Boundedness of the Hilbert Transform on Weighted Lorentz Spaces. Elona Agora

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1 Boundedness of the Hilbert Transform on Weighted Lorentz Saces Elona Agora Programa de Doctorat de Matemàtiques Universitat de Barcelona Barcelona, abril 212

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3 Memòria resentada er a asirar al grau de Doctora en Matemàtiques er la Universitat de Barcelona Barcelona, abril 212. Elona Agora María Jesús Carro Rossell i Francisco Javier Soria de Diego, rofessors del Deartament de Matemàtica Alicada i Anàlisi de la Universitat de Barcelona CERTIFIQUEN: Que la resent memòria ha estat realitzada, sota la seva direcció, er Elona Agora i que constitueix la tesi d aquesta er a asirar al grau de Doctora en Matemàtiques. María Jesús Carro Rossell Francisco Javier Soria de Diego

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5 Stouc goneíc mou, sta adérfia mou 'Olga kai Dhm htrh, sthn josha, ston Jorge

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7 Contents Acknowledgements Resum Notations iii v xiii 1 Introduction 1 2 Review on weighted Lorentz saces Weighted Lorentz saces Duality Several classes of weights The Muckenhout A class of weights The B and B, classes of weights The Ariño-Muckenhout B class of weights The B, class The B class The B (u) and B, (u) classes of weights The AB class Necessary conditions for the boundedness of H on Λ u(w) Restricted weak-tye boundedness on intervals Restricted weak-tye boundedness Necessary conditions and duality The case u A Comlete characterization of the boundedness of H on Λ u(w) Necessary conditions involving the A condition Necessity of the B condition Necessity of the weak-tye boundedness of M Sufficient conditions Comlete characterization i

8 Contents ii Geometric conditions Remarks on the Lorentz-Shimogaki and Boyd theorems Further results and alications on L,q (u) saces Non-diagonal roblem Background of the roblem in the non-diagonal case Basic necessary conditions in the non-diagonal case Necessity of the weak-tye boundedness of M Alications on L,q (u) saces Bibliograhy 13

9 Acknowledgements As you set out for Ithaca hoe that your journey is a long one, full of adventure, full of discovery. Laistrygonians and Cyclos, angry Poseidon, do not be afraid of them: you ll never find things like that on your way as long as you kee your thoughts raised high... Hoe that your journey is a long one. May there be many summer mornings when, with what leasure, what joy, you come into harbors seen for the first time... Konstantinos Petrou Kavafis With what leasure, what joy, I can see the harbor now. I can smell the land. And behind me, and inside me, many stories. Stories and eole. Peole that have made me feel in any moment that I am not alone... María Jesús Carro and Javier Soria are the first ersons, to whom I would like to exress my deeest gratitude. Both of them have been always close to me, suorting, advicing, encouraging, motivating, collaborating with me. Under their guidance I have grown. Through them I have learned to think, to overcome the fears... lots of them, and I have gained a further understanding of mathematics. Definitely, without them, without their atience, their ideas, I would not have been here. I am also grateful to the members of GARF (The Real and Functional Analysis Grou) for the oortunities to attend talks, conferences, meet other eole with similar interests from all over the world, and discover so many wonderful things, which has been a great motivation all these years. I also thank the eole of the Deartment of Alied Mathematics and Analysis of the Universitat de Barcelona, for making all these years so comfortable, and making me feel art of this deartment. I also extend my gratitude to the members of the analysis grous of the UB/UAB/CRM for what I have learned from them all these years. I owe my sincere gratitude to Rodolfo Torres and Estela Gavosto for giving me the oortunity to visit the Deartment of Mathematics of the University of Kansas. Secially I would like to thank Andrea and Kabe Moen for making this visit so exciting. iii

