0, δ1 and δ2 small enough we have that 1) The union of central cubes is a connected set. 5.1. ORIENTED WHITNEY COVERINGS 119

2) Every peripheral cube is contained in a window canvas. The subcollections Wk are not disjoint and, if two peripheral cubes Q and S are not contained in any common Wk, then R À distpQ, Sq À diampΩq.

3) For each peripheral cube Q P Wk there exists a cube S Qk above Q which is central. Furthermore, 4) All the central cubes have comparable side-length. Next we provide a tree-like structure to a particular family of cubes contained in a single window. Deﬁnition 5.6. We say that a Whitney covering is properly oriented with respect to a window Qk if the cubes in the Whitney covering have sides parallel to the faces of Qk. Given Whitney cubes Q, S Qk, we will say that Q P SHvpSq if S is above Q. When the Whitney covering is properly oriented, this property is transitive and deﬁnes a partial order relation. We want to have a somewhat rigid structure to gain some control on the chains we use, so we need to introduce a chain function.

Definition 5.7. Given two different cubes Q, S P Wk (where W is a properly oriented Whitney covering with respect to the k-th window), we define rQ, Ss : rQ, QSs Y rSQ,Ss to the chain such that r s p q r s p q • the cubes in the subchains Q, QS Q1,...,QM1 and SQ,S SM2 ,...,S1 satisfy that Qj¡1 P SHvpQjq and Sj¡1 P SHvpSjq for j ¡ 1 with respect to the “vertical” direction induced by the window, and p q • the only pair of cubes Qj1 ,Sj2 which has a vertical segment contained in the boundaries of p q p q both of them is QM1 ,SM2 , that is, QS,SQ .

For δ0 small enough, this definition makes sense, that is, the chain rQ, Ss exists and is unique. We define the collection of central cubes in a given window c t P X P X p qu Wk Q W0 : Q Qk Ω and there exists S Wk SHv Q , and we call the collection of central and peripheral cubes Wcp : W Y Wc. Then, r¤, ¤s : Wcp ¢ k k k k cp Ñ p cpqM Wk M Wk satisfying the properties above is called (the) chain function. In the next sections we will make use of the following technical results, specific for Lipschitz domains, which sharpen the results of Lemma 1.18 for g constant. ¡ ¡ P cp Lemma 5.8. Let a d 1 and Q Wk a Whitney cube. Then ¸ `pSqa `pQqa

SPSHvpQq with constants depending only on a and d. Proof. Selecting the cubes by their side-length, we can write ¸ ¸8 ¸ ¸8 a ¡j a a ¡ja ¡j `pSq p2 `pQqq `pQq 2 #tS P SHvpQq : `pSq 2 `pQqu. SPSHvpQq j 0 SPSHvpQq j 0 `pSq2¡j `pQq 120 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

Since the domain is Lipschitz and the Whitney covering is oriented, the number of cubes with a given side-length intersecting a vertical line is uniformly bounded. Thus, we get that ¡j pd¡1qj #tS P SHvpQq : `pSq 2 `pQqu ¤ C2 and therefore ¸ ¸8 `pSqa À `pQqa 2¡jpa¡pd¡1qq. SPSHvpQq j 0 This is bounded if a ¡ d ¡ 1. ° Remark 5.9. By the same token, given an R-window Q , we get `pSqa À Ra (see Remark k SQk 5.5, property 3)). Thus, the lemma is also valid for Q central writing ¸ `pSqa `pQqa SPW by the last statement of Remark 5.5. Lemma 5.10. Let b ¡ a ¡ d ¡ 1 and Q a Whitney cube. Then ¸ `pSqa ¤ C`pQqa¡b, DpQ, Sqb SPW with C depending only on a, b and d.

Proof. Let us assume that Q P Wk. First of all we consider the cubes contained in Qk and we classify those cubes by their side-length and their long distance to Q: ¸ p qa ¸8 ¸8 ¸ p i p qqa ` S ¤ 2 ` Q p qb p j p qqb D Q, S ¡8 i 2 ` Q SQk i j 0 S:`pSq2 `pQq 2j `pQq¤DpS,Qq 2j 1`pQq 8 8 ¸ ¸ 2ia ¤ `pQqa¡b #tS : `pSq 2i`pQq,DpS, Qq 2j 1`pQqu. 2jb i¡8 j0 Note that the value of j in the last sum must be greater or equal than i because, otherwise, the last cardinal would be zero. Using again the fact that the number of cubes with a given side-length intersecting a vertical line is uniformly bounded, we can see that ¢ ¡ p2j 1q`pQq d 1 #tS P W : `pSq 2i`pQq,DpS, Qq 2j 1`pQqu ¤ C C2pj¡iqpd¡1q. k 2i`pQq Thus, 8 ¸ `pSqa ¸ ¸j À `pQqa¡b 2ipa 1¡dq¡jpb 1¡dq ¤ C `pQqa¡b DpQ, Sqb a,b,d SQk j0 i¡8 as soon as b ¡ a ¡ d ¡ 1. On the other hand, when S Qk for every window containing Q. Then the long distance DpQ, Sq is always bounded from below by a constant times R (because Q δ0Qk), so separating W in subcollections Wk and using Remark 5.9, ¸ `pSqa ¸ pdiamΩqa ¸ ¸ `pSqa À À Ra¡b À `pQqa¡b. (5.1) DpQ, Sqb Rb Rb SQk SPW0 jk SPWj

To prove the lemma for a central cube Q P W0, just apply an argument analogous to (5.1). 5.2. CARLESON MEASURES 121

5.2 Carleson measures

Next we recall some useful results from [ARS02]. First we need to introduce some notation.

o x

Figure 5.1: y P ShT pxq.

Deﬁnition 5.11. We say that a connected, loopless graph T is a tree, and we will ﬁx a vertex o P T and call it its root. This choice induces a partial order in T , given by x ¤ y if x P ro, ys where ro, ys stands for the geodesic path uniting those two vertices of the graph (see Figure 5.1). We call shadow of x in T to the collection

ShT pxq ty P T : y ¥ xu.

We say that a function ρ : T Ñ R is a weight if it takes positive values (by a function we mean a function deﬁned in the vertices of the tree).

Deﬁnition 5.12. Given h : T Ñ R, we call the primitive Ih the function ¸ Ihpyq hpxq. xPro,ys Theorem 5.13. [ARS02, Theorem 3] Let 1 p 8 and let ρ be a weight on T . For a nonnegative measure µ on T , the following statements are equivalent: i) There exists a constant C Cpµq such that } } ¤ } } Ih Lppµq C h Lppρq

ii) There exists a constant C Cpµq such that for every r P T one has ¸ p1 1¡p1 µpShT pxqq ρpxq ¤ CµpShT prqq.

xPShT prq

For every 1 ¤ p ¤ 8, we say that a non-negative measure µ is a p-Carleson measure for pI, ρ, pq if there exists a constant C Cpµq such that the condition i) is satisfied. To use the techniques on Carleson measures introduced in [ARS02] we need to have some tree structure coherent with the chain function from Definition 5.7. Note that this structure can be given in terms of the vertical shadow SHv as long as the Whitney covering is properly oriented. We also define the (almost) continuous version of the shadow: Definition 5.14. Given an R-window Q of a Lipschitz domain Ω with a properly oriented Whitney P p 1 q P d¡1 ¢ covering W, for every x Q, we write x x , xd R R and, if x is contained in a semi-open Whitney cube Q P W, we define the vertical shadow of x as " * 1 Sh pxq : y P Q X Ω: y x and x1 ¡ y1 ¤ `pQq . v d d 8 2 122 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

Note that if x is the center of the upper pd ¡ 1q-dimensional face of Q, the vertical projection of Shvpxq (which is a pd ¡ 1q-dimensional square) coincides with the vertical projection of Q (see Figure 5.2). Finally, we deﬁne the vertical extension of Shvpxq, " * 1 Sh pxq : y P Q X Ω: y x 2`pQq and x1 ¡ y1 ¤ `pQq . v d d 8 2 More generally, given a set U Q we call its shadow ( 1 1 ShvpUq : y P Q X Ω: there exists x P U such that yd xd and x y . Note that Sh pQq P for any Whitney cube Q Q (see Figure 5.2). v P PSHvpQq

x Q

Figure 5.2: The shadows Shvpxq and ShvpQq coincide when x is the center of the upper face of the cube. Furthermore, P ShvpQq if and only if P P ShT pQq SHvpQq.

