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Holder continuity of green's functions

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Title:
Holder continuity of green's functions
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Tookos, Ferenc
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University of South Florida
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Subjects / Keywords:
Logarithmic capacity
Newtonian potential
Equilibrium measure
Boundary behavior
Wiener's criterion
Dissertations, Academic -- Mathematics -- Doctoral -- USF   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
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Abstract:
ABSTRACT: We investigate local properties of the Green function of the complement of a compact set$E$. First we consider the case $E\subset 0,1$ in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the H\"older-$1/2$ condition locally at the origin, then the density of $E$ at $0$, in terms of logarithmic capacity, is the same as that of the whole interval $0,1$. We give an integral estimate on the density in terms of the Green function, which also provides a necessary condition for the optimal smoothness. Then we extend the results to the case $Esubset -1,1. In this case the maximal smoothness of the Green function is "older-1 and a similar integral estimate and necessary condition hold as well.In the second part of the paper we consider the case when $E$ is acompact set in R, > 2. We give a Wiener type characterization for the "older continuity of the Green function, thus extending a result of L.
Thesis:
Thesis (Ph.D.)--University of South Florida, 2004.
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Includes bibliographical references.
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by Ferenc Tookos.
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Holder continuity of green's functions
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ABSTRACT: We investigate local properties of the Green function of the complement of a compact set$E$. First we consider the case $E\subset [0,1]$ in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the H\"older-$1/2$ condition locally at the origin, then the density of $E$ at $0$, in terms of logarithmic capacity, is the same as that of the whole interval $[0,1]$. We give an integral estimate on the density in terms of the Green function, which also provides a necessary condition for the optimal smoothness. Then we extend the results to the case $Esubset [-1,1]. In this case the maximal smoothness of the Green function is "older-1 and a similar integral estimate and necessary condition hold as well.In the second part of the paper we consider the case when $E$ is acompact set in R, > 2. We give a Wiener type characterization for the "older continuity of the Green function, thus extending a result of L.
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Adviser: Vilmos Totik, Ph.D.
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Logarithmic capacity.
Newtonian potential.
Equilibrium measure.
Boundary behavior.
Wiener's criterion.
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x Mathematics
Doctoral.
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by FerencTookos Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:VilmosTotik,Ph.D. MouradE.H.Ismail,Ph.D. EvgueniiA.Rakhmanov,Ph.D. BorisShekhtman,Ph.D. DateofApproval: October1,2004 Keywords:logarithmiccapacity,Newtonianpotential,equilibriummeasure,boundarybehavior,Wiener'scriterion WorkwassupportedbyNSFDMS-0097484andNSFDMS-040650 cCopyright2005,FerencTookos

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Iextendmythankstoallthesupervisingcommitteemembers,MouradE.H.Is-mail,EvgueniiA.RakhmanovandBorisShekhtmanforcreatinganexcellentresearchatmosphereinApproximationTheory. IalsothankmyprofessorsattheUniversityofSzegedforprovidingmewithastrongeducation.Specialthanksareduetomyanalysisprofessors,L.Hatvani,J.Heged}us,L.LeindlerandL.Kerchy. Finally,Iowemuchgratitudetomyparentsandmybrotherfortheirunerringsupportfromafar.

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Abstractiii 1Introduction1 2OptimalSmoothnessforE[0;1]5 2.1Notations,Denitions...........................5 2.2Results...................................6 2.3ProofofTheorem2.2.1..........................8 2.4ProofofTheorem2.2.3..........................14 2.5Lemmas..................................18 3AWiener-typeConditioninRd29 3.1Preliminaries...............................29 3.2Results...................................34 3.3ProofofTheorem3.2.1..........................36 Proofofa)inTheorem3.2.1.......................36 Proofofb)inTheorem3.2.1......................40 3.4ProofofLemma3.2.2...........................54 References57 AbouttheAuthorEndPagei

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2.2Themappings 1and 2.........................24ii