10 Acknowledgements iv I would like to thank the members of the Deartment of Mathematics of Karlstad University, and I warmly thank Sorina and Ilie Barza and their family for their great hositality. I also thank Viktor Kolyada and Martin Lind. I would like to mention that morning, when together with Bharti Pridhnani, Jorge Antezana, Salvador Rodríguez and Amadeo Irigoyen we started our FOCA S seminar (Seminario de Análisis Comlejo, Armónico y Funcional) with the idea to share what we know, but also what we do not know... We have discussed mathematics together, we have shared our enthusiasm, our dee assion for mathematics, our hainess, but also our frustrations... They have been a great suort during all these years as collegues and as well as friends. I extend my gratitude to my other friends and collegues, secially to Nadia Clavero, Marta Canadell, Jerry Buckley, Daniel Seco, Pedro Tradacete, Jordi Marzo, Gerard Ascensi, Asli Deniz, Jordi Lluís... who gave me a great suort all these years, each one on his own way. Without them, this journey wouldn t have been that interesting and exciting. I warmly thank Anca and Liviu Marcoci, and Carmen Ortiz for our nice relationshi during their visits in the Universitat de Barcelona. Particularly, I would like to thank Isabel Cerdà and Victor Rotger for making me feel like at home since the very beginning in Barcelona. I esecially thank Ana (Coma) and Laura (Huguet), with whom I have shared all my us and downs... and very downs. I warmly thank Bahoz, Pilar (Sancho), Julianna, Kristynaki, Sofaki, Annoula, Giwrgakis (and of course I would like to mention his arents in Rhodes), Katerinaki (Kwsta), Giwrgakis (Maglaras), Peri, Elena, Bea, Kumar, Itziar, Iker, Joan, Divina, Nadine, Didac, Leila, Jesús, Núria, Elia, Tedi, Mireia, Achilleas, Dhimitris, Pantelis, Prodromos... This journey has been marvelous with them. I thank all of them for their wishes, their advices, their hugs, their smiles, their jokes, for making me forget my roblems, for sending me always ost-cards or ictures from wherever they are, for their atience, for taking me to Ikastola, to CCCB or to the gelateria italiana, or for just being there. Fusiká ja hjela na euqarist hsw touc goneíc mou Pétro kai Qruσánjh gia thn agáh kai storg h touc, kai ou ánw aó ta diká touc óneira ébazan ánta ta diká mac. Euqaristẃ eíshc ta adérfia mou 'Olga kai Dhm htrh gia thn agáh kai thn áeirh uost hrixh touc óla autá ta qrónia kajẃc kai th giagiá mou gia th metadotik h thc qará. Qwríc autoúc den ja hmoun edẃ. Se autoúc kai afierẃnw th douleiá kai ton kóo mou. Eíshc euqaristẃ ton jeío Fẃth. I esecially thank Jorge, for being always here, and to whom, together with my arents, my sister, my brother, and my grandmother (josha) I dedicate all of my efforts. The revision of the text in catalan has been a courtesy of Ana Coma and Divina Huguet and the ictures are a courtesy of Jorge Antezana. It has been a honour for me to obtain the scholarshi of the Foundation Ferran-Sunyer i Balaguer for my visit to the University of Karlstad, Sweden. I also would like to mention that this work has been financially suorted by the Ministry of Education, Culture and Sorts of Sain, recisely by the scholarshi FPU, and then by the State Scholarshi Foundation I.K.Y., of Greece. H olokl hrwsh thc ergaσíac aut hc égine sto laísio thc ulooíhshc tou metatuqiakoú rográmmatoc ou sugqrhmatodot hjhke mésw thc Práxhc <<Prógramma qor hghshc uotrofiẃn I.K.U. me diadikaσía exatomikeuménhc axiológhshc akad. étouc >> aó órouc tou E.P. <<Ekaídeush kai dia bíou májhsh>> tou Eurwaïkoú koinwnikoú tameíou (EKT) kai tou ESPA, tou

