; TeX output 2002.11.21:1353 v ~ 9N cmbx12VECTORٚg cmmi12A|{Y cmr82 WEIGHTS 7XANDٚAHARD Y-LITTLEWOODٚMAXIMALFUNCTION w-K`y cmr10MICHAELUUCHRISTANDMICHAELGOLDBERG `6 - cmcsc10Abstract.n`AnanalogueoftheHardy-LittlewoGodmaximalfunctionisintroGduced, 6 forfunctionstakingvqaluesinnite-dimensionalHilbGertspaces.Itisshowntobe6 b> cmmi10L^ٓR cmr72TbGoundedבwithrespecttoweightsבintheclassA2TofT*reil, therebyextendinga6 theoremUUofMuckenhouptUUfromthescalartothevectorcase.S XQ cmr12Abasiccrhapterofthesub jectofsingularintegralopSeratorsistheweightednorm theoryV,swhicrhVprovidesanecessaryandsucientconditiononanonnegativefunction wfor[sucrhopSerators,>andfortheHardy-LittlewoSod[maximalfunctionM@,>tobSe bSounded̤inL22 cmmi8pwithrespecttothemeasurewR(x)dx..See[3],ҥ[6],[12]..More̤recenrtlyV, aspSectsofthistheoryharvebeenextendedrstbryTVreilandVolbSerg[17,18v],then bryotherauthors,3tofunctionstakingvXaluesinnite-dimensionalHilbSertspaces, withwreightstakingvXaluesinthecorrespSondingspacesofHermitianforms.These extensionsejharvereliedonnewideas,quitedierentfromthoseemployedinthescalar casebryearlierauthors,andhaveshednewlightonthescalartheoryV. Nonetheless,psomeaspSectsofthescalartheoryhadapparenrtlynotbeensuccessfully generalized,including M"itself.Indeed,doubtsharvebSeenexpressed[20,17]asto whethertherecouldexistanryusefulanalogueofM@. In,thisnotewreintroSduceavectoranalogueoftheHardy-LittlewoSod,maximal function,and:prorveitsbSoundedness,inthesimplestcase,pJ=2.Indoingso,wre seek\rstlyV,ٞtoclarifytherelationshipsbSetrween\thenewvrectortheory,ٞandthemore familiarxscalarcase,YandsecondlyV,toopSenupanalternativreapproachtothesub ject, whicrhwmightleadtovectoranaloguesofotherfeaturesofthescalartheoryV.Thesecond authorm[5]hascarriedthisprogramfurtherbrydemonstratingthattheanalogueof ourUtheoremholdsforall1,