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1、BULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume15,Number2,October1986POTENTIALTHEORYFORTHESCHRODINGEREQUATIONM.CRANSTON,E.FABES,ANDZ.ZHAORecentlytherehasbeenawaveofresults[2,4,5,11,15,16,17],onwhatisnowreferredtoastheconditionalgaugetheorem.Theseworkswereinspiredby[1an
2、d6].Weprovethisresultingreatergeneralitythanbeforeandderiveinterestingnewconsequences.LetbeauniformlyellipticoperatorwhosecoefficientsareboundedmeasurablefunctionsonaboundedLipschitzdomainDÇRd.DefinetheKatoclassKdastheclassoffunctionsonDsuchthatlimsup/,
3、7(^2dy=0.Ourapproachistop
4、roveresultsabouttheoperatorL=A+VbyusingknownresultsforAandstudyingtheprobabilisticquantity,theconditionalgauge.Inordertointroducetheconditionalgaugeletp(t,x,y)betheGreenfunc•tionfortheparabolicequationA—djdtonDx(0,oo).Let(Xt,Px)bethediffusion,killedattheexittimeTD=inf{£>0:XtGJ9}
5、,whosetransitiondensityisp(t,x,y).TheanalysisinvolvesthediffusionXtconditionedonitsexitposition.Thisconditioneddiffusion,see[10],hastransitiondensitypz(t,x,y)=KA{X)Z)~1p(t,Xiy)KA(y,z),whereKAisthekernelfunctionforAonD,x,yGZ>,zGdD.WeshallwriteP*()=PX(-XTD=z)ande^(£)=exp{/0V(x3)d
6、s}.Theso-calledgaugeisthefunctionon£>,F(l;x)=Ex[ev{rD)andtheconditionalgaugeisdefinedonDxdDbyF(l;x,z)=££[ev(T£>)].Theorem1wasfirstprovenin[12]whenA—A,VisboundedanddDisC2,laterwhenA=A,VGKdanddDisC1,1in[16]andrecentlywhenA=A,VGLpforsomep>d/2anddDisLipschitzin[13].Ourmainresultist
7、hefollowing.THEOREM1.SupposetheuniformlyellipticA=£^(M*)5?)hasboundedmeasurablecoefficients,VGKd,andDÇRdisboundedandLipschitz.ThenF(l;x)8、tion(1985Revision).Primary60J45,31B25.©1986AmericanMathematicalSociety0273-0979/86$1.00+$.25perpage213214M.CRANSTON,E.FABES,ANDZ.ZHAOSKETCHFORPROOFOFTHEOREM1.Theprooffollows[16]andrequiresaGreenfunction-kernelfunctionestimate.LetthenGAbetheGreenfunctionforAandthedomainD.Whatisre
9、quiredare(a)forsomepositiveconstantcandx,y€D,z€