Recall that we have a proper orientation in the Whitney covering. Thus, given a Whitney cube Q, we call the father of Q, FpQq the neighbor Whitney cube which is immediately on top of Q with respect to the vertical direction. This parental relation is coherent with the order relation in Deﬁnition 5.6 (P P SHvpQq if P is a descendant of Q). This would provide a tree structure to the Whitney covering W if there was a common ancestor Q0 for all the cubes. This does not happen, but we can add a “formal” cube Q0 (root of the tree) and then we can write p q t P cp u SHv Q0 : Q Wk : Q Q . If we call T to the tree with the Whitney cubes as vertices complemented with Q0 and the structure given by the aforementioned order relation, then for every Whitney cube Q Q, we have that SHvpQq ShT pQq. Since we will only consider functions and measures supported in the window canvas δ0Q X Ω, we can extend all of them formally in Q0 as the null function. Now, some minor modiﬁcations in the proof of [ARS02, Proposition 16] allow us to rewrite this proposition in the following way.

Proposition 5.15. Given 1 p 8, a positive number s P R and an R-window Q of a Lipschitz domain Ω with a properly oriented Whitney covering W, consider the weights ρpxq d¡sp d¡sp distpx, BΩq , ρW pQq `pQq . For a positive Borel measure µ supported on δ0Q X Ω, the following are equivalent:

1. For every a P δ0Q X Ω one has » 1¡p1 p1 dx ρpxq pµpShvpxq X Shvpaqqq ¤ CµpShvpaqq. p B qd Shvpaq dist x, Ω 5.3. INTEGER SMOOTHNESS: A SUFFICIENT CONDITION 123

2. For every Whitney cube P Q one has ¸ p1 1¡p1 µpShvpQqq ρW pQq ¤ CµpShvpP qq. (5.2)

QPSHvpP q

In virtue of [ARS02, Theorem 1], when d 2, s 1 and the domain Ω is the unit disk in the plane, the first condition is equivalent to µ being a Carleson measure for the analytic Besov space p q Bp ρ , that is, for every analytic function defined on the unit disc D, » p q } }p À} }p | p q|p p ¡ | |2qp| 1p q|p p q dm z f pp q f p q : f 0 1 z f z ρ z . L µ Bp ρ p1 ¡ |z|2q2 D Definition 5.16. We say that a measure satisfying the hypothesis of Proposition 5.15 is an s, p- Carleson measure for Q (or simply p-Carleson measure if s 1). We say that a positive and finite Borel measure µ is an s, p-Carleson (or p-Carleson) measure for a Lipschitz domain Ω if it is an s, p-Carleson (resp. p-Carleson) measure for every R-window of the domain.

5.3 Integer smoothness: a suﬃcient condition for p ¤ d

We will use local versions of the Key Lemmas in Chapter 2 in order to get rid of some technical diﬃculties: Lemma 5.17. Let Ω Rd be a bounded Lipschitz domain, T an admissible convolution Calder´on- Zygmund operator of order n P N and 1 p 8. Then the following statements are equivalent. i) For every f P W n,ppΩq one has } } ¤ } } TΩf W n,ppΩq C f W n,ppΩq. (5.3)

n,p P p q | c ii) For every window Q and every f W Ω with f pδ0Qq 0 one has ¡ © ¸ p n n¡1 ¤ } }p ∇ TΩ P3Q f C f W n,ppΩq, LppQq QPWQ

where the Whitney covering WQ is properly oriented with respect to Q, that is, with the dyadic grid parallel to the local coordinates (see Deﬁnition 5.6). Proof. To see that i) implies ii) just use the Key Lemma 2.2 with an appropriate dyadic grid. t uN d¡1pB q{ d¡1 To see the converse, choose a ﬁnite a collection of windows Qk k1 with N H Ω R δ0 z δ0 such that 4 Qk is a covering of the boundary of Ω, call Q0 to the inner region Ω 2 Qk, and t uN 8 t u Y t uN let ψk k0 C be a partition of the unity related to the covering Q0 δ0Qk k1. Consider a function f P W n,ppΩq. Note that hypothesis ii) does not give information about the inner P n,pp dq region, but since ψ0 is compactly supported in Ω, ψ0f W R and by Remark 1.29 also p q P n,pp dq T ψ0f W R , so

}TΩpψ0fq} n,pp q }T pψ0fq} n,pp q ¤ }T pψ0fq} n,pp dq ¤ C}ψ0f} n,pp q. W Ω W Ω W R W Ω

Now, replacing f by ψkf and W by a properly oriented Whitney covering Wk with respect to Qk in (2.2) we get ¡ © ¡ © ¸ p ¸ p } n p q}p ¤ n ¡ n¡1p q n n¡1p q ∇ TΩ ψkf LppΩq ∇ TΩ ψkf P3Q ψkf ∇ TΩ P3Q ψkf LppQq LppQq QPWk QPWk ¤ } }p C ψkf W n,ppΩq. 124 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

Thus, by the triangle inequality we have that

¸N ¸N } } ¤ } p q} ¤ } } TΩf W n,ppΩq TΩ ψkf W n,ppΩq C ψkf W n,ppΩq. k0 k0 j À ¡j Choosing ψk as bump functions with the usual estimates on the derivatives ∇ ψk L8 R , we get (5.3) using the Leibniz formula: £ ¸N ¸N ¸n } } À } } À j À} } TΩf n,pp q ψkf n,pp q ∇ f p f n,pp q. W Ω W Ω L pδ0Qkq W Ω k0 k0 j0

We are ready to prove Theorem 5.1. This proof is very much in the spirit of Theorem 2.1. λ Again we fix a point x0 P Ω and we use the polynomials Pλpxq px ¡ x0q for every multiindex |λ| n, but now the key point is to use the Poincaréinequality instead of the Sobolev Embedding n p Theorem. The hypothesis in Theorem 5.1 is equivalent to dµλpxq |∇ TΩPλpxq| dx being a p-Carleson measure for Ω for every |λ| n. Proof of Theorem 5.1. Consider a fixed R-window Q and a properly oriented Whitney covering W, that is, with dyadic grid parallel to the window faces. Making use of Lemma 5.17, we only need to show that ¡ © ¸ p n n¡1 ¤ } }p ∇ TΩ P3Q f C f W n,ppΩq LppQq QPSHvpQ0q n,p P p q | c for every f W Ω with f pδ0Qq 0. Fix such a function f. Using the expression (1.31) and expanding it as in (2.10) at a fixed point x0 P Ω, we have ¡ © ¸ p ¸ ¸ ¸ n n¡1 À | |p} n }p ∇ TΩ P3Q f Cγ,λ,Ω mQ,γ ∇ TΩPλ LppQq. LppQq QPSHvpQ0q |γ| n ~0¤λ¤γ QPSHvpQ0q

Moreover, by induction on (1.35), the coeﬃcients are bounded by § § § § ¸ § § ¸ § § |β¡γ| § β § § β § |mQ,γ | À `pQq Cβ,γ § D f dm§ À Cβ,γ,R § D f dm§ ,

|β| n: β¥γ 3Q |β| n: β¥γ 3Q so ¡ © § § ¸ p ¸ ¸ § §p n n¡1 À § β § p q ∇ TΩ P3Q f § D f dm§ µλ Q . LppQq 3Q QPSHvpQ0q |β| n QPSHvpQ0q ~0¤λ¤β

