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1、ThreeregularityresultsinHarmonicAnalysisTerenceTaoADissertationPresentedtotheFacultyofPrincetonUniversityinCandidacyfortheDegreeofDoctorofPhilosophyRecommendedforAcceptanceBytheDepartmentofMathematicsJune,1996⃝cCopyrightbyTerenceTao,1998.AllRightsReservedAbstractInt
2、hisdissertationweprovecertainregularitypropertiesofthreeunrelatedfamiliesofoperatorsarisingfromseparateproblemsinharmonicanalysis.ThefirstresultconcernstheclassicalBochner-RieszoperatorsSδonEu-nclideanspacesR(aswellasmoregeneralRieszmeansonmanifolds).BytheworkofC.Feff
3、erman,P.Tomas,E.Stein,andM.Christ,onecanob-tainregularityresultsontheseoperatorsfromL2restrictiontheory.Weencompasstheseresultsbyusingtherestrictiontheoremtoproveanoptimalweak-typeestimatefortheindexδ=n−1,whichisthesharpestpossible2(n+1)resultonecanobtainfromL2restr
4、ictiontheoryalone.Thesecondresultaddressesthequestionofthepointwiseconvergenceofvariouswaveletsamplingmethods.Inapplicationsoneoftensamplesafunc-∑tionf=j,kaj,kψj,kbydiscardingallbutthelargestwaveletco-efficients,∑leavingareconstructedfunctionoftheform
5、aj;k
6、>λaj,kψj,k.
7、Weshowthatundergeneralconditionsthesampledfunctionconvergespointwisealmosteverywheretotheoriginalfunctionasλ→0.Thisisachievedbyapprox-imatingthesamplingoperatorbyalinearsamplingmethodwhoseconver-gencewasestablishedbyKelly,Kon,andRaphael.Ourfinalresultextendsrecentwor
8、kbyM.Christ,J.L.RubiodeFran-cia,andA.Seeger,onweak(1,1)estimatesforroughoperatorsinEuclideanspaces,tomoregeneralhomogeneousgroups.Thisisstillaworkinprogressbytheauthor,andtheresultspresentedherearesomewhatpartialinna-ture.Weshowthatahomogeneoussingularintegralconvol
9、utionoperatorisofweak-type(1,1)ifitisboundedonL2andthekernelisLlogLonthesphere.iiiAcknowledgementsTheauthorisdeeplyindebtedtohisadvisor,EliasStein,forhisconstantguidance,advice,andsuggestions.TheauthoralsowishestothankCharlesFefferman,AnnaGilbert,IngridDaubechies,and
10、AndreasSeegerforseveralusefuldiscussions.PartsofthisresearchwereconductedwhiletheauthorwassupportedbyaFulbrightGraduateawardandaSloanGradu
