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1、OntheparabolickerneloftheSchr6dingeroperatorbyPETERLI(l)andSHINGTUNGYAUUniversityofUtahUniversityofCalifornia,SanDiegoSaltLakeCity,UT,U.S.A.LaJolla,CA,U.S.A.Tableofcontentsw0.Introduction.............................153w1.Gradientestimates.........................155w2.Harnackinequalities..........
2、..............166w3.Upperboundsoffundamentalsolutions..............170w4.Lowerboundsoffundamentalsolutions..............181w5.HeatequationandGreen'skernel.................190w6.TheSchr6dingeroperator......................196Appendix.................................199References.....................
3、...........200w0.IntroductionInthispaper,wewillstudyparabolicequationsofthetype(A-q(x,t)-~)u(x,t)=O(0.1)onageneralRiemannianmanifold.Thefunctionq(x,t)isassumedtobeC2inthefirstvariableandC1inthesecondvariable.Inclassicalsituations[20],aHarnackinequalityforpositivesolutionswasestablishedlocally.Howev
4、er,thegeometricdependencyoftheestimatesiscomplicatedandsometimesunclear.OurgoalistoproveaHarnackinequalityforpositivesolutionsof(0.1)(w2)byutilizingagradientestimatederivedinw1.Themethodofproofisoriginatedin[26]and[8],wheretheyhavestudiedtheellipticcase,i.e.thesolutionistimeindependent.Insomesituat
5、ions(Theorems2.2and(1)ResearchpartiallysupportedbyaSloanfellowshipandanNSFgrant.11-868283ActaMathematica156.Imprim6le15mai1986154PETERLIANDSHINGTUNGYAU2.3),theHarnackinequalityisvalidglobally,whichenablesustorelatetheglobalgeometrywiththeanalysis.Inw3,weapplytheHarnackinequalitytoobtainupperestimat
6、esforthefunda-mentalsolutionoftheequation(A-q(x)-~tt)u(x,t)=0,(0.2)whereqisafunctiononMalone.Weshallpointoutthatfortheheatequation(q=0),upperestimatesfortheheatkernelwereobtainedin[7]and[5].Howevertheestimatewhichweobtainissofarthesharpest,especiallyforlargetime.WhentheRiccicurvatureisnonnegativeth
7、esharpnessisapparent,sinceacomparablelowerboundisalsoobtainedinw4.Alowerboundforthefundamentalsolutionof(0.2)isalsoderivedforsomespecialsituations.Applicationsoftheseestimatesfortheheatkernelarediscussedinw