11 Resum L objectiu rincial d aquesta tesi és unificar dues teories conegudes i aarentment no relacionades entre elles, que tracten l acotació de l oerador de Hilbert, sobre esais amb esos, definit er Hf(x) = 1 π lim ε + x y >ε f(y) x y dy, quan aquest límit existeix gairebé a tots els unts. Per una banda, tenim l acotació de l oerador H sobre els esais de Lebesgue amb esos i la teoria desenvoluada er Calderón i Zygmund. Per altra banda, hi ha la teoria de l acotació de l oerador H desenvoluada al voltant dels esais invariants er reordenació. El marc natural er unificar aquestes teories consisteix en els esais de Lorentz amb esos Λ u(w) i Λ, u (w), els quals van ser definits er Lorentz a [68] i [67] de la següent manera: on { ( } 1/ Λ u(w) = f M : f Λ u (w) = (fu(t)) w(t)dt) <, (1) { Λ, u (w) = f M : f Λ, u (w) = su f u(t) = inf{s > : u({x : f(x) > s}) t} i W (t) = t> } W 1/ (t)fu(t) <, (2) t w(s)ds. Més concretament, estudiarem l acotació de l oerador H sobre els esais de Lorentz amb esos: H : Λ u(w) Λ u(w), (3) i la seva versió de tius dèbil H : Λ u(w) Λ, u (w). (4) Abans de descriure els nostres resultats, resentem una breu revisió històrica de l oerador H. Aquest oerador va ser introduït er Hilbert a [48] i [49]. Però, no va ser fins el 1924 quan Hardy el va anomenar oerador de Hilbert en honor a les seves contribucions (veure [43], i [44]). v

12 Resum vi L oerador H sorgeix en molts contextos diferents, com l estudi de valors de frontera de les arts imaginàries de funcions analítiques i la convergència de sèries de Fourier. Entre els resultats clàssics, esmentem el teorema de Riesz: H : L L, és acotat, quan 1 < < (veure [85], i [86]). Tot i que l acotació en L 1 no és certa, Kolmogorov va rovar a [58] l estimació de tius dèbil següent: H : L 1 L 1,. (5) Per a més informació en aquests temes veure [4], [94], [36], i [8]. Els resultats més rellevants que van servir er motivar aquest estudi són: (I) Si w = 1, aleshores (3) i (4) corresonen a l acotació i la seva versió dèbil H : L (u) L (u), (6) H : L (u) L, (u), (7) resectivament. Aquestes desigualtats sorgeixen naturalment quan en el teorema de Riesz, la mesura subjacent es canvia er una mesura general u. Aleshores, el roblema és estudiar quines són les condicions sobre u que ermeten que l oerador H sigui acotat a L (u). Aquesta nova aroximació va donar naixement a la teoria de les desigualtats amb esos, la qual juga un aer imortant en l estudi de roblemes de valor de la frontera er l equació de Lalace en dominis Lischitz. Altres alicacions inclouen desigualtats vectorials, extraolació d oeradors, i alicacions a equacions no lineals, de derivades arcials i integrals (veure [36], [41], [56], i [57]). L estudi de (6) i (7) roorciona juntament amb l acotació de l oerador maximal de Hardy-Littlewood en els mateixos esais, la teoria clàssica de esos A. L oerador sublineal M, introduït er Hardy i Littlewood a [45], es defineix er Mf(x) = su x I 1 I I f(y) dy, on el suremum es considera en tots els intervals I de la recta real que contenen x R. Per a més referències veure [38], [36], [41], [4], i [94]. Diem que u A si, er a > 1, tenim: ( ) ( su u(x)dx u (x)dx) 1/( 1) <, (8) I I I I I on el suremum es considera en tots els intervals I de la recta real, i u A 1 si Mu(x) u(x) a.e x R. (9)

13 vii Resum Muckenhout va rovar a [71] que, si 1, la condició A caracteritza l acotació i si > 1 també caracteritza M : L (u) L, (u), M : L (u) L (u). Hunt, Muckenhout i Wheeden van rovar a [54] que, si 1, la condició A caracteritza (7) i si > 1 la mateixa condició caracteritza també (6). Per una rova alternativa d aquests resultats veure [26]. Per exonents < 1 no hi ha ca es u que comleixi (6) i (7). (II) El cas u = 1 correson a l acotació de l oerador H en els esais de Lorentz clàssics, i va ser solucionat er Sawyer a [9]: H : Λ (w) Λ (w). (1) Una caracterització simlificada dels esos els quals l acotació és certa, es resenta en termes de la condició B B introduïda er Neugebauer a [8]. Diem que w B si er a tot r >, i (11) caracteritza l acotació rovat a [5]. La condició w B és r r 1 t ( r ) r w(t) dt w(t)dt, (11) t M : Λ (w) Λ (w), t w(s)ds dt r w(s)ds, (12) er a tot r >. Si > 1 la condició B B caracteritza també la versió de tius dèbil H : Λ (w) Λ, (w), (13) mentre que el cas 1, es caracteritza er la condició B, B. Diem que w B, si, i només si M : Λ (w) Λ, (w) (14) es acotat. Precisament, tenim que: (α) Si > 1, B, = B. (β) Si 1, aleshores w B, si, i només si w és quasi-concava: er a tot < s r <, W (r) r W (s) s. (15)