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FerencTookos FirstweconsiderthecaseE[0;1]intheextendedcomplexplane.WeextendaresultofV.AndrievskiiwhichclaimsthatiftheGreenfunctionsatisestheHolder-1=2conditionlocallyattheorigin,thenthedensityofEat0,intermsoflogarithmiccapacity,isthesameasthatofthewholeinterval[0;1].WegiveanintegralestimateonthedensityintermsoftheGreenfunction,whichalsoprovidesanecessaryconditionfortheoptimalsmoothness.ThenweextendtheresultstothecaseE[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1].InthiscasethemaximalsmoothnessoftheGreenfunctionisHolder-1andasimilarintegralestimateandnecessaryconditionholdaswell. InthesecondpartofthepaperweconsiderthecasewhenEisacompactsetinRd,d>2.WegiveaWienertypecharacterizationfortheHoldercontinuityoftheGreenfunction,thusextendingaresultofL.CarlesonandV.Totik.Theobtaineddensityconditionisnecessary,anditissucientaswell,providedEsatisestheconecondition.ItisalsoshownthattheHolderconditionfortheGreenfunctionataboundarypointcanbeequivalentlystatedintermsoftheequilibriummeasureandthesolutiontothecorrespondingDirichletproblem.Theresultssolvealongstandingopenproblem-raisedbyMaz'jainthe1960's-underthesimpleconecondition.iii

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SupposethatECisacompactsetwithpositivelogarithmiccapacitycap(E)>0.Let:= Supposethat0isaregularpointofE,i.e.,g(z)iscontinuousat0andg(0)=0.FirstconsiderthecaseE[0;1].ThemonotonicityoftheGreenfunctionyieldsg(z)g

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(c.f.[1,Theorem3.6]).Varioussucientconditionsfor(1.0.1)intermsofmetricpropertiesofEarestatedin[4],wherethereadercanalsondfurtherreferences. TherearecompactsetsE[0;1]oflinearLebesguemeasure0withproperty(1.0.1)(seee.g.[4,Corollary5.2]),hence(1.0.1)mayhold,thoughthesetEisnotdenseat0intermsoflinearmeasure.Onthecontrary,V.Andrievskii[2]provedthatifEsatises(1.0.1)thenitsdensityinasmallneighborhoodof0,measuredintermsoflogarithmiccapacity,isarbitraryclosetothedensityof[0;1]inthatneighborhood,i.e.(1.0.1)implieslimr!0cap(E\[0;r]) 4:(1.0.2) InChapter2wewillproveanintegralestimateforthedensityviatheGreenfunction,fromwhich(1.0.2)easilyfollows. AndrievskiialsoconstructedaregularcompactsetE[0;1]suchthatlimr!0g()]TJ/F4 11.95 Tf 9.3 0 TD[(r) 2 holdsbutliminfr!0cap(E\[0;r]) Furthermoreheprovedthatconversely,(1.0.2)doesnotimply(1.0.1). Nowlet'sturntothecaseE[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1].Inthiscaseg(ir)g

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becauseg(x+iy)ismonotoneiny.Aswewillsee,thenecessaryconditionfortheoptimalsmoothnesscanbegeneralizedtothiscase,aswell. LetusconsidernowthemoregeneralsettingwhenEisanarbitrarycompactsubsetofC.Assumethat0isaboundarypointof.Severalequivalentconditionsareknownfortheregularityof0(seee.g.([13,AppendixA2.]).OneofthemisduetoWiener.ItcharacterizestheregularitywiththecapacityofthesetsEn=E\( withsomepositivenumbersC;. For">0setNE(")=fn2N:cap(En)"2)]TJ/F6 7.97 Tf 6.58 0 TD[(ng;(1.0.7) andwesaythatasubsequenceN=fn1
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Totiksuggestedthattheseresultscouldbeextendedtothehigherdimensionalcase,i.e.whenERd.ForthiscaseaWienertypeconditionlikeinTheorem1.0.2wasalreadydenedbyMaz'ja(see[7]-[10]).Maz'japroveditssuciencyfortheHoldercontinuityofthesolutiontotheDirichletproblemandshowedthatingeneralitisnotnecessary.InChapter3wewillprovethesuciencyofthisconditionfortheHoldercontinuityoftheGreenfunctionandshowthatitisalsonecessaryprovidedEsatisestheconecondition.Wealsogiveanequivalentcharacterizationintermsoftheequilibriummeasure.Inotherwords,undertheconeconditionwecompletelycharacterizeHoldercontinuity,whichhasbeenalongstandingopenproblem.4

