PhD Dissertation Singular Integral Operators on Sobolev Spaces on Domains and Quasiconformal Mappings Mart´ı Prats Soler Mem`oriapresentada per a obtenir el grau de Doctor en Matem`atiquesper la Universitat Aut`onomade Barcelona 2015 Advisor: Dr. Xavier Tolsa Domenech` Facultat de Ciencies` Departament de Matematiques` Plus Ultra - Versi´odi`adica All`adell`ade 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 s´ola darrera, no, no s´om´esque 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 ´esinfinit, pertot acaba i comen¸ca, i en¸c`a,enll`a,amunt i avall, la immensitat ´esoberta, i a on tu veus l'hiperpl`a eixams de cubs formiguegen. Dels camins de l’infinit s´onlos 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 m`asterfins ara ha estat un far en el cam´ı,donant-me la corda necess`ariaper tirar endavant i mostrant-me una manera intuitiva i geom`etricad'entendre l’an`alisimatem`atica.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’an`alisiper aquest ambient tant agradable que han aconseguit crear. I, evidenment, als meu germans i cosins matem`atics,companys de despatx i de caf`esdurant tots aquests anys, amb especial menci´oal Daniel i l'Antonio per les vides paral¤leles i en Vasilis i l'Albert pels consells i les visions de futur. Seuraavaksi, haluaisin l¨ahett¨a¨aparhaimmat kiitokset Eero Saksmanille, joka oli Helsingin vierai-luni ajan yhteisty¨okumppanini, opettajani ja yst¨av¨ani.Holistisella n¨ak¨okulmallaanmatem- atiikkaan, h¨anmuistutti minua siit¨a,ett¨aparhaiten eteenp¨ainp¨a¨asee,kun on hieman sivussa oikealta polulta. Marraskuiseen Muumilaaksoon v¨ari¨atoivat Riikka, Mikko, Timo, Matteo, Miren, Teemu ja Marco sek¨amahtavat T¨o¨ol¨ontornit ja tietysti Helsingin yliopiston Matematiikan ja tilastotieteen laitos, jonka k¨ayt¨av¨anjokaisessa ovessa on tunnetun kirjan kirjoittajan nimi. My acknowledgement also to Dr. Antti V¨ah¨akangas, 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-Syrj¨anenfor 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 tambi´enal IMUS, por organizar un seminario extraordinario en Sevilla y M´alaga,con estupendas partidas de ping-pong y esas ca~nasal lado del campo del mejor equipo de f´utbol del sur de la pen´ınsula(a quien espero que esa tesis sea de ayuda), extensivo a la organizaci´on del II Congreso de J´ovenes 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 tamb´eal professorat de la Universitat de Barcelona, on vaig obtenir la llicen- ciatura, amb especial menci´oa 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 Call´us, Moi`a,Sant Feliu del Rac´oi 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 matem`atica,aix´ıcom en Gran´e,i el Jose Luis Diaz per la seva persever`anciai per no donar-me per perdut quan portava els cabells massa llargs. Finalment, el m´esimportant del m´on,pares, germanes, nebodets, cosins, tiets i `aviaCarme per aguantar les meves abs`encies,de cos i d’`anima,durant tots aquests anys i, sobretot, l'Aura, que es mereix la meitat o m´esdel t´ıtol. I a la Tieta, l'avi Ramon, la Iaia i el Quim per la seva guia des d'una mica m´esenll`ade la frontera dels dominis coneguts. i ii A Els Navegants, per la ter`apia,l'Ateneu i la CUP per la lluita contra corrent, i, ja que ens posem nost`algics, al guru Kaaraikkudi R. Mani, Sinera, Cantiga, el Virolet, Yarak, l'ARC, Fidelio, secci´od'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 Gr`aciaper 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 Calder´on-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 Calder´on-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 |rjKpxq| ¤ 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 " 0 |w¡z|¡" z w is an admissible convolution Calder´on-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 Calder´on-Zygmund op- erator. 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 .