14 Resum viii (III) Recentment, Carro, Raoso i Soria van estudiar a [2] l anàleg de la relació (3), erò er a l oerador M, en comtes de l oerador H M : Λ u(w) Λ u(w), i la solució és la classe de esos B (u) definida com: ( W u ( W u ( J j=1 I j ( J j=1 S j )) )) C max 1 j J ( ) ε Ij, (16) S j er a algun ε > i er cada família finita d intervals disjunts, i oberts (I j ) J j=1, i també cada família de conjunts mesurables (S j ) J j=1, amb S j I j, er a cada j J. Aquesta classe de esos recuera els resultats ben coneguts en els casos clàssics; és a dir, si w = 1 llavors (16) és la condició A i si u = 1, llavors és la classe de esos B (veure [2]). En el mateix treball es va considerar la versió de tius dèbil del roblema, M : Λ u(w) Λ, u (w). (17) Tanmateix, la caracterització geomètrica comleta de l estimació (17) no es va resoldre er a 1. En aquesta tesi, caracteritzem totalment les acotacions (3) i (4), quan > 1, donant una versió estesa i unificada de les teories clàssiques. També caracteritzem (17) er la condició B (u) quan > 1. Els resultats rincials d aquesta tesi roven que els enunciats següents són equivalents er a > 1 (veure el Teorema 6.19): Teorema. Sigui > 1. Els enunciats següents són equivalents: (i) H : Λ u(w) Λ u(w) és acotat. (ii) H : Λ u(w) Λ, u (w) és acotat. (iii) u A, w B (iv) u A, w B i M : Λ u(w) Λ u(w) és acotat. i M : Λ u(w) Λ, u (w) és acotat. (iv) Existeix ε >, tal que er a cada família finita d intervals disjunts, oberts (I j ) J j=1, i cada família de conjunts mesurables (S j ) J j=1, amb S j I j, er a cada j J, es verifica que: ( min log I ) j j S j ( W u ( W u ( J j=1 I j ( J j=1 S j )) )) max j ( ) ε Ij. S j