β |p qc Taking into account that f δ0Q 0, we have 3P D f dm 0 for P close enough to the root Q0. Thus, £ ffl ¸ Dβf dm Dβf dm ¡ Dβf dm . 3Q 3P 3FpP q P PrQ,Q0q With a slight modification of the proof of Poincaréinequality in [Eva98, Theorem 5.8.1/1], one can see that } ¡ } À p q} } f f3Q Lpp3QX3FpQqq ` Q ∇f Lpp3Qq, 5.4. FRACTIONAL SMOOTHNESS: A SUFFICIENT CONDITION 125 and, by the same token, f ¡ f p q À `pQq}∇f} . Therefore, 3F Q Lpp3QX3FpQqq Lpp3FpQqq ¤ ¡ © p ¸ p ¸ ¸ ¸ n n¡1 À ¥ p q | β | p q ∇ TΩ P3Q f ` P ∇D f dm µλ Q . LppQq 3P QPSHvpQ0q |β| n QPSHvpQ0q P PrQ,Q0s ~0¤λ¤β (5.4)

By assumption, µλ is a p-Carleson measure for every |λ| n, that is, it satisﬁes both conditions p of Proposition 5.15. By Theorem 5.13, we have that, for every h P l pρW q, ¤ p ¸ ¸ ¸ ¥ p d¡p hpP q µλpQq ¤ C hpQq `pQq , (5.5)

QPSHvpQ0q P PrQ,Q0s QPSHvpQ0q

d¡p where ρW pQq `pQq . p q p q | β | Let us fix β and λ momentarily and take h P ` P 3P ∇D f dm in (5.5). Using Jensen’s inequality and the finite overlapping of the quintuple cubes,ffl we have ¤ p ¢ ¸ ¸ ¸ p ¥ β β d `pP q |∇D f| dm µλpQq ¤ C |∇D f| dm `pQq 3P 3Q QPSHvpQ0q P PrQ,Q0s QPSHvpQ0q ¸ À |∇Dβf|p dm `pQqd P p q 3Q »Q SHv Q0 À |∇Dβf|p dm. (5.6) Ω Plugging (5.6) into (5.4) for each β and λ, we get ¡ © ¸ p n n¡1 ¤ } }p ∇ TΩ P3Q f C f W n,ppΩq. LppQq QPSHvpQ0q

5.4 Fractional smoothness: a suﬃcient condition for sp ¤ d

Lemma 2.11 can be rewritten for fractional Triebel-Lizorkin spaces: 8 8 ¡ d ¡ d d Lemma 5.18. Let 1 p , 1 q and 0 s 1 with s p q , let Ω R be a bounded Lipschitz domain, and let T be a p, q-admissible convolution Calder´on-Zygmundoperator of order s. Then the following statements are equivalent. P s p q i) For every f Fp,q Ω one has

}TΩf} s p q ¤ C}f} s p q. Fp,q Ω Fp,q Ω

s P p q | c ii) For every window Q and every f Fp,q Ω with f pδ0Qq 0 one has ¸ | |p s p ¤ } }p fQ ∇ TΩ1 p C f s p q, q L pQq Fp,q Ω QPWQ 126 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

where the Whitney covering WQ is properly oriented with respect to Q, that is, with the dyadic s grid parallel to the local coordinates (see Deﬁnition 5.6), and the gradient ∇qf is deﬁned as

£» 1 | p q ¡ p q|q q s p q f x f y ∇qf x sq d dy , p qX |x ¡ y| Bρ1δpxq x Ω

with ρ1 big enough (see Corollary 2.22).

Proof. It is trivial that i) implies ii) by the Key Lemma 2.11, increasing ρε if necessary. 8 To see the converse, use the same partition of the unity tψku C as in the proof of Lemma 5.17. Again, for the central region, we have that

}TΩpψ0fq} s p q }T pψ0fq} s p q ¤ }T pψ0fq} s p dq ¤ C}ψ0f} s p dq. Fp,q Ω Fp,q Ω Fp,q R Fp,q R

By means of (2.13), (2.14) and (2.47), it is immediate to see that }ψ0f} s p dq À}ψ0f} s p q. Fp,q R Fp,q Ω Let 1 ¤ k ¤ N and consider a properly oriented Whitney covering Wk with respect to Qk. By (2.15), we have that ¸ s pp q ¡ p q qp À} }p ∇ TΩ ψkf ψkf Q p ψkf s p q q L pQq Fp,q Ω QPWk and, using the hypothesis ii), we get ¸ ¸ s p qp À |p q |p s p } }p À} }p ∇ TΩ ψkf p p ψkf Q ∇ TΩ1 p ψkf s p q ψkf s p q. q L pQq q L pQq Fp,q Ω Fp,q Ω QPWk QPWk Now we cannot use Leibniz’ rule. Instead, for x P Ω we have that

£» 1 | p q p q ¡ p q p q|q q sp qp q f x ψk x f y ψk y ∇q fψk x sq d dy p qX |x ¡ y| Bρ1δpxq x Ω £» 1 | p q ¡ p q|q q ¤ | p q| ψk x ψk y f x sq d dy p qX |x ¡ y| Bρ1δpxq x Ω £» 1 | p q ¡ p q|q q 1¡s | p q| f x f y À R | p q| sp qp q ψk y sq d dy f x ∇q f x p qX |x ¡ y| R Bρ1δpxq x Ω and, by the triangle inequality, the previous statements and (2.14), we get

¸N ¸N

}TΩf} s p q ¤ }TΩpψkfq} s p q À }ψkf} s p q Fp,q Ω Fp,q Ω Fp,q Ω k0 k0 ¸N ¡ © ¸N ¸N }ψ f} ∇spψ fq À }f} ∇sf . k LppΩq q k LppΩq LppΩq q LppΩq k0 k0 k0

Proof of Theorem 5.2. As before, consider Q and a properly oriented Whitney covering W. By Lemma 5.18, we only need to show that ¸ | |p s p ¤ } }p fQ ∇ TΩ1 p C f s p q q L pQq Fp,q Ω QPSHvpQ0q 5.5. SMOOTHNESS ONE: A NECESSARY CONDITION 127

s P p q | c for every f Fp,q Ω with f pδ0Qq 0. Fix one such a function f. Then ¸ ¸ p |f |p∇sT 1 |f |pµpQq. Q q Ω LppQq Q QPSHvpQ0q QPSHvpQ0q p | c | | Taking into account that f pδ0Qq 0, we have fP 0 for P close enough to the root Q0. Thus, By Jensen’s inequality § § § § » » § ¸ ¨§ ¸ s § § `pP q |fpxq ¡ fpyq| |f | f ¡ f p q À dy dx Q § P F P § p qd p qd s § § ` P P 5P ` P P PrQ,Q0q P PrQ,Q0q ¢ ¸ » » 1 ¸ » `pP qs |fpxq ¡ fpyq|q q À dy dx ¤ `pP qs¡d ∇sfpxq dx, p qd p qd sq q ` P P 5P ` P P P PrQ,Q0q P PrQ,Q0q and, adding on Q, we get ¤ » p ¸ ¸ ¸ p |f |p∇sT 1 À ¥ `pP qs¡d ∇sfpxq dx µpQq. Q q Ω LppQq q P QPSHvpQ0q QPSHvpQ0q P PrQ,Q0q Since µ is an s, p-Carleson measure, it satisﬁes both conditions of Proposition 5.15. By Theorem 5.13, we have that, for every function h : W Ñ C, ¤ p ¸ ¸ ¸ ¥ hpP q µpQq ¤ C hpQqp`pQqd¡sp.

