A priori estimates for weak solutions of complex Monge-Amp`e

A priori estimates for weak solutions of complex Monge-Amp`ere equations

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aAPRIORIESTIMATESFORWEAKSOLUTIONSOFCOMPLEXMONGE-AMPERE`EQUATIONSS.BENELKOURCHI&V.GUEDJ&A.ZERIAHIAbstract.LetXbeacompactK¨ahlermanifoldandωasmoothclosedformofbidegree(1,1)whichisnonnegativeandbig.WestudytheclassesEχ(X,ω)ofω-plurisubharmonicfunctionsof niteweightedMonge-Amp`ereenergy.Whentheweightχhasfastgrowthatin nity,thecorrespondingfunctionsareclosetobebounded.WeshowthatifapositiveRadonmeasureissuitablydominatedbytheMonge-Amp`erecapacity,thenitbelongstotherangeoftheMonge-Amp`ereoperatoronsomeclassEχ(X,ω).Thisisdonebyestablishingaprioriestimatesonthecapacityofsublevelsetsofthesolutions.OurresultextendsU.Cegrell’sandS.Kolodziej’sresultsandputsthemintoaunifyingframe.ItalsogivesasimpleproofofS.T.Yau’scelebratedaprioriC0-estimate.2000MathematicsSubjectClassi cation:32W20,32Q25,32U05.1.IntroductionLetXbeacompactconnectedK¨ahlermanifoldofdimensionn∈N .Throughoutthearticleωdenotesasmoothclosedformofbidegree(1whichisnonnegativeandbig,i.e.suchthatthecomplex ,1)Monge-Amp`Xωn>0.Wecontinuethestudystartedin[GZ2],[EGZ]ofereequation(MA)µ(ω+ddc )n=µ,where ,theunknownfunction,isω-plurisubharmonic:thismeansthat ∈L1(X)isuppersemi-continuousandω+ddc ≥0isapositivecurrent.WeletPSH(X,ω)denotethesetofallsuchfunctions(see[GZ1]fortheirbasicproperties).Hereµisa xedpositiveRadonmeasureoftotalµ(X)= massXωn,andd= +2iπ(

A priori estimates for weak solutions of complex Monge-Amp`ere equations

2S.BENELKOURCHI&V.GUEDJ&A.ZERIAHI

Wethenlookforconditionsonthemeasureµwhichinsurethatthesolution isalmostbounded.FollowingtheseminalworkofS.Kolodziej

[K2,3],wesaythatµisdominatedbytheMonge-Amp`ereCapacityCapωifthereexistsafunctionF:R+→R+suchthatlimt→0+F(t)=0and( )µ(K)≤F(Capω(K)),forallBorelsubsetsK X.

HereCapωdenotestheglobalversionoftheMonge-Amp`erecapacityintro-ducedbyE.BedfordandA.Taylor[BT](seesection2).

ObservethatµdoesnotchargepluripolarsetssinceF(0)=0.WhenF(x) xαvanishesatorderα>1andωisK¨ahler,S.Kolodziejhasproved[K2]thatthesolution ∈PSH(X,ω)of(MA)µiscontinuous.Theboundednesspartofthisresultwasextendedin[EGZ]tothecasewhenωismerelybigandnonnegative.IfF(x) xαwith0<α<1,twoofushaveprovedin[GZ2]thatthesolution has niteχ energy,whereχ(t)= ( t)p,p=p(α)>0.Thisresultwas rstestablishedbyU.Cegrellinalocalcontext[Ce].

Anotherobjectiveofthisarticleisto llinthegapinbetweenCegrell’sandKolodziej’sresults,byconsideringallintermediatedominatingfunctionsF.WriteFε(x)=x[ε( ln(x)/n)]nwhereε:R→[0,∞[isnonincreasing.Oursecondmainresultis:

THEOREMB.Ifµ(K)≤Fε(Capω(K))forallBorelsubsetsK X,thenµ=(ω+ddc )nwhere ∈PSH(X,ω)satis essupX =0and

Capω( < s)≤exp( nH 1(s)). xHereH 1isthereciprocalfunctionofH(x)=e0ε(t)dt+s0,wheres0=

s0(ε,ω)≥0onlydependsonεandω.