15 ix Resum En articular, recuerem els casos clàssics w = 1, i u = 1. A més, reescrivim els nostres resultats en termes d alguns índex de Boyd generalitzats. Lerner i Pérez van estendre a [66] el teorema de Lorentz-Shimogaki en esais funcionals quasi-banach, no necessàriament invariants er reordenació. Motivats els seus resultats, donem una extensió del teorema de Boyd, en el context dels esais de Lorentz amb esos (veure el Teorema 6.26). També hem solucionat el cas de tius dèbil, 1 amb alguna condició addicional en w (veure el Teorema 6.2). Els caítols són organitzats de la següent manera: Per tal de dur a terme aquest rojecte com a monografia auto-continguda, en el Caítol 2 estudiem totes les roietats bàsiques dels esais de Lorentz amb esos. Aquest caítol també conté un resultat de densitat nou: rovem que l esai de funcions C amb suort comacte, Cc, és dens en els esais de Lorentz amb esos Λ u(w) en el cas que u i w no són integrables (veure el Teorema 2.13). Això serà imortant er solucionar alguns roblemes tècnics de la definició de l oerador de Hilbert en Λ u(w). El Caítol 3 recull totes les classes de esos que aareixen en aquesta monografia. Primer estudiem les classes de esos A i A. A continuació, estudiem les classes de esos B i B, que caracteritzen l acotació de M en els esais de Lorentz clàssics. Com ja hem mencionat, la classe dels esos B no és suficient er obtenir l acotació de l oerador H sobre els esais Λ u(w), i es requereix també la condició B. Desrés, investiguem la condició B (u) i trobem algunes exressions equivalents noves estudiant el comortament asimtòtic a l infinit d una funció submultilicativa (veure el Corol lari 3.38). Finalment, definim i estudiem una classe nova de esos AB, que combina les classes A i B (veure Proosició 3.46 er a més detalls). En el Caítol 4 trobem condicions necessàries er l acotació de tius dèbil de l oerador H i obtenim algunes conseqüències útils. Si restringim l acotació H : Λ u(w) Λ, u (w) a funcions característiques d intervals, tenim: ( ) bν W u(s) ds ( bν su ( ) b> b log 1 + ν ), W u(s) ds ν b er a cada ν (, 1] (veure el Teorema 4.4). En articular, això imlica u L 1 (R) i w L 1 (R + ) (veure la Proosició 4.5). A més a més si restringim l acotació de tius dèbil a funcions característiques de conjunts mesurables (veure el Teorema 4.8), obtenim ( ) W (u(i)) I W (u(e)), E i er això W u satisfà la condició doblant i w és quasi-còncava. Finalment, l acotació de tius dèbil imlica, alicant arguments de dualitat, que es comleix: u 1 χ I (Λ u (w)) χ I Λ u (w) I,

16 Resum x er a tots els intervals I de la recta real (veure el Teorema 4.16). Estudiem aquesta condició i, en conseqüència, obtenim l acotació de tius dèbil de l oerador H en els esais Λ (w). En el Caítol 5 caracteritzem l acotació de tius dèbil en els esais de Lorentz clàssics er a >, sota la suosició que u A 1 : H : Λ u(w) Λ, u (w) w B, B, (veure el Teorema 5.2). A més a més rovem que si u A 1 i > 1 tenim que H : Λ u(w) Λ u(w) w B B, (veure Teorema 5.4), mentre en el cas 1 tenim el mateix resultat amb una condició addicional en els esos (veure el Teorema 5.5). Per això, si u A 1, l acotació del tius fort (res. del tius dèbil) H : Λ u(w) Λ u(w) (res. H : Λ u(w) Λ, u (w)) coincideix amb l acotació del mateix oerador er u = 1. El Caítol 6 conté la solució comleta del roblema quan > 1; és a dir, la caracterització del tius dèbil de l acotació de l oerador H en els esais de Lorentz amb esos (veure el Teorema 6.13) i també la seva versió de tius fort (Teorema 6.18). A més, les condicions geomètriques que caracteritzen ambdós, les acotacions dels tius dèbil i fort de l oerador H en Λ u(w) es donen er al Teorema 6.19 quan > 1, i al Teorema 6.2 er l acotació del tius dèbil en el cas < 1. Finalment, reformulem els nostres resultats en termes del teorema de Boyd (veure el Teorema 6.26). Alguns dels resultats tècnics més significatius que hem utilitzat er rovar els nostres teoremes rincials són els següents: (a) Hem caracteritzat la condició A, en termes de l oerador H de la manera següent (veure el Teorema 6.3): H(uχ I )(x) dx u(i), i així obtenim que (4) imlica la necessitat de la condició AB. (b) Provem que si > 1, llavors H : Λ u(w) Λ, u I (w) M : Λ u(w) Λ, (w), (veure el Teorema 6.8) que, en articular, roorciona una rova diferent del fet ben conegut, que correson al cas w = 1: sense fer servir exlícitament la condició A. H : L (u) L, (u) M : L (u) L, (u), (c) Solucionem comletament l acotació de (17), si > 1 i la solució és la classe B (u) (veure el Teorema 6.17). En articular, mostrem que si > 1, llavors M : L (u) L, (u) M : L (u) L (u), u

