; TeX output 1998.05.21:1704 sK83"V ff cmbx10BestAppro=ximationintheMeandō=b=yAnalyticandHarmonicFunctions/6@ K`y cmr10DmitryUUKhavinson*,JohnE.McqCarthy#!", cmsy10yT,andHaroldS.Shapiro lMayUU21,1998/ V2"V cmbx101." Introb"duction.{ F*or,b> cmmi10n2,4let"V cmbx10B 0er cmmi7n*denotetheunitballinR^nq~,andforp1letL^pdenotetheBanach space#ofp-summablefunctionsonBnq~. *0LetL1ɍpvh.(Bn)denotethesubspaceofharmonicfunctionsgonBnthatarep-summable.Whenn=2,l"wegoftenwriteDinsteadofBٓR cmr72|s,andweUUletA^pdenotetheBergmanspaceofanalyticfunctionsinL^pR.K LetС!,zbGeafunctioninL^pR.EW*eareinterestedinndingthebestapproximationto!,zinA^pBandgL1ɍpvh.(Bnq~).[ExistenceofabGestapproximantgisstraighforward;qthispaperconsiderstheUUfollowingtwoqualitativepropGerties:q(i) UniquenessUUofbGestapproximants,UUwhenp=1.(ii) HereditaryFLregularityofthebGestapproximantf^? inheritedfromthatof![٫,e.g., whether2continuity*,Holder&continuity,real-analyticity2of!intheclosedunitdisk enforceUUthosepropGertiesin![٫'sbestapproximant. TheseandmanyothersimilarquestionshavebGeenwell-studiedforthecasewhenthenormalizedareameasuredA:=<$J1"w fe (֍sdxdy)isreplacedbydȫ=<$4d"w fe (֍2KontheunitcircleTandthespacesA^p6arereplaced,3Gaccordingly*,bythefamiliarHardy#MspacesH ^p v4(cf.,3Ge.g.,[Ak],m[D],f5[Ka],[Kh2{6]&,[RS],[W],andreferencescitedtherein).hInthatsituation,theapproachbasedonHahn&{Banach#A'dualityandtheF.UUandM.RieszLtheoremidentifyingtheannihilator($ ': cmti10AnnB(bѱH ^poP)1in(L^q=( T;dG)(asH1ɍ qxݍ0=JxfJyfڧ2H ^q:f(0)=0gZqp,ݱqQ=<$ p}w fe @ߟ (֍p8 1turnsouttobGequitesuccessfulandanswersanumbGerofquestions.ThedicultywiththisapproachwhenvusingareameasureisthetremendoussizeoftheannihilatorAnnR(5A^pR)/ofA^pOinL^qj.TheUUfollowingresult,whichweshallcallKhavin's-?lemma,characterizesAnn0o(A^pR),. [ ff xs * PartiallyUUsuppGortedbyNSFGrantDMS97-03915 9y PartiallyUUsuppGortedbyNSFGrantDMS95-31967 ;1 * s F*or^ űp:1