Graduate Texts in Mathematics Loukas Grafakos Classical Fourier Analysis Third Edition Graduate Texts in Mathematics 249 Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA GraduateTextsinMathematicsbridgethegapbetweenpassivestudyandcreative understanding, offering graduate-level introductions to advanced topics in mathe- matics.Thevolumesarecarefullywrittenasteachingaidsandhighlightcharacter- istic features of the theory. Although these books are frequently used as textbooks ingraduatecourses,theyarealsosuitableforindividualstudy. Forfurthervolumes: http://www.springer.com/series/136 Loukas Grafakos Classical Fourier Analysis Third Edition 123 LoukasGrafakos DepartmentofMathematics UniversityofMissouri Columbia,MO,USA ISSN0072-5285 ISSN2197-5612(electronic) ISBN978-1-4939-1193-6 ISBN978-1-4939-1194-3(eBook) DOI10.1007/978-1-4939-1194-3 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2014946585 MathematicsSubjectClassification(2010):42Axx,42Bxx ©SpringerScience+BusinessMediaNewYork2000,2008,2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ToSuzanne Preface The great response to the publication of my book Classical and Modern Fourier Analysisin2004hasbeenespeciallygratifyingtome.IwasdelightedwhenSpringer offered to publish the second edition in 2008 in two volumes: Classical Fourier Analysis,2ndEdition,andModernFourierAnalysis,2ndEdition.Iamnowelated to have the opportunity to write the present third edition of these books, which Springer has also kindly offered to publish. The third edition was born from my desire to improve the exposition in several places, fix a few inaccuracies, and add some new material. I have been very fortunate to receive several hundred e-mail messages that helped me improve the proofs and locate mistakes and misprints in thepreviouseditions. Inthisedition,Imaintainthesamestyleasinthepreviousones.Theproofscon- taindetailsthatunavoidablymakethereadingmorecumbersome.Althoughitwill behoove manyreaderstoskimthroughthemoretechnicalaspectsofthepresenta- tion and concentrate on the flow of ideas, the fact that details are present will be comforting to some. (This last sentence is based on my experience as a graduate student.) Readers familiar with the second edition will notice that the chapter on weightshasbeenmovedfromthesecondvolumetothefirst. This first volume Classical Fourier Analysis is intended to serve as a text for a one-semester course with prerequisites of measure theory, Lebesgue integration, and complex variables. I am aware that this book contains significantly more ma- terial than can be taught in a semester course; however, I hope that this additional informationwillbeusefultoresearchers.Basedonmyexperience,thefollowinglist of sections (or parts of them) could be taught in a semester without affecting the logicalcoherenceofthebook:Sections1.1,1.2,1.3,2.1,2.2.,2.3,3.1,3.2,3.3,4.4, 4.5,5.1,5.2,5.3,5.5,5.6,6.1,6.2. Alonglistofpeoplehaveassistedmeinthepreparationofthisbook,butIremain solely responsible for any misprints, mistakes, and omissions contained therein. Pleasecontactmedirectly([email protected])ifyouhavecorrectionsorcom- ments.Anycorrectionstothiseditionwillbepostedtothewebsite http://math.missouri.edu/˜loukas/FourierAnalysis.html vii viii Preface whichIplantoupdateregularly.Ihavepreparedsolutionstoalloftheexercisesfor thepresenteditionwhichwillbeavailabletoinstructorswhoteachacourseoutof thisbook. Athens,Greece, LoukasGrafakos March2014 Acknowledgments Iamextremelyfortunatethatseveralpeoplehavepointedouterrors,misprints,and omissionsinthepreviouseditionsofthebooksinthisseries.Alltheseindividuals have provided me with invaluable help that resulted in the improved exposition of thetext.Forthesereasons,Iwouldliketoexpressmydeepappreciationandsincere gratitudetotheallofthefollowingpeople. Firsteditionacknowledgements: GeorgiosAlexopoulos,Nakhle´Asmar,Bruno Calado, Carmen Chicone, David Cramer, Geoffrey Diestel, Jakub Duda, Brenda Frazier, Derrick Hart, Mark Hoffmann, Steven Hofmann, Helge Holden, Brian Hollenbeck, Petr Honz´ık, Alexander Iosevich, Tunde Jakab, Svante Janson, Ana Jime´nezdelToro,GregoryJones,NigelKalton,EmmanouilKatsoprinakis,Dennis Kletzing, Steven Krantz, Douglas Kurtz, George Lobell, Xiaochun Li, Jose´ Mar´ıa Martell, Antonios Melas, Keith Mersman, Stephen Montgomery-Smith, Andrea Nahmod, Nguyen Cong Phuc, Krzysztof Oleszkiewicz, Cristina Pereyra, Carlos Pe´rez, Daniel Redmond, Jorge Rivera-Noriega, Dmitriy Ryabogin, Christopher Sansing, Lynn Savino Wendel, Shih-Chi Shen, Roman Shvidkoy, Elias M. Stein, Atanas Stefanov, Terence Tao, Erin Terwilleger, Christoph Thiele, Rodolfo Torres, DeanieTourville,NikolaosTzirakis,DonVaught,IgorVerbitsky,BrettWick,James Wright,andLinqiaoZhao. Second edition acknowledgements: Marco Annoni, Pascal Auscher, Andrew Bailey, Dmitriy Bilyk, Marcin Bownik, Juan Cavero de Carondelet Fiscowich, Leonardo Colzani, Simon Cowell, Mita Das, Geoffrey Diestel, Yong Ding, Jacek Dziubanski, Frank Ganz, Frank McGuckin, Wei He, Petr Honz´ık, Heidi Hulsizer, PhilippeJaming,SvanteJanson,AnaJime´nezdelToro,JohnKahl,CorneliaKaiser, Nigel Kalton, Kim Jin Myong, Doowon Koh, Elena Koutcherik, David Kramer, Enrico Laeng, Sungyun Lee, Qifan Li, Chin-Cheng Lin, Liguang Liu, Stig-Olof Londen, Diego Maldonado, Jose´ Mar´ıa Martell, Mieczysław Mastyło, Parasar Mohanty, Carlo Morpurgo, Andrew Morris, Mihail Mourgoglou, Virginia Naibo, TadahiroOh,MarcoPeloso,MariaCristinaPereyra,CarlosPe´rez,HumbertoRafeiro, MariaCarmenRegueraRodr´ıguez,AlexanderSamborskiy,AndreasSeeger,Steven Senger,SumiSeo,ChristopherShane,ShuShen,YoshihiroSawano,MarkSpencer, Vladimir Stepanov, Erin Terwilleger, Rodolfo H. Torres, Suzanne Tourville, ix