17 xi Resum sense utilitzar la desigualtat de Hölder inversa. Les tècniques que vam fer servir er obtenir la caracterització de l acotació H : Λ u(w) Λ u(w), i la seva versió de tius dèbil H : Λ u(w) Λ, u (w), quan > 1 ens ermeten aconseguir algunes condicions necessàries er l acotació de tius dèbil de l oerador H en el cas no diagonal: H : Λ u (w ) Λ 1, u 1 (w 1 ), que serà també necessari er la versió de tius fort H : Λ u (w ) Λ 1 u 1 (w 1 ). En el Caítol 7 estudiem aquestes condicions. En rimer lloc, resentem una breu revisió en els casos clàssics, on, er una banda, tenim el conegut roblema de dos esos er l oerador de Hilbert, H : L (u ) L, (u 1 ) i H : L (u ) L (u 1 ), que es va lantejar als anys 197, erò no s ha resolt comletament, i d altra banda, tenim el cas no-diagonal de l acotació de l oerador H en els esais de Lorentz clàssics. Finalment, resentem algunes alicacions resecte a la caracterització de l acotació H : L,q (u) L r,s (u), er alguns exonents, q, r, s >. En articular, comletem alguns dels resultats obtinguts a [25] er Chung, Hunt, i Kurtz. Els resultats d aquesta memoria están inclosos a [1, 2, 3].

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19 Notations Throughout this monograh, the following standard notations are used: The letter M is used for the sace of measurable functions on R, endowed with the measure u = u(x)dx. Moreover, u and w will denote weight functions; that is, ositive, locally integrable functions defined on R and R + = [, ), resectively. If E is a measurable set of R, we denote u(e) = u(x)dx E and we write W (r) = r w(t)dt, for r. For < <, L denotes the usual Lebesgue sace and L dec the cone of ositive, decreasing functions belonging to L. The limit case L is the set of bounded measurable functions defined on R, while L (u) refers to the sace of functions belonging to L, whose suort has finite measure with resect to u. The letter denotes the conjugate of ; that is 1/ + 1/ = 1. In addition, Cc refers to the sace of smooth functions defined on R with comact suort. We denote by S the class of simle functions S = {f M : card(f(r)) < }. The class of simle functions with suort in a set of finite measure is: S (u) = {f S : u({f }) < }. Furthermore, we write S c for the sace of simle functions with comact suort. The distribution function of f M is λ u f (s) = u({x : f(x) > s}), the non-increasing rearrangement with resect to the measure u is t f u(t) = inf{s > : λ u f(s) t}, and fu (t) = 1 f t u(s)ds. The rearrangement of f with resect to the Lebesgue measure is denoted as f (t). Finally, letting A and B be two ositive quantities, we say that they are equivalent (A B) if there exists a ositive constant C, which may vary even in the same theorem and is indeendent of essential arameters defining A and B, such that C 1 A B CA. The case A CB is denoted by A B. For any other ossible definition or notation, we refer to the main reference books (e.g. [8], [36], [4], [41], [87], [94]). xiii

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21 Chater 1 Introduction The main urose of this work is to unify two well-known and, a riori, unrelated theories dealing with weighted inequalities for the Hilbert transform, defined by Hf(x) = 1 π lim f(y) ε + x y dy, x y >ε whenever this limit exists almost everywhere. On the one hand, we have the Calderón- Zygmund theory of the boundedness of H on weighted Lebesgue saces. On the other hand, there is the theory develoed around the boundedness of H on classical Lorentz saces in the context of rearrangement invariant function saces. A natural unifying framework for these two settings consists on the weighted Lorentz saces Λ u(w) and Λ, u (w) defined by Lorentz in [68] and [67] as follows: { ( } 1/ Λ u(w) = f M : f Λ u (w) = (fu(t)) w(t)dt) <, (1.1) and { Λ, u (w) = f M : f Λ, u (w) = su t> } W 1/ (t)fu(t) <. (1.2) More recisely, we will study the boundedness of H on the weighted Lorentz saces: H : Λ u(w) Λ u(w), (1.3) and its weak-tye version H : Λ u(w) Λ, u (w). (1.4) Before describing our results, we resent a brief historical review on the Hilbert transform. This oerator was introduced by Hilbert in [48] and [49], and named Hilbert transform by Hardy in 1924, in honor of his contributions (see [43] and [44]). It arises in many different contexts such as the study of boundary values of the imaginary arts of analytic functions 1