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Forthenotionsoflogarithmicpotentialtheoryseee.g.[12]or[13].InwhatfollowsEdenotestheequilibriummeasureofE,U(z):=Zlog1 thelogarithmicpotentialofthemeasure,gG(z;a)theGreenfunctionofthedomainGwithpoleata,!(x;H;G)theharmonicmeasureinGcorrespondingtothesetH@G.WeshallfrequentlyusetherelationgCnE(z)=log1 cap(E))]TJ/F4 11.95 Tf 11.95 0 TD[(UE(z);z2CnE(2.1.1)5

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LetGbeadomainwithcompactboundaryandwithcap(@G)>0,andletbeameasuresupportedon forz2@Gwiththeexceptionofasetofcapacity0.ForregularGtheexceptionalsetisempty.IfGisbounded,thentheconstantis0([13,Ch.II,Theorem4.1]),andifGisunbounded,thenitis([13,Ch.II,Theorem4.4])const=ZGgG(a;1)d(a):(2.1.3) WeshallusethenotationBal;Gforthebalayagemeasure Thereisaconnectionbetweenharmonicandbalayagemeasures:ifK@Garecompactsets,thenforx2GtheequalityBalx;G(K)=!(x;K;G)(2.1.4) holds,wherexdenotesthepointmass(Diracmeasure)placedatthepointx(seee.g.[13,AppendixA3,(3.3)]).Therefore,inwhatfollowsweshallinterchangeablyusetheharmonicmeasureandbalayagenotations. Recallthatcap(I)=jIj=4foranyintervalI,wherejIjdenotesthelength(Lebesguemeasure)ofI.6

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4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) Sincecap(E")jE"j=4,condition(2.2.7)issomewhatweakerthanZ101 4)]TJ 13.15 8.08 TD[(jE"(t)j whichisknowntobesucientfor(1.0.1)(see[4,Theorem2.1]). Andrievskii'stheoremisaconsequenceofCorollary2.2.2(seeLemma2.5.1). Condition(2.2.7)isnotsucientforHoldercontinuity,itdoesnotimply(1.0.1).Indeed,letPk2k=1butPk3k<1,andconsiderasetEoftheformE=[0;1]n1[k=1((1)]TJ/F4 11.95 Tf 11.95 0 TD[(k)2)]TJ/F6 7.97 Tf 6.58 0 TD[(nk;2)]TJ/F6 7.97 Tf 6.59 0 TD[(nk) withsomeveryfastincreasingsequencefnkg(saynk+1>k2nk).Onecanverifythatforthisset(2.2.8)istruebecauseofPk3k<1,butitwasshownin([4,Corollary3.3])that(1.0.1)doesnothold,duetoP2k=1. ThemethodusedintheproofofTheorem2.2.1canbeappliedtothecaseE[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]aswell.ThehighestsmoothnessoftheGreenfunctionattheorigin(Lipschitz7

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4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 2:(2.2.11)

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4)]TJ/F1 11.95 Tf 13.14 8.08 TD[(cap(Fj) Fortheproofrstofallnoticethat(1.0.1)impliesE[[0;r]([0;r])C2g()]TJ/F4 11.95 Tf 9.3 0 TD[(r);00(recallthatE[[0;r]denotestheequilibriummeasureofE[[0;r]).Thisisimmediate,since(see(2.1.1))g()]TJ/F4 11.95 Tf 9.29 0 TD[(r)g cap(E[[0;r]))]TJ/F4 11.95 Tf 11.95 0 TD[(UE[[0;r]()]TJ/F4 11.95 Tf 9.29 0 TD[(r)=UE[[0;r](0))]TJ/F4 11.95 Tf 11.96 0 TD[(UE[[0;r]()]TJ/F4 11.95 Tf 9.3 0 TD[(r)=Zlogt+r tdE[[0;r](t)log2Zr0dE[[0;r](t)=(log2)E[[0;r]([0;r]): 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(Fj)

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andtherelation(2.3.13)follows.

