; TeX output 2003.03.14:1020 hGS 9K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen28,2003,223{238%`K7"V cmbx10THEMAXIMALFUNCTION>ߍeHONVwxARIABLELb> cmmi10LL2M0er ff cmmi7Mp?SPٚACESy7?M"V cmbx10D.TCruz-UribQe,A.Fiorenza,andC.J.NeugebauernTN8rinitĽyCollege,DepartmentofMathematics CUHartford,CT06106-3100,U.S.A.;daĽvid.cruzurib;Be@mail.trincoll.edu -UnivĽersitUadiNap;Boli,DiptodiCostruzionieMetodiMatematiciinArcĽhitetturaPOViaMonĽteoliveto,3,I-80134Nap;Boli,Italy;orenza@unina.it/andConsiglioNazionaledelleRicercĽhe,Istitutop;BerleApplicazionidelCalcolo}݁\MauroPicone"{sezionediNap;BolihHviaPietroCastellino,111,I-80131Nap;Boli,ItalyhY[PurdueUnivĽersityN8,DepartmentofMathematicsKIYWN8estLafaĽyette,IN47907-1395,U.S.A.;neug@math.purdue.eduD"V UU cmbx10DAbstract.?WN8eSgivĽecontinuityconditionsontheexp;BonentfunctionS<b> UU cmmi10p(x)whichSaresu- cienĽt fortheHardy{Littlewo;Bod maximalop;Beratortobeboundedonthev{ariableLebesguespaceL=0er U cmmi7p:ٓR U cmr7(x).( ) ,&where isanĽyop;BensubsetofDRn.vFN8urther,ourconditionsarenecessaryonDR .vOurresult2extendstherecenĽtworkofPickandR0uezick{a2[20],(Diening[3]andNekvinda[19].gWN8ealsoshoĽw3thatundermuchweakerassumptionson3p(x) ,Othemaximalop;Beratorsatisesaweak-typ;Bemo;BdularinequalitĽyN8.z 1.pIn9troQduction K`y cmr10Given>anopGenset> E!", cmsy10R^ 0er cmmi7nq~,18and>ameasurablefunctionb> cmmi10p:UP E![1;1) ,18let L^pٓR cmr7(x)h8( )FdenoteFtheBanachfunctionspaceofmeasurablefunctionsfZon suchthatUUforsomeUU>0 ,J܍ hcu cmex10Z y kjf(x)=jp(x)dx<1;܍withUUnormQkfk:p(x); D=infƟ^qDZ>0:cZUR y ^<$Pjf(x)jPw fe Z (֍ 1Rݟ^8Q۴p(x)I1dx1^:ThesespacesareaspGecialcaseoftheMusielak{Orliczspaces(cf.Musielak[18]).Whenkp(x)@=p0 Mޫiskconstant,0pL^p(x)h8( )bGecomesthestandardLebesguespaceL^p Z cmr50r( ) . ff xsUU2000MathematicsSubvjectClassication:%Primary42B25,42B35.DTheauthorswĽouldliketothankLarsDieningandAlehEsNekvindaforsharingwithuspreprintsoftheirwĽorkonthisproblem. * 224=-?!p0J cmsl10D.UUCruz-UribGe,A.Fiorenza,andC.J.NeugebauerhGF*unctionsinthesespacesandtheassoGciatedSobolevspacesWc^k+B;p(x)W( )have bGeenconsideredbyanumbGerofauthors:see,"forexample,[1],[6]{[9],[11]{[17],[21], [22]and[24].yTheyappGearinthestudyofvqariationalintegralsandpartialdierentialUUequationswithnon-standardgrowthconditions.SomeZofthepropGertiesoftheLebesguespacesreadilygeneralizetothespacesL^p(x)h8( ) :Ysee,iforeexample,KovaGcikandRakosnqk[15].Ontheotherhand,iele-mentarypropGerties,suchasthecontinuityoftranslation,oftenfailtohold(see[15]or[10]),andforapplicationsitisanimpGortantandopenproblemtodeterminewhichŢresultsfromharmonicanalysisremaintrueinthevqariableexpGonentsetting.InUUthispapGerweconsidertheHardy{LittlewoGodUUmaximalopGerator,ߍ(1:1)uNMf(x)=g:supdBW=O! cmsy73x<$E1rw fe s (֍jB qj&ucZ, yBW=\ ?jf(y[٫)jdy;,wherevthesupremumistakenoverallballsvBwhichcontainvxandvforwhichjBR\ j>0 .iIt;Uiswellknown(cf.DuoandikoGetxea[5])thatthemaximaloperatorsatisesUUthefollowingweakandstrong-typGeinequalities:%yXc@ jfx2 :Mf(x)>tgj '$<$)DZCKw fe ;ğ (֍trpZ$ y #jf(y[٫)jpIdy;1p<1;dbɡZi y qMf(y[٫)pIdy '$CaġZ y jf(y[٫)jpIdy;1