22 Introduction 2 and the convergence of Fourier series. Among the classical results, we mention Riesz theorem which states that H : L L is bounded, whenever 1 < < (see [85] and [86]). Although the L 1 boundedness for H fails to be true, Kolmogorov roved in [58] the following weak-tye estimate: H : L 1 L 1,. (1.5) For further information on these toics see [4], [94], [36] and [8]. The following examles, involving weighted inequalities, have been historically relevant to motivate our study. (I) If w = 1, then (1.3) and (1.4) corresond to the boundedness and its weak-tye version H : L (u) L (u), (1.6) H : L (u) L, (u), (1.7) resectively. These inequalities arise naturally when in the Riesz theorem, the underlying measure is changed from Lebesgue measure to a general measure u. Then, the roblem is to study which are the conditions over u that allow the Hilbert transform to be bounded on L (u). This new aroach gave birth to the theory of weighted inequalities, which lays a large art in the study of boundary value roblems for Lalace s equation on Lischitz domains. Other alications include vector-valued inequalities, extraolation of oerators, and alications to certain classes of nonlinear artial differential and integral equations (see [36], [41], [56], and [57]). The study of (1.6) and (1.7) yield together with the boundedness of the Hardy-Littlewood maximal function M, on the same saces, the classical theory of the Muckenhout A weights. The sublinear oerator M, introduced by Hardy and Littlewood in [45], is defined by Mf(x) = su x I 1 f(y) dy, I I and the suremum is considered over all intervals I of the real line containing x R. For further references see [38], [36], [4], [41], and [94]. We say that u A if, for > 1, the following holds: su I ( 1 I I ) ( 1 u(x)dx I I u 1/( 1) (x)dx) 1 <, (1.8) and the suremum is considered over all intervals of the real line, and u A 1 if Mu(x) u(x) a.e x R. (1.9) Muckenhout showed in [71] that, if 1, the A condition characterizes the boundedness M : L (u) L, (u),

23 3 Introduction and if > 1 it also characterizes M : L (u) L (u). Hunt, Muckenhout, and Wheeden roved in [54] that, for 1, the A condition characterizes (1.7) and for > 1 it also characterizes (1.6). For an alternative roof of these results see [26]. For < 1 there are no weights u such that (1.6) or (1.7) hold. (II) The case u = 1 corresonds to the boundedness of the Hilbert transform on the classical Lorentz saces, solved by Sawyer in [9]. A simlified characterization of the weights for which the boundedness H : Λ (w) Λ (w) (1.1) holds, whenever >, is given in terms of the B B condition, introduced by Neugebauer in [8]. We say that w B if the following condition holds: for all r >, and (1.11) characterizes the boundedness r ( r ) r w(t) dt w(t)dt, (1.11) t M : Λ (w) Λ (w), roved in [5]. The condition w B is given by r 1 t t w(s)ds dt r w(s)ds, (1.12) for all r >. If > 1 the B B class characterizes also the weak-tye version H : Λ (w) Λ, (w), (1.13) whereas the case 1 is characterized by the B, B condition. We say that w B, if and only if M : Λ (w) Λ, (w) (1.14) is bounded. It holds that: (α) If > 1, B, = B. (β) If 1, then w B, if and only if w is quasi-concave: for every < s r <, W (r) r W (s) s. (1.15) (III) Recently, Carro, Raoso and Soria studied in [2] the analogous of relation (1.3), but for the Hardy-Littlewood maximal function, instead of H M : Λ u(w) Λ u(w),