5.5 Smoothness n 1: a necessary condition

The implication 2. ùñ 1. in Theorem 5.3 is a consequence of Theorem 5.1, so we only need to prove that 1. ùñ 2.. To do so, we will solve a Neumann problem by means of the Newton P 1p dq potential: given an integrable function with compact support g Lc R , its Newton potential is » » |x ¡ y|2¡d log |x ¡ y| Ngpxq gpyq dy if d ¡ 2, Ngpxq gpyq dy if d=2 p2 ¡ dqwd 2π P d d for a.e. x R , where wd stands for the surface measure of the unit sphere in R . Recall that the gradient of Ng is the pd ¡ 1q-dimensional Riesz transform of g, » ¡ p q pd¡1q p q x y p q ∇Ng x R g x d g y dy. wd|x ¡ y|

It is well known that ∆Ngpxq gpxq for x P Rd (see [Fol95, Theorem 2.21] for instance). 128 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

P 1p d q ¡ Remark 5.19. Given g Lc R and d 2, consider the function ¡ © » pd¡1q ¡ © 2 R g pyq p q pd¡1q p q d p q P d F x : N 2 Rd g dσ x d¡2 dσ y for x R , (5.7) B d p2 ¡ dqw |x ¡ y| R d

pd¡1q pd¡1q where Rd stands for the vertical component of R and dσ is the hypersurface measure in d BR . This function is well deﬁned since » » p ¡ q z d 1 ¤ d | p q| p q Rd g d g z dz dσ y L1pσq B d d |y ¡ z| R R » £» zd p q | p q| } } d dσ y g z dz g 1 d B d |y ¡ z| R R

d and, thus, the right-hand side of (5.7) is an absolutely convergent integral for each x P R , with }g} | p q| ¤ 2 1 8p d q F x ¡ | |d¡2 . By the same token, all the derivatives of F are well deﬁned, F is C , d 2 xd ¡ © R p q pd¡1q pd¡1q p q harmonic and ∇F x R 2 Rd g dσ x . When d 2 we have to make the usual modiﬁcations.

d ¡ Lemma 5.20. Consider a ball¨B1 R centered¨ at the origin and a real number ε 0. Let P 8 d X 1 z d¡1 ¢ p q g L supported in R 4 B1 R 0, ε and deﬁne ¡ © p q pd¡1q p q ¡ p q h x : N 2 Rd g dσ x Ng x .

d P 8p d q Then h has weak derivatives in R and for every φ Cc R , » » ∇φ ¤ ∇h dm φ g dm. (5.8) d d R R

P d z Furthermore, if B1 has radius r1 then for every x R B1 we have $ 1 &' }g} if d ¡ 2, | |d¡2 1 | p q| À ¢x h x ' x | log x | (5.9) % |log |x|| 1 r 2 2 }g} if d 2, 1 |x|2 1 and ¢ § § § § 1 § xd § |∇hpxq| À 1 §log § }g} . (5.10) |x|d¡1 |x| 1 Remark 5.21. Note that h can be understood as a weak solution to the Neumann problem # d ¡∆hpxq gpxq if x P R , B p q PB d dh y 0 if y R .

Proof of Lemma 5.20. Let us deﬁne F as in (5.7). Then, ¡ © pd¡1q pd¡1q ∇F R 2 Rd g dσ 5.5. SMOOTHNESS ONE: A NECESSARY CONDITION 129

8 and h F ¡ Ng. Since g P L , it follows that Ng is C1pRdq (in fact it is harmonic¡ © out of the pd¡1q pd¡1q support of g) and it satisfies that BdNg R g. Moreover, we have that R g |B d is in ¡ © d d R p ¡ q L1 and it is C8, so B F N 2B R d 1 g dσ P C8 up to the boundary as well. The remaining j j d ¡ © B pd¡1q pd¡1q pd¡1qp ¤q partial derivative of F satisfies dF Rd 2 Rd g dσ . Note that the kernel of Rd 2 , d when acting on functions defined on BR , coincides with the Poisson kernel

2xd d wd|x|

8 d d and, therefore, it maps L X CpBR q functions to continuous functions in R , and it satisﬁes the pd¡1q p 1 q p 1q P 8 X pB d q 1 P d pointwise identity limxdÑ0 Rd g x , xd g x for every g L C R and every x R B p 1 q pd¡1q p 1q (see [Fol95, Theorem 2.44]). In particular, limxdÑ0 dF x , xd Rd g x . P 8p dq d Consider φ Cc R . Using the Green identities, since F is harmonic in R , we have » » » » » » ¤ ¡ ¤ B ¡ pd¡1q ∇φ ∇F dm ∇φ ∇Ng dm φ dF dσ φ Rd g dσ φg dm φg, d d B d B d d d R R R R R R proving (5.8). To prove the pointwise bounds for ∇h, recall that ¡ © p q pd¡1q pd¡1q p q ¡ pd¡1q p q ∇h x R 2 Rd g dσ x R g x .

P d z p q 1 Given x R B1, since supp g 4 B1, §» § § gpzqpx ¡ zq § }g} |Rpd¡1qgpxq| c § dz§ À 1 . (5.11) § | ¡ |d § | |d¡1 B1 x z x P p q 1 R PB d X p | |{ q On the other hand, consider z supp g 4 B1 and x B1. Then, for y R B 0, x 2 d one has |x ¡ y| |x|, for y PBR X Bp0, 2|x|qzBp0, |x|{2q one has |y ¡ z| |x| and otherwise |y ¡ x| |y ¡ z| |y|. Thus, § § § ¡ © § §» ¢» § § p ¡ q § § gpzqz dz px ¡ yqdσpyq§ §Rpd¡1q 2 R d 1 g dσ pxq§ c § d § d § | ¡ |d | ¡ |d § B d B y z x y » R 1 ¢» | p q| p q À g z zd dz dσ y | ¡ |d | |d¡1 B d XBp0,|x|{2q B y z x »R 1 ¢» | p q| p q g z zd dz dσ y | |d | ¡ |d¡1 B d XBp0,2|x|qzBp0,|x|{2q B x x y » R ¢» 1 p q | p q| dσ y g z zd dz 2d¡1 . (5.12) B d z p | |q |y| R B 0,2 x B1 ³ }g} p q 1 dσ y 1 The ﬁrst term can be bounded by C | |d¡1 because B d | ¡ |d C . The second can be § § x R y z zd r }g} § | | § r }g} 1 1 § xd § 1 1 bounded by C |x|d log |x| using polar coordinates and the last one can be bounded by C |x|d trivially. Thus, § § § ¡ © § } } } } § § } } § pd¡1q pd¡1q § g 1 r1 g 1 § xd § r1 g 1 §R 2 R g dσ pxq§ À §log § , d |x|d¡1 |x|d |x| |x|d 130 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

proving (5.10) since r1 ¤ |x|. To prove the pointwise bounds for h, recall that ¡ © p q pd¡1q p q ¡ p q h x N 2 Rd g dσ x Ng x .