Thisgeneralstatementhasseveralusefulconsequences: +∞ +∞ if0ε(t)dt<+∞,thenH 1(s)=+∞fors≥s∞:=e0ε(t)dt+

s0,henceCapω( < s)=0.Thismeansthat isboundedfrombelowby s∞.ThisresultisduetoS.Kolodziej[K2,3]whenωisK¨ahler,and[EGZ]whenω≥0ismerelybig;

thecondition( )iseasytocheckformeasureswithdensityinLp,p>

1.Ourresultthusgivesasimpleproof(Corollary3.2),followingtheseminalapproachofS.Kolodziej([K2]),oftheC0-aprioriestimateofS.T.Yau[Y],whichiscrucialforprovingtheCalabiconjecture(see[T]anoverview); +for∞ when0ε(t)dt=+∞,thesolution isgenerallyunbounded.Thefasterε(t)decreasestowardszero,thefasterthegrowthofH 1atin nity,hencethecloseris frombeingbounded;

thespecialcaseε≡1isofparticularinterest.Hereµ(·)≤Capω(·),andourresultshowsthatCapω( < s)decreasesexponentiallyfast,hence has“loglog-singularities”.Thesearethetypeofsin-gularitiesofthemetricsusedinArakelovgeometryinrelationwithmeasuresµ=fdVwhosedensityhasPoincar´e-typesingularities(see[Ku],[BKK]).

WeproveTheoremBinsection3,afterestablishingTheoremAinsection

2.1andrecallingsomeusefulfactsfrom[GZ2],[EGZ]insection2.2.We

A priori estimates for weak solutions of complex Monge-Amp`ere equations

`APRIORIESTIMATESFORSOLUTIONSOFMONGE-AMPEREEQUATIONS3

thentestthesharpnessofourestimatesinsection4,wherewegiveexamplesofmeasuresful llingourassumptions:theseareabsolutelycontinuouswithrespecttoωn,andtheirdensitydonotbelongtoLp,foranyp>1.

2.Weaklysingularquasiplurisubharmonicfunctions

TheclassE(X,ω)ofω-pshfunctionswith niteweightedMonge-Amp`ereenergyhasbeenintroducedandstudiedin[GZ2].ItisthelargestsubclassofPSH(X,ω)onwhichthecomplexMonge-Amp`ereoperator(ω+ddc·)niswell-de nedandthecomparisonprincipleisvalid.Recallthat ∈E(X,ω)ifandonlyif(ω+ddc j)n( ≤ j)→0,where j:=max( , j).

2.1.TherangeoftheMonge-Amp`ereoperator.Therangeofthecnoperator(ω+dd·)actingonE(X,ω)hasbeencharacterizedin[GZ2]whenωisaK¨ahlerform.Weextendherethisresulttothecasewhenωismerelynonnegativeandbig.

Theorem2.1.Assumeωisasmoothclosednonnegative(1,1)formonX, andµisapositiveRadonmeasuresuchthatµ(X)=Xωn>0.

Thenthereexists ∈E(X,ω)suchthatµ=(ω+ddc )nifandonlyifµdoesnotchargepluripolarsets.

Proof.Wecanassumewithoutlossofgeneralitythatµandωarenormalized sothatµ(X)=Xωn=1.Consider,forA>0,

CA(ω):={νprobabilitymeasure/ν(K)≤A·Capω(K),forallK X},whereCapωdenotestheMonge-Amp`erecapacityintroducedbyE.BedfordandA.Taylorin[BT](see[GZ1]forthiscompactsetting).Recallthat (ω+ddcu)n/u∈PSH(X,ω),0≤u≤1.Capω(K):=sup

K

Indeed, xν∈CA(ω),0<p<1,andωj:=ω+εj ,where isak¨ahlerfo …… 此处隐藏：15702字，全部文档内容请下载后查看。喜欢就下载吧 ……