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4C1p 4C1p 4C1p Nowsince[0;r]isoutsideDj,Bal[0;1]

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Fory62E[[0;r]thequantityBaly;Cn(E[[0;r])([0;r])=!y;[0;r];Cn(E[[0;r]) withsomeabsoluteconstantc1>0becausedist(Jj;0)dist(Jj;[0;1])jIjj=dj.By([13,Ch.II,(4.47)])wehaveBal)]TJ/F6 7.97 Tf 6.58 0 TD[(dj;Cn(E[[0;r])([0;r])Bal)]TJ/F6 7.97 Tf 6.58 0 TD[(dj;Cn[0;1]([0;r])(2.3.19)=1

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"1 4)]TJ/F1 11.95 Tf 13.14 8.08 TD[(cap(E"(qm)) "1 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(qjM+l)) ButE"(t)=[0;"u=2][(E"(u)\["u=2;u])[[u;t],i.e.E"(t)isobtainedfromE"(t)\["u=2;u]byattachingone-oneintervalstotherightandtotheleft.Therefore,wecanapplyLemma2.5.4below(2.5.42),twicetoconcludecap(E"(u)\(["u=2;u])) 4)]TJ/F1 11.95 Tf 13.14 8.09 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.14 8.09 TD[(cap(E"(u)) Thisistrueforallutu(1)]TJ/F4 11.95 Tf 12.15 0 TD[("=2)=(1)]TJ/F4 11.95 Tf 12.14 0 TD[("),thereforeifwesquarebothsides,13

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4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(E"(t)) 1)]TJ/F4 11.95 Tf 11.95 0 TD[("1 4)]TJ/F1 11.95 Tf 13.14 8.08 TD[(cap(E"(u)) Letkbethelargestintegerforwhichqk>2r ".Summingup(2.3.24)forthevaluesu=q;q2;q3;:::;qkandmakinguseof(2.3.21)weobtainZ1qk1 4)]TJ/F1 11.95 Tf 13.14 8.09 TD[(cap(E"(t)) "1 "1)]TJ/F6 7.97 Tf 13.3 4.7 TD[(" "; ".Then,changing4r "forrwecanuseHarnack'sinequalitytoobtaing()]TJ/F4 11.95 Tf 10.49 8.09 TD[("r WearegoingtousethenotationsofStepI.Insteadof(2.3.14)nowwehaveE[[0;r]([0;r])C2g(ir);0
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tdE[[0;r](t)logp holdswithsomepositiveconstantc. Now(c.f.(2.3.16))wehaveBal[)]TJ/F8 7.97 Tf 6.58 0 TD[(1;1] sinced[)]TJ/F8 7.97 Tf 6.58 0 TD[(1;1](t)=1 In(2.3.19)weused)]TJ/F6 7.97 Tf 6.59 0 TD[(dj.Now,since)]TJ/F4 11.95 Tf 9.3 0 TD[(djmaybeinE,letuschangeitforidj.ByHarnack'sinequalitywehavefory2JjBaly;Cn(E[[0;r])([0;r])c1Balidj;Cn(E[[0;r])([0;r])c1Balidj;Cn[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]([0;r])=c1!idj;[0;r];Cn[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]:(2.4.28) Applyingthetransformation'(z)=z)]TJ 12.55 9.83 TD[(p

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2 22)]TJ/F8 7.97 Tf 6.59 0 TD[(arccosrZ3 1)]TJ/F1 11.95 Tf 11.95 0 TD[(2(q dj(q 1)]TJ/F1 11.95 Tf 11.96 0 TD[(2(q 1)]TJ/F1 11.95 Tf 11.95 -0.01 TD[((q 1)]TJ/F1 11.95 Tf 11.96 0 TD[((p