When d ¡ 2 we use the same method as in (5.11) and (5.12) using Newton’s potential instead of the vectorial pd ¡ 1q-dimensional Riesz transform to get }g} r x }g} r }g} |hpxq| À 1 1 d 1 1 1 . |x|d¡2 |x|d |x|d¡1 When d 2 the Newton potential is logarithmic, but the spirit is the same. In this case, arguing as before, |x| |x| log |x| x log x |hpxq| À log |x|}g} r }g} 2 2 . 1 1 1 |x|2

Proposition 5.22. Let 1 p 8. Given a window Q centered at the origin of a δ-special Lipschitz domain Ω (see Deﬁnition 1.4) with a Whitney covering W and given f P W 1,ppΩq, deﬁne the Whitney averaging function ¸ Afpxq : χ pxq fpyqdy. (5.13) Q QPW 3Q

If µ is a ﬁnite positive Borel measure supported on δ0Q with

d¡p µpShvpQqq ¤ C`pQq for every Whitney cube Q Q, (5.14) and A : W 1,ppΩq Ñ Lppµq is bounded, then µ is a p-Carleson measure. p R q Proof. We will argue by duality. Let us assume that the window Q Q 0, 2 is of side-length R and centered at the origin, which belongs to BΩ. Note that the boundedness of A is equivalent to the boundedness of its dual operator

1 A¦ : Lp pµq Ñ pW 1,ppΩqq¦. We also assume that µ 0 in a neighborhood of BΩ. One can prove the general case by means of truncation and taking limits since the constants of the Carleson condition (5.2) and the the norm of the averaging operator will not get worse by this procedure. Fix a cube P . Analogously to [ARS02, Theorem 3], we apply the boundedness of A¦ to the test function g χShvpP q to get

1 ¦ p p1 }A g} ¦ À }g} 1 µpSh pP qq. pW 1,ppΩqq Lp pµq v Thus, it is enough to prove that ¸ 1 p¡d ¦ p1 p p qqp p q p¡1 À} } p p qq µ Shv Q ` Q A g pW 1,ppΩqq¦ µ Shv P . (5.15) QPSHvpP q Given any f P W 1,ppΩq, using (5.13) and Fubini’s Theorem, » » £ » ¸ χ xA¦g, fy g Af dµ f 3Q g dµ dm, mp3Qq Ω QPW Q 5.5. SMOOTHNESS ONE: A NECESSARY CONDITION 131 where we wrote x¤, ¤y for the duality pairing. Consider » ¸ p q ¸ p q rp q χ3Q x p q µ Q g x : p q g dµ χ3Q x p q. (5.16) m 3Q Q m 3Q QPW QPSHvpP q

Then, » xA¦g, fy f gr dm. Ω Note that gr is in L8 with norm depending on the distance from the support of µ to BΩ by (5.14), but the norm of gr in L1 is }r} p p qq g L1 µ Shv P .

Qω ÝÑω

ShvpQq

ShωpQq

d Figure 5.3: We divide R in pre-images of Whitney cubes.

d Ñ d p 1 q p 1 p 1qq Consider also the change of variables ω : R R , ω x , xd x , xd A x where A is the Lipschitz function whose graph coincides with BΩ, and to every Whitney cube Q assign the set ¡1p q p q ¡1p p qq P d Qω ω Q and its shadow Shω Q ω Shv Q (see Figure 5.3). Then, for every x R we deﬁne g0pxq : grpωpxqq|detpDwpxqq|, (5.17) p p¤qq } } where det Dw stands for the determinant of the Jacobian matrix. Note that still g0 L1 }r} p p qq p d q g L1 µ Shv P , and, since ω R Ω, » » ¦ xA g, fy f gr dm f ¥ ω ¤ g0 dm. (5.18) d Ω R The key of the proof is using ¡ © p q pd¡1q p q ¡ p q h x : N 2 Rd g0 dσ x Ng0 x , (5.19)

1,1p d q which is the Wloc R solution of the Neumann problem » » ¤ P 8p d q ∇φ ∇h dm φ g0 dm for every φ Cc R , (5.20) d d R R provided by Lemma 5.20. We divide the proof in four claims. 132 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

P 8p d q Claim 5.23. If φ Cc R , then » xA¦g, φ ¥ ω¡1y ∇φ ¤ ∇h dm. d R Proof. Since ω is bilipschitz, the Sobolev W 1,p norms before and after the change of variables ω P 8p d q ¥ ¡1 P 1,pp q are equivalent (see [Zie89, Theorem 2.2.2]). In particular, for φ Cc R , φ ω W Ω and we can use (5.18) and (5.20).

Now we look for bounds for }B h} p1 . The H¨olderinequality together with a density d L pShω pP qq argument would give us the bound

¦ } } À} } 1 p p qq A g pW 1,ppΩqq¦ ∇h Lp µ Shv P , with constants depending on the window size R, but we shall need a kind of converse. Claim 5.24. One has

¦ }B h} p1 À}A g} 1,p ¦ µpSh pP qq. d L pShω pP qq pW pΩqq v

¡1 p p1 Proof. Take a ball B1 containing ω p4Qq. The duality between L and L gives us the bound §» § § § § § }B h} p1 À sup φ B h dm . d L pShω pP qq § d § P 8p X d q φ Cc B1 R } } ¤ φ p 1 To use the full potential of the Fourier transform, consider hs to be the symmetric extension of s 1 1 s h with respect to the hyperplane xd 0, h px , xdq hpx , |xd|q. It is immediate that h has s s s 1 1 global weak derivatives Bjh pBjhq for 1 ¤ j ¤ d ¡ 1 and Bdh px , xdq ¡Bdhpx , ¡xdq for every xd 0. Thus, §» § § § s }B } 1 À § B § dh Lp pSh pP qq sup § φ dh dm§ . (5.21) ω P 8p q φ Cc B1 } } ¤ φ p 1

P 8p q rp q p q ¡ p ¡ q Given φ Cc B1 , consider the function φ x φ x φ x 2 r1 ed , where ed denotes the 1 p q unit vector in the d-th direction and r1 2 diam B1 , and take » xd r 1 Iφpxq φpx , tqdt. (5.22) ¡8 P 8p q B }B }p } }p Then, we have Iφ Cc 3B1 with dIφ φ in the support of φ and dIφ p 2 φ p. Thus, » » s s s φ Bdh dm xBdIφ, Bdh y ¡ BdIφ Bdh dm, (5.23) 3B1zB1 where we use the brackets for the dual pairing of test functions and distributions. Using H¨older’s inequality and the estimate (5.10) one can see that the error term in (5.23) is bounded by » s s |B I B h |dm ¤ }B I } }B h } p1 ¤ C}φ} p µpSh pP qq. (5.24) d φ d d φ p d L p3B1zB1q L v 3B1zB1

Note that C only depends on r1, which can be expressed as a function of the Lipschitz constant δ0 and the window side-length R. 5.5. SMOOTHNESS ONE: A NECESSARY CONDITION 133

It is well known that the vectorial d-dimensional Riesz transform, » ¡ pdq p q 1 x y p q P R f x p.v. d 1 f y dy for every f S 2w d |x ¡ y| d 1 R is, in fact, a Calder´on-Zygmund operator and, thus, it can be extended to a bounded operator in p pdq pdq pdq ¥ pdq L . Writing Ri for the i-th component of the transform and Rij : Ri Rj for the double B pdq pdq Riesz transform in the i-th and j-th directions, one has iiIφ Rii ∆Iφ ∆Rii Iφ by a simple pdq B Fourier argument (see [Gra08, Section 4.1.4]). Thus, writing fφ Rdd Iφ, we have ∆fφ ddIφ, so

s s s xBdIφ, Bdh y ¡xBddIφ, h y ¡x∆fφ, h y. (5.25)

8p p qq ¤ ¤ Let fr ϕrfφ with ϕr a bump function in Cc B2r 0 such that χBr p0q ϕr χB2r p0q, 2 |∇ϕr| À 1{r and |∆ϕr| À 1{r . We claim that

s s s ¡x∆fφ, h y ¡ lim x∆fr, h y lim x∇fr, ∇h y, (5.26) rÑ8 rÑ8

The advantage of fr is that it is compactly supported, while only the Laplacian of fφ is compactly B P 8p dq supported. Recall that ∆fφ ddIφ Cc R so, by the hypoellipticity of the Laplacian operator, P 8p dq fφ C R itself (see [Fol95, Corollary (2.20)]). Thus, the second equality in (5.26) comes from the deﬁnition of distributional derivative. It remains to prove

s rÑ8 x∆fr ¡ ∆fφ, h y ÝÝÝÑ 0. (5.27)