24 Introduction 4 and the solution is the B (u) class of weights, defined as follows: ( ( J )) W u ( )) C max 1 j J W u j=1 I j ( J j=1 S j ( ) ε Ij, (1.16) S j for some ε > and for every finite family of airwise disjoint, oen intervals (I j ) J j=1, and also every family of measurable sets (S j ) J j=1, with S j I j, for every j. This class of weights recovers the well-known results in the classical cases; that is, if w = 1 then (1.16) is the A condition and if u = 1, then it is the B condition (see [2]). In the same work, the weak-tye version of the roblem was also considered M : Λ u(w) Λ, u (w). (1.17) However, the comlete geometric characterization of the estimate (1.17) was not obtained for 1. In this work, we totally solve the roblem of the boundedness (1.3) and its weak-tye version (1.4), whenever > 1 giving a unified version of the classical theories. We also characterize (1.17) by the B (u) condition, since it will be involved in the solution of (1.3) and (1.4). We will see that this solution is given in terms of conditions involving both underlying weights u and w in a rather intrinsic way. Summarizing, the main results of this thesis rove that the following statements are equivalent for > 1 (see Theorem 6.19): Theorem. If > 1, then the following statements are equivalent: (i) H : Λ u(w) Λ u(w) is bounded. (ii) H : Λ u(w) Λ, u (w) is bounded. (iii) u A, w B (iv) u A, w B and M : Λ u(w) Λ u(w) is bounded. and M : Λ u(w) Λ, u (w) is bounded. (iv) There exists ε >, such that for every finite family of airwise disjoint, oen intervals (I j ) J j=1, and every family of measurable sets (S j ) J j=1, with S j I j, for every j J, it holds that: ( ( ( min log I ) J )) W u j j=1 I ( ) j ε Ij ( ( j S j J )) max. j W u S j j=1 S j Furthermore, we reformulate our results in terms of some generalized uer and lower Boyd indices. Lerner and Pérez extended in [66] the Lorentz-Shimogaki theorem in quasi- Banach function saces, not necessarily rearrangement invariant. Motivated by their results, we define the lower Boyd index and give an extension of Boyd theorem, in the context of weighted Lorentz saces (see Theorem 6.26).

25 5 Introduction Moreover, we have solved the weak-tye boundedness of H on Λ u(w) for 1, with some extra assumtion on w (see Theorem 6.2). The chaters are organized as follows: In order to carry out this roject as a self-contained monograh, we study in Chater 2 all the basic roerties of the weighted Lorentz saces. This chater also contains a new density result: we rove that the C functions with comact suort, C c, is dense in weighted Lorentz saces Λ u(w), rovided u and w are not integrable (see Theorem 2.13). This will be imortant in order to solve technical roblems, since the Hilbert transform is well-defined on C c. In Chater 3 we summarize all the classes of weights that aear throughout this work. First we study the Muckenhout A class of weights and the A condition. Then, we study the B and B, conditions that characterize the boundedness of M on classical Lorentz saces, introducing the Hardy oerator. Since, as we have already mentioned, the B (res. B, ) condition is not sufficient for the strong-tye (res. weak-tye) boundedness of the Hilbert transform on Λ (w), we introduce and study the B condition. Next, we investigate the B (u) condition, and find some new equivalent exressions studying the asymtotic behavior of some submultillicative function at infinity (see Corollary 3.38). Finally, we define and study a new class of airs of weights AB, that combines the already known A and B classes (see Proosition 3.46 for more details). This new class of weights is involved in the study of the boundedness of the Hilbert transform on weighted Lorentz saces (see Chater 6). In Chater 4 we find necessary conditions for the weak-tye boundedness of the Hilbert transform on weighted Lorentz saces and obtain some useful consequences. If we restrict the weak-tye boundedness of the Hilbert transform, H : Λ u(w) Λ, u (w), to characteristic functions of intervals, we have that ( ) bν W u(s) ds ( bν su ( ) b> b log 1 + ν ), W u(s) ds ν b for every ν (, 1] (see Theorem 4.4). In articular, this imlies that u L 1 (R) and w L 1 (R + ) (see Proosition 4.5). We also show that, if we restrict the weak-tye boundedness of H to characteristic functions of measurable sets (see Theorem 4.8), we obtain ( ) W (u(i)) I W (u(e)), E and hence W u satisfies the doubling condition and w is quasi-concave. In articular w 2. Thus, in what follows after Corollary 4.9, we shall assume, without loss of generality, that u L 1 (R), w L 1 (R + ), and w 2.

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