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21 4r dj: 1222jr; TheproofofCorollary2.2.5isimmediatefromLemmas2.5.1and2.5.4.Firstofall,Lemma2.5.1implies(1.0.2)andlimr!0cap(E\[)]TJ/F4 11.95 Tf 9.3 0 TD[(r;0]) 4: 2:(2.4.29) Next,takingI=[0;r],J=[)]TJ/F4 11.95 Tf 9.3 0 TD[(r;0],F=E\[0;r]andG=E\[)]TJ/F4 11.95 Tf 9.29 0 TD[(r;0]in(2.5.41)wecaninfercap(E\[)]TJ/F4 11.95 Tf 9.29 0 TD[(r;r]) cap(E\[)]TJ/F4 11.95 Tf 9.3 0 TD[(r;0])[[0;r]4cap(E\[0;r]) Finally,(2.2.11)isadirectconsequenceof(2.4.29)and(2.4.30). 17

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4)]TJ/F1 11.95 Tf 13.15 8.51 TD[(cap(E"=2(u)) 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) log(1+)Ztt=(1+)1 4)]TJ/F1 11.95 Tf 13.14 8.52 TD[(cap(E"=2(u)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 4:(2.5.31) Nowletftngbeanarbitrarypositivesequencetendingto0andsetFn=E"(tn)=tn;n=1

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cap(Fn)(1+")log4;x2Fn;holds,whichimpliesfornmax(n0;n1)Un(x)1 1)]TJ/F1 11.95 Tf 11.95 0 TD[(2p cap(Fn\([";1)]TJ/F4 11.95 Tf 11.95 0 TD[("])ZUndn1 1)]TJ/F1 11.95 Tf 11.95 0 TD[(2p 4(1+")=(1)]TJ/F8 7.97 Tf 6.59 0 TD[(2p Intheprecedingargumentweusedthatasn!1,wehaveFn![0;1]intheweaktopologyonmeasures.Infact,letbeaweaklimitofsomesubsequence,sayFnl!asl!1.Thenissupportedin[0;1],hastotalmass1,andallwehavetoshowisthat=[0;1].WeknowthatUFn(x)=log1 cap(Fn)(2.5.32) forx2Fnwiththeexceptionofasetofcapacity0,andthesameistruefor[0;1].SinceFn[0;1],itfollowsthatUFn(x)U[0;1](x)+logcap([0;1]) cap(Fn) forx2Fnwiththeexceptionofasetofcapacity0,andsinceeverysetofzerocapacityhaszeroFn-measure(see[13,RemarkI.1.7,p.28]),itfollowsthatthisinequalityis19

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However,thefunctionU[0;1](x))]TJ/F4 11.95 Tf 11.95 0 TD[(U(x) vanishesatinnity,soitisharmonicthere,andanappealtotheminimumprincipleonthedomain 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(F) 2 Proof.Itisenoughtoshowthatif1=8andH=fx2[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]:!(x;T;DnF)>2=3g;

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Letthusbe1=8.Supposetothecontrarythat(2.5.33)isnottrue,i.e.jHj<4.Thencap([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH)j[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nHj LetD2denotetheopendiskabouttheoriginandofradius2.Ifx2[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH,wehave!(x;@D2;D2nF)!(x;T;DnF)<2 cap(F)+log3; cap(F))]TJ/F4 11.95 Tf 11.95 0 TD[(UF(x)2=log1 cap([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nH))]TJ/F4 11.95 Tf 11.95 0 TD[(U[)]TJ/F13 5.98 Tf 5.76 0 TD[(1;1]nH(x)+2: cap(F)log1 cap([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH)+2;21

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2)]TJ/F1 11.95 Tf 11.95 0 TD[(cap(F)1 2)]TJ/F1 11.95 Tf 11.95 0 TD[(cap([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH)e)]TJ/F8 7.97 Tf 6.59 0 TD[(2<1 2)]TJ/F1 11.95 Tf 13.15 8.09 TD[(1)]TJ/F1 11.95 Tf 11.95 0 TD[(2 2(1)]TJ/F4 11.95 Tf 11.96 0 TD[(e)]TJ/F8 7.97 Tf 6.58 0 TD[(2)+2