Since ∆fφ is compactly supported, taking r big enough we can assume that

∆rpϕr ¡ 1qfφs p∆ϕrqfφ 2∇ϕr ¤ ∇fφ, so » ¢ | || s| | || s| |x ¡ sy| À fφ h ∇fφ h ∆fr ∆fφ, h 2 dm. B2r p0qzBr p0q r r If the dimension is d 2 and |x| large enough, equation (5.9) reads as

| p q| À | | || } } h x log x g0 1. ¡ | p q| À 1 } } If d 2, we can use the same bound, since h x |x|d¡2 g0 1. Using H¨older’sinequality in (5.22) } } ¤ } } 8 B B pdq pdqB we have that Iφ q C φ q for any given 1 q . Now, jfφ jRdd Iφ Rdj dIφ, so using the boundedness of the d-dimensional Riesz transform in Lq, we get

} } } } } } ¤ p} } }B } q ¤ } } fφ W 1,q fφ Lq ∇fφ Lq Cq Iφ q dIφ q Cq φ q. (5.28) Thus, for r large enough and choosing q d we get » ¢ » | || s| | || s| p q fφ h ∇fφ h À log r p| | | |q } } 2 dm fφ ∇fφ dm g0 1 B2r p0qzBr p0q r r r B2r p0qzBr p0q logprq d rÑ8 À q1 } } } } ÝÝÝÑ r fφ 1,q p dq g0 0, r W R 1 proving (5.27). 134 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

sp 1 q p 1 ¡ q Back to (5.26), we can use fr x , xd : fr x , xd by a change of variables to obtain » » » ¤ s ¤ s ¤ x ¦ p sq ¥ ¡1y ∇fr ∇h dm ∇fr ∇h dm ∇fr ∇h dm A g, fr fr ω (5.29) d d R R by means of Claim 5.23. Summing up, by (5.23), (5.24), (5.25), (5.26) and (5.29) and letting r tend to inﬁnity, we get §» § § § § § § B s § À §x ¦ p sq ¥ ¡1y§ } } p p qq § φ dh dm§ A g, fφ fφ ω φ Lp µ Shv P . (5.30)

Summing up, by (5.21), (5.30) and the estimate (5.28) with q p, we have got that § § ¦ ¡1 }B } 1 À §x ¥ y§ p p qq dh Lp pSh pP qq sup A g, f ω µ Shv P . ω } } ¤ f 1,pp dq 1 W R ¥ ¡1 } } On the other hand, by [Zie89, Theorem 2.2.2] f ω 1,p f 1,pp d q for every f, so W pΩq W R we have

¦ ¦ }B } 1 À |x y| p p qq } } p p qq dh Lp pSh pP qq sup A g, f µ Shv P A g pW 1,ppΩqq¦ µ Shv P , ω } } ¤ f W 1,ppΩq 1 that is Claim 5.24. Next we establish the relation between (5.15) and Claim 5.24. Claim 5.25. One has ¸ 1 p¡d p1 p p qqp p q p¡1 À }B } µ Shv Q ` Q dh p1 (5.31) L pShω pP qq QPSHvpP q £ 1 » » p ¸ z ¡ x d d grpωpzqqdz dx : 1 2 . | ¡ |d Qω tz:zd¡xdu x z QPSHvpP q

Proof. Note that in (5.19) we have deﬁned h in such a way that ¡ © B p q pd¡1q pd¡1q p q ¡ pd¡1q p q dh x Rd 2 Rd g0 dσ x Rd g0 x » £ » ¡ p q ¡ 1 2xdzd dσ y xd zd p q d d d g0 z dz. w d w B d |y ¡ z| |x ¡ y| |x ¡ z| d R d R ¡ © P d pd¡1q pd¡1qp¤q ¡ pd¡1qp¤q Given x, z R , consider the kernel of Rd 2 Rd dσ Rd , » p q ¡ p q 2xdzd dσ y xd zd G x, z d d d , w B d |y ¡ z| |x ¡ y| |x ¡ z| d R so that » ¡1 Bdhpxq Gpx, zqg0pzq dz. (5.32) w d d R We have the trivial bound zd ¡ xd Gpx, zq χt ¡ upzq ¥ 0, (5.33) |x ¡ z|d zd xd 5.5. SMOOTHNESS ONE: A NECESSARY CONDITION 135

but given any Whitney cube Q P SHvpP q, if x P Qω and z P ShωpQq we can improve the estimate. In this case, » » p q p q dσ y Á dσ y 1 d d B d X p q |y ¡ z| B d X ¡1p p p qqq |y ¡ z| z R Shω Q R ω Shv ω z d and, thus, » zd ¡ xd 2xdzd dσpyq Gpx, zq χt ¡ upzq ¥ d zd xd d d |x ¡ z| wd B d |y ¡ z| |x ¡ y| »R p q p q p q Á ` Q zd dσ y Á ` Q d d d . (5.34) `pQq B d X p q |y ¡ z| `pQq R Shω Q

d By the Lipschitz character of Ω we know that | det Dωpzq| 1 for every z P R . Thus, by (5.16) and (5.17), given Q P SHvpP q we have ¸ » » µpShvpQqq µpSq À grpwqdw g0pzqdz. ShvpQq Shω pQq SPSHvpQq

1 Then, raising to the power p , averaging with respect to x P Qω, we get that

£ 1 » p 1 1 1 1 z ¡ x µpSh pQqqp `pQqp ¡dp À |B hpxq|p dx d d grpωpzqqdz dx. v d | ¡ |d Q Q tz:z ¡x u x z ¤ ω ω d d £ 1 » » p 1 ¡ ¥}B }p zd xd rp p qq p q¡d dh p1 g ω z dz dx ` Q L pQω q | ¡ |d Qω tz:zd¡xdu x z

1 p ¡ 1q p ¡ d P p q and, since p d 1 p p¡1 p¡1 , summing with respect to Q SHv P we get the estimate (5.31), that is, Claim 5.25.

Finally, we bound the negative contribution of the pd¡1q-dimensional Riesz transform in (5.31), that is we bound 2 . Claim 5.26. One has

£ 1 » » p ¸ z ¡ x 2 d d grpωpzqqdz dx À µpSh pP qq. (5.35) | ¡ |d v Qω tz:zd¡xdu x z QPSHvpP q 136 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

P d P Proof. Consider x, z R with xd zd and two Whitney cubes Q and S such that x Qω and z P ω¡1p3Sqzω¡1p3Qq. Then

z ¡ x distpωpzq, BΩq `pSq d d À . |x ¡ z|d DpS, Qqd DpS, Qqd

On the other hand, when 3S X 3Q H, » |z ¡ x | d d dz À `pQq `pSq. | ¡ |d ω¡1p3Qq x z

° p q From the deﬁnition of gr in (5.16) it follows that grpωpzqq χ pωpzqq µ L . Bearing LPSHvpP q 3L mp3Lq all these considerations in mind, one gets

¤ 1 p ¸ ¸ µpSq`pSq 2 À `pQqd ¥ . DpS, Qqd QPSHvpP q SPSHvpP q

Consider a fixed ¡ 0. One can apply first the Hölderinequality and then (5.14) to get

1 ¤ ¤ p 1 p ¸ ¸ µpSq`pSq1¡p ¸ µpSq`pSq1 p 2 À `pQqd ¥ ¥ DpS, Qqd DpS, Qqd QPSHvpP q SPSHvpP q SPSHvpP q 1 ¤ ¤ p 1 p ¸ ¸ µpSq`pSq1¡p ¸ `pSqd¡p 1 p À `pQqd ¥ ¥ . DpS, Qqd DpS, Qqd QPSHvpP q SPSHvpP q SPSHvpP q

By Lemma 5.10, the last sum is bounded by C`pQq¡p 1 p with C depending on as long as ¡ ¡ ¡ ¡ p¡2 p¡1 d d p 1 p d 1, that is, when p p . Thus,

1 1 1 1 ¸ ¸ µpSq`pSq1¡p `pQqd¡p p {p p 2 À DpS, Qqd QPSHvpP q SPSHvpP q ¸ ¸ d¡1 p1 1 `pQq µpSq`pSq1¡p . DpS, Qqd SPSHvpP q QPSHvpP q

1 Again by Lemma 5.10, the last sum does not exceed C`pSq¡1 p with C depending on as long ¡ ¡ 1 ¡ ¡ 1 p¡1 as d d 1 p d 1, that is when 0 p1 p . Summing up, we need " * p ¡ 2 p ¡ 1 max , 0 . p p

Such a choice of is possible for every p ¡ 1. Thus, ¸ 2 À µpSq µpShvpP qq.

