Riemann流形上的A-调和形式

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1、Chapter1IntroductionChapter1IntroductionInthepresentpaper,wemainlydealwithA-harmonicformsonRiemannianman-ifolds.Asweallknow,manyinterestingresultsconcerninggeometricandanalyticpropertiesofharmonicformshasbeenfoundin[1-9].Amongtheseresults,theSobolev-Poincar´eimbeddinginequa

2、lityandPoincar´einequalityaretwocriticaloneswhichhaveengagedtheattentionofmanymathematicians.Ontheotherhand,theLaplace-Beltramioperator,Green’soperator,gradientoperatorandthehomotopyoperatorareeffectivetoolsindifferentfields,includingphysics,potentialtheory,nonlinearanalysisan

3、dthetheoryofelasticity.Inthispaper,weprovelocalAr(λ1,λ2,Ω)-weightedSobolev-Poincar´eimbeddinginequalitiesandAr(λ1,λ2,Ω)-weightedPoincar´einequalitiesforthecomposi-tionoftheLaplace-Beltramioperator,homotopyoperatorandGreen’soperatorappliedntoA-harmonicformsonmanifoldsinR,n≥2

4、.TheseresultscanbeusedtostudytheintegrabilityofA-harmonicformsandthepropertiesoftherelatedoperatorswhichareappliedtoA-harmonicformsonmanifolds.Wenowintroducesomenotationsusedinthispaper.Throughoutthispaper,wealwaysassumethatMisaRiemannian,compact,orientedandC∞smoothmanifold

5、nn1withoutboundaryonR,andΩisanopensubsetofR,n≥2.WewriteR=R.VVnllnLete1,e2,···,enbethestandardunitbasisofR.Let=(R)bethelinearspaceofl-vectors,spannedbytheexteriorproductseI=ei1∧ei2∧···∧eil,correspondingtoallorderedl-tuplesI=(i1,i2,···,il),1≤i1

6、Grassmanalgebra=isagradedalgebrawithrespecttotheexteriorproducts.PVPVVForα=αIe∈andβ=βIe∈,theinnerproductinisgivenbyIIXIIhα,βi=αβwithsummationoveralll-tuplesI=(i1,i2,···,il)andallintegersl=0,1,···,n.VVWedefinetheHodgestaroperator∗:→bytherule∗1=e1∧e2∧···∧enandVVα∧∗β=β∧∗α=hα,βi

7、(∗1)forallα,β∈.Thenormofα∈isgivenbytheformula-1-:)W)
)`2V0V

8、α

9、=hα,αi=∗(α∧∗α)∈=R.TheHodgestarisanisometricisomorphismonwith∗:Vl→Vl−1and∗∗(−1)l(n−l):Vl→Vl.Adifferentiall-formωonMisadeRhamcurrent(see[10,ChapterIII])onMVVlnlwithvaluesin(R).LetMbethelthexteriorpowerofthecota

10、ngentbundleand∞Vl′VlC(M)bethespaceofsmoothl-formsonM.WeuseD(M,)todenotethespaceVof