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2ZJ001)]TJ/F4 11.95 Tf 11.95 0 TD[(y2 2ZA001)]TJ/F4 11.95 Tf 11.96 0 TD[(y2 whichisclearwithsomec">0,sincey2[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]andonthetwosidesduringintegrationrunstroughtwoarcsofcomparablelengthbothofwhichlieofdistanced"=4from[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1].Thus,(2.5.36)istruewithsomec">0,andthisgives(2.5.35). Nextweturntothegeneralcase,i.e.when[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nF=[)]TJ/F1 11.95 Tf 9.3 0 TD[(1+";1)]TJ/F4 11.95 Tf 12.92 0 TD[("]nFisanarbitraryopenset.Sincetheconstantc"shouldbeindependentofthesetF(dependingonlyon"with[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;)]TJ/F1 11.95 Tf 9.29 0 TD[(1+"][[1)]TJ/F4 11.95 Tf 11.26 0 TD[(";1]F),withoutlossofgeneralitywemayassumeFtoconsistofnitelymanyintervals,inwhichcase[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nFconsistsofnitelymanyopenintervals,sayI1;:::;Im.23

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Weshowthattheconstantc"veriedaboveforthespecialcasewhen[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nFwasaninterval,isappropriate.Tothisend,startingfrom0=x,wesuccessivelydenethemeasuresnbyn+1=Baln;Dn([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nIjn);

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NowBalx;DnFistheuniquemeasuresupportedonT[Fwhichhasmass1anditslogarithmicpotentialislog1=jz)]TJ/F4 11.95 Tf 12.29 0 TD[(xj,thustheproofwillbecompleteifweshowthathasmass1,i.e.(T[F)=1,whichisthesameaslimn!1n([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nF)=0 whichwewantedtoproveanyway.Thiswillbedonebyshowingthatineachstep25

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LetIj=[aj;bj],andlet>0besosmallthatalltheintervals[aj)]TJ/F4 11.95 Tf 12.41 0 TD[(;aj]and[bj;bj+]arepartof()]TJ/F1 11.95 Tf 9.29 0 TD[(1;1)andtheyaredisjoint.ForI=Ijnandy2IthevalueBaly;Dn([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nI)([ajn)]TJ/F4 11.95 Tf 11.95 0 TD[(;ajn][[bjn;bjn+]); withsomepositiveconstantfollowsfromthefactthatherethelefthandsideis!(z;A)]TJ/F2 11.95 Tf 9.74 1.79 TD[([A+;D)=1 2ZA)]TJ/F9 7.97 Tf 6.25 1.07 TD[([A+1)]TJ/F4 11.95 Tf 11.95 0 TD[(z2 Weobtainfrom(2.5.40)Baly;Dn([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nI)(F)Baly;Dn([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nI)([ajn)]TJ/F4 11.95 Tf 11.95 0 TD[(;ajn][[bjn;bjn+]);26

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4cap(I[G):(2.5.41)

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cap(F))]TJ/F1 11.95 Tf 11.96 0 TD[(log4 TheanalogousformulaforF[GandI[Greadsaslog1 cap(F[G))]TJ/F1 11.95 Tf 11.95 0 TD[(log1 cap(I[G)=Z(I[G)n(F[G)g whereweusedthat(I[G)n(F[G)=InF,sotheintegrationisoverthesamesetontherighthandsidesof(2.5.43)and(2.5.44).SincethemeasureIisthebalayageofI[GontoI(see[13,TheoremIV.1.6,(e)]),wehaveonInFtheinequalitydI[G(a)dI(a).Atthesametimeg cap(F[G))]TJ/F1 11.95 Tf 11.95 0 TD[(log1 cap(I[G)log1 cap(F))]TJ/F1 11.95 Tf 11.95 0 TD[(log4 28

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ForthenotionsofclassicalpotentialtheoryinRdseee.g.[6].TheNewtonianpotentialofthemeasureisdenedasU(x):=Z1 infI(); whereSristhe(d)]TJ/F1 11.95 Tf 11.95 0 TD[(1)-dimensionalnormalizedsurfaceareameasureonSr.29