SPSHvpP q 5.5. SMOOTHNESS ONE: A NECESSARY CONDITION 137

Now we can ﬁnish the proof of Proposition 5.22. The ﬁrst term in the right-hand side of (5.31) is bounded due to Claim 5.24 by

1 1 }B }p À} ¦ }p p p qqp1 1 dh p1 A g p 1,pp qq¦ µ Shv P . (5.36) L pShω pP qq W Ω

p1 p1¡1 Being µ a ﬁnite measure, µpShvpP qq ¤ µpShvpP qqµpδ0Qq and, thus, the bounds (5.35) and (5.36) combined with (5.31) prove (5.15), leading to ¸ 1 p¡d p ¡ µpShvpQqq `pQq p 1 À µpShvpP qq.

QPSHvpP q

For the sake of clarity, we restate Theorem 5.3 in terms of Carleson measures.

Theorem 5.27. Given an admissible convolution Calder´on-Zygmundoperator of order 1, a Lips- chitz domain Ω and 1 p 8, the following statements are equivalent:

1. Given any window Q with a properly oriented Whitney covering, and given any Whitney cube P δ0Q, one has

£ 1 ¸ » p » p¡d p ¡ p |∇TΩpχΩq| dm `pQq p 1 ¤ C |∇TΩpχΩq| dm. ShvpQq ShvpP q QPSHvpP q

1,p 2. TΩ is a bounded operator on W pΩq. Proof. The implication 1 ùñ 2 is Theorem 5.1. To prove that 2 ùñ 1 we will use the previous proposition. Let us assume that we have a properly oriented Whitney covering W associated to an R-window Q of a Lipschitz domain Ω, p R q where we assume that the window Q Q 0, 2 is of side-length R and centered at the origin. 1,p Note that since TΩ is bounded on W pΩq then, by the Key Lemma 2.2, ¸ » | |p | p qp q|p À} } P 1,pp q f3Q ∇TΩ χΩ x dx f W 1,ppΩq for f W Ω . (5.37) QPW Q

¡ Consider the Lipschitz function A : Rd 1 Ñ R whose graph coincides with the boundary of Ω in Q. We say that Ωr is the special Lipschitz domain deﬁned by the graph of A that coincides with Ω in the window Q. One can consider a Whitney covering W associated to Ωr such that it coincides with W in δ0Q. Consider the averaging operator ¸ 1,p r Afpxq : χQpxqf3Q for f P W pΩq. QPW

p q | p qp q|p p q ¤ ¤ Writing dµ x : ∇T χΩ x χδ0Q x dx and taking a bump function χδ0Q ψQ χQ, we have that every f P W 1,ppΩrq satisﬁes that ¸ ¸ }Af}p µpQqf p µpQqpfψ qp À}fψ }p À}f}p . Lppµq 3Q Q 3Q Q W 1,ppΩq W 1,ppΩr q P QPW Q W by (5.37) and the Leibniz formula. That is, A : W 1,ppΩrq Ñ Lppµq is bounded. 138 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

d¡p In order to apply Proposition 5.22, we only need to show that µpShvpQqq ¤ C`pQq for every Whitney cube Q Q, which in particular implies that µ is ﬁnite. Fix a Whitney cube Q Q and ¤ ¤ | | À 1 consider a bump function ϕQ such that χShvp2Qq ϕQ χShvp4Qq with ∇ϕQ `pQq . Then, » p µpShvpQqq |∇T χΩpxq| dx p qX »Shv Q δ0Q » p p ¤ |∇T pχΩ ¡ ϕQqpxq| dx |∇T ϕQpxq| dx. (5.38) ShvpQq Ω P p q p p ¡ qq ¡ 1 p q With respect to the ﬁrst term, notice that given x Shv Q , dist x, supp χΩ ϕQ 2 ` Q so Lemma 1.31 together with (1.41) allows us to write » 1 1 |∇T pχ ¡ ϕ qpxq| À dy À . Ω Q | ¡ |d 1 p q ΩzShvp2Qq y x ` Q

d Being Ω a Lipschitz domain, mpShvpQqq `pQq , so » p d¡p |∇T pχΩ ¡ ϕQqpxq| dx À `pQq . ShvpQq The second term in the right-hand side of (5.38) is bounded by hypothesis by a constant times } }p ϕQ W 1,ppΩq, and } }p } }p } }p À p qd p qd¡p À p p q p qd¡p ϕQ W 1,ppΩq ϕQ LppΩq ∇ϕQ LppΩq ` Q ` Q R 1 ` Q , where R is the side-length of the R-window Q, proving that µ satisﬁes (5.14).

5.6 On the complex plane

Remark 5.28. The article of Arcozzi, Rochberg and Sawyer [ARS02] has been the cornerstone in our quest for necessary conditions related to Carleson measures. In fact their article provides a quick shortcut for the proof of Theorem 5.27 (avoiding Proposition 5.22) for simply connected domains of class C1 in the complex plane, and we believe it is worth to give a hint of the reasoning. Sketch of the proof. In the case of the unit disk, we found in the Key Lemma 2.2 that if T is an admissible convolution Calder´on-Zygmund operator of order 1 bounded on W 1,ppDq, then § § » ¸ § §p § § | p q|p p q À } }p § f dm§ ∇T χ z dm z f 1,pp q (5.39) D W D QPW 3Q Q

P 1,pp q p q | p q|p p q p q p ¡ | |2q2¡p for all f W D . If one considers dµ z ∇T χD z dm z and ρ z 1 z , then, when f is in the Besov space of analytic functions on the unit disk Bppρq, we have that » p q } }p | p q|p | 1p q|pp ¡ | |2qp p q dm z } }p f p q : f 0 f z 1 z ρ z f 1,pp q. Bp ρ p1 ¡ |z|2q2 W D D Using the mean value property (and (5.14) for the error terms), one can see that if T is bounded, then for every holomorphic function f the bound in (5.39) is equivalent to » |fpzq|p|∇T χ pzq|pdmpzq À }f}p , D Bppρq D 5.6. ON THE COMPLEX PLANE 139

i.e., }f} p À}f} . Following the notation in [ARS02], the measure µ is a Carleson measure L pµq Bppρq for pBppρq, pq, establishing Theorem 5.27 for the unit disk by means of Theorem 1 in that article. For Ω C Lipschitz and f analytic in Ω, we also have that, if T : W 1,ppΩq Ñ W 1,ppΩq is bounded, then » | p q|p| p q|p p q À } }p f z ∇T χΩ z dm z f W 1,ppΩq. Ω

If Ω is simply connected, considering a Riemann mapping F : D Ñ Ω, and using it as a change of variables, one can rewrite the previous inequality as » » |f ¥ F pωq|pµpF pωqq|F 1pωq|2dmpωq À |fpF p0qq|p |pf ¥ F q1pωq|p|F 1pωq|2¡pdmpωq. D D Writing dµ˜pωq µpF pωqq|F 1pωq|2dmpωq, and ρpωq |F 1pωqp1 ¡ |ω|2q|2¡p, one has that given any g analytic on D, }g} p À }g} . L pµ˜q Bppρq

So far so good, we have seen thatµ ˜ is a Carleson measure for pBppρq, pq, but we only can use [ARS02, Theorem 1] if two conditions on ρ are satisﬁed. The ﬁrst condition is that the weight ρ is “almost constant” in Whitney squares, that is

for x1, x2 P Q P W ùñ ρpx1q ρpx2q, and this is a consequence of Koebe distortion theorem, which asserts that for every w P D we have |F 1pωq|p1 ¡ |ω|2q distpF pωq, BΩq

(see [AIM09, Theorems 2.10.6 and 2.10.8], for instance). The second condition is the Bekoll´e- Bonami condition, which is ¢ » » p¡1 ¨1¡p1 p1 ¡ |z|2qp¡2ρpzqdmpzq p1 ¡ |z|2qp¡2ρpzq dmpzq À mpQqp. Q Q

If the domain Ω is Lipschitz with small constant depending on p (in particular if it is C1), then this condition is satisﬁed (see [B´ek86,Theorem 2.1]).