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whereq.e.means\quasi-everywhere",i.e.withtheexceptionofasetofzerocapacity. IfEisofpositivecapacity,thenEhasniteenergy.HenceasetofzerocapacityhaszeroE-measure,andsoifapropertyholdsquasi-everywhere,i.e.withtheexceptionofasetofzerocapacity,thenitalsoholdsE-almosteverywhere. IfisameasuresupportedonthecompactsetFandU(x)1forallx2Rd,thenthesetK:=fx:U(x)g(3.1.3) hascapacityatmost(1=)cap(F).Infact,ifKisofpositivecapacity,thentheinequalityU(x) cap(F)UK(x) cap(K)+ cap(K) holdstrueforquasi-everyx2K.HencethisistrueforK-almostallx,andthentheprincipleofdomination([6,Theorem1.27])givesthesameinequalityforallx2Rd.Nowcap(K)1 followsifweletxtendtoinnity. Weshallalsoneedthefollowingresult.ThereisapositiveconstantcsuchthatifAS1and(A)denotesthe(d)]TJ/F1 11.95 Tf 11.96 0 TD[(1)-dimensionalsurfaceareameasureofAthen(A)cp Indeed,ifdenotesthenormalizedsurfaceareameasureonS1thenbasedonthedenitionofcapacity:1 cap(A)1

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ZS1ZS11 k ForregularGtheexceptionalsetisempty.IfGisbounded,then Gandleavetherestofunchanged.InthissenseifG1G2,thentakingbalayageoutofG2canbedoneintwosteps:rsttakebalayageoutofG1,andthentakethebalayageoftheresultingmeasureoutofG2. Perhapsthemostimportantconnectionbetweenequilibriumandbalayagemea-suresisthefactthatifEFarecompactsetsofpositivecapacity,thenEisthebalayageofFontoE(i.e.outoftheunboundedcomponentof IfK@Garecompactsetsofpositivecapacity,thentheharmonicmeasure!(x;K;G)istheuniquesolutionofthegeneralizedDirichlet-probleminGcorre-spondingtothecharacteristicfunctionofKin@G.Thereisaconnectionbetween31

PAGE 38

holds,whereadenotesthepointmass(Diracmeasure)placedatthepointaand Green'sfunctionofGwithpoleaty2GisdenedasgG(x;y)=Uy(x))]TJ/F4 11.95 Tf 11.95 0 TD[(U x(y) wherey2SrandSristhenormalizedsurfaceareameasureonSr.Indeed,Poisson'sformula(seee.g.[3,Section1.3,(1.3.1)])givesd x(y) x(y) MultiplyingbyRd)]TJ/F8 7.97 Tf 6.59 0 TD[(2andlettingR!1wegetthat

PAGE 39

foranyalyingintheunboundedcomponentofRdnEwithsomeconstantsca,Ca. LetbeameasureonSr.ThelowerRadon-Nikodymderivative(density)ofwithrespecttonormalizedsurfaceareameasureonSrisdenedasfollows(seee.g.[5,Chapter3]or[11,ChapterVII]).Letx02Srand0<<1.ThentheconeC(x0;):=fx2Rd:hx;x0i Finally,letusrecallthattheNewtoniancapacityissubadditive:ifF=[ki=1Fi,thencap(F)kXi=1cap(Fi):(3.1.11) Inparticular,oneofthesetsFimusthavecapacitycap(F)=k.Ontheotherhand,ifthedistancebetweenthesetsF1andF2isatleastl,thencap(F1[F2)cap(F1)+cap(F2) 1+2cap(F1)cap(F2) Indeed,set=1)]TJ/F4 11.95 Tf 11.95 0 TD[(t

PAGE 40

LetBr=Br(0)betheballofradiusrabouttheorigin,andweshalldenoteitsclosureby

PAGE 41

withsomepositivenumbersC;. Followingthedenitionsin[4],for">0setNE(")=fn2N:cap(En)"2)]TJ/F6 7.97 Tf 6.58 