Conclusions

In this dissertation we have found answers to different problems related to the boundedness of convolution Calderón-Zygmund operators on Sobolev (and Triebel-Lizorkin) spaces on domains. It is worthy to finish pointing out possible further steps in this research. Chapters 2 and 5: Putting together Theorems 2.1 and 2.8, we have obtained the following T pP q-theorem:

Theorem. Let Ω Rd be a bounded uniform domain, T an admissible convolution Calderón- Zygmund operator of order s P N or 0 s 1, 8 ¡ d rss¡1 d p with s p and let P stand for the polynomials of degree smaller than s. Then } } 8 P rss¡1 TΩP W s,ppΩq for every P P if and only if s,p TΩ is bounded on W pΩq. On the other hand, putting together Theo- s s rems 5.1 and 5.2 (writing ∇ for ∇2) we have shown the following theorem (see Figure 5.4): Theorem. Let Ω be a bounded Lipschitz domain, let 1 p ¤ d and let T be an admissible convolution Calderón-Zygmundoperator of or- P ¡ d ¡ d p q der s N or 0 s 1 (with s p 2 in the Figure 5.4: Indices for which the T P -theorem s p fractional case). If the measure |∇ TΩP pxq| dx (in green) and the Carleson theorem (in red) are is a p-Carleson measure for every polynomial P valid in W s,ppΩq for uniform domains in R3. of degree smaller than s, then TΩ is a bounded operator on W s,ppΩq. I expect that, combining the techniques of the integer orders of smoothness and the fractional ones, the first theorem above can be extended to the green sawtooth region (see Figure 5.5a) and the second theorem above to the red one, using £» 1 | n¡1 p q ¡ n¡1 p q|q q s p q ∇ f x ∇ f y ∇qf x sq d dy . p qX |x ¡ y| Bρ1δpxq x Ω ¡ d ¡ d Moreover, I believe that the restriction s p 2 is rather unnatural. I expect that using other expressions for the gradient in terms of means on balls, for instance, this restriction can be avoided (see Figure 5.5b).

141 142 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

In fact, there is hope that, using higher order diﬀerences, a T pP q-theorem can be obtained for s p q all the supercritical range, and for Fp,q Ω in general. Even more challenging is the question of whether Theorem 5.3 has a counterpart for other orders of smoothness or not, that is, if there is a necessary Carleson condition when s 1.

(a) Natural extension of our techniques. (b) Other expressions for the gradient.

Figure 5.5: Conjectures on the indices where our results are valid for higher orders of fractional smoothness, in R3.

8 1 Remark 5.29. For 1 p, q and 0 s p , we have that the multiplication by the charac- s p dq teristic function of a half plane is bounded in Fp,q R . This implies that for domains Ω whose boundary consists of a ﬁnite number of polygonal boundaries, the pointwise multiplication with χΩ is also bounded in this space and, using characterizations by diﬀerences, this property can be seen to be stable under bi-Lipschitz changes of coordinates. Summing up, given a Lipschitz domain Ω P s p dq and a function f Fp,q R , we have that

}χΩ f} s p dq À}f} s p dq. Fp,q R Fp,q R

s s p q Ñ s p dq Moreover, if T is an operator bounded in Fp,q, using the extension E : Fp,q Ω Fp,q R (see P s p q [Ryc99], for instance), for every f Fp,q Ω we have that

}TΩf} s p q }T pχΩ Efq} s p q ¤ }T pχΩ Efq} s ¤ }T } s Ñ s }χΩ Ef} s À}Ef} s Fp,q Ω Fp,q Ω Fp,q Fp,q Fp,q Fp,q Fp,q

À}f} s p q. Fp,q Ω

That is, given a p, q-admissible convolution Calder´on-Zygmundoperator T and a Lipschitz domain s p q 1 Ω we have that TΩ is bounded in Fp,q Ω for any 0 s p .

Chapter 3: Characteristic functions of planar domains s,p In this chapter we have studied when BχΩ P W pΩq. By the previous remark, when sp 1 s,p and Ω is a Lipschitz domain BχΩ P W pΩq regardless of any other consideration in the smoothness of the boundary (see Figure 5.6). 5.6. ON THE COMPLEX PLANE 143

Conjecture 5.30. Let 0 s 8, let 1 p 8 and let Ω be a bounded pδ, Rq-Crss¡1,1 domain s 1¡1{p P s,pp q ¡ 1 with parameterizations in Bp,p . Then, we have that BχΩ W Ω and, if s p , then } s }p À} }p ∇ BχΩ LppΩq N s¡1{p . Bp,p pBΩq P 1 Theorem 3.27, proves the case s N, and the result in [CT12] proves the case p s 1. I expect the cases 1 s 8 with s R N to be proven without much eﬀort using the techniques of both. The case sp 1 may also be of some interest. There is no need to say that further improvements in the T pP q-theorems discussed above could help to deduce if, given a domain Ω, s,p the truncated Beurling transform BΩ is bounded on W pΩq. I conclude the discussion on the results of this chapter with two open questions. Remark 5.31. In [Tol13] it is seen that this result is sharp for s 1. It remains to see if this is 1 8 the case for p s 1 and for 1 s . At some point in my PhD studies I got interested in ﬁnding a counterpart of this theorem in higher dimensions. The main obstruction is Proposition 3.17, which depends strongly on complex analysis. It remains to see if this technique can be bypassed using real analysis tools for any operator in higher dimensions.

Figure 5.6: Indices considered on Conjecture 5.30. For s 1 the result is sharp (blue segments). s,p For sp 1 (in yellow), every Lipschitz domain satisﬁes that BχΩ P W pΩq as a consequence of Remark 5.29.

Chapter 4: An application to quasiconformal mappings In relation to quasiconformal mappings, I expect the following conjecture to be true.

Conjecture 5.32. Let n P N, and let 0 tsu ¤ 1 and s n ¡ 1 tsu. Let Ω be a bounded s¡1{p Lipschitz domain with outward unit normal vector N in Bp,p pBΩq for some p 8 with tsu p ¡ 2 P s,pp q } } p q and let µ W Ω with µ L8 1 and supp µ Ω. Then, the principal solution f to the Beltrami equation is in the Sobolev space W s 1,ppΩq (see Figure 5.7).

This conjecture for s P N is exactly Theorem 4.1. I expect this result to be proven straight ahead using the techniques exposed in this chapter when 0 s 1 (Conjecture 5.30 holds by [CT12]). The main diﬃculties will be to prove the compactness of rµ, Bs and Rm. For higher fractional orders of smoothness I expect some complications but the core of the proof should remain the same. However, to study this case, we will need a quantitative version of Conjecture 5.30 to be proven for the corresponding indices. 144 CHAPTER 5. CARLESON MEASURES ON LIPSCHITZ DOMAINS

Figure 5.7: Conjectures on the indices where the application to quasiconformal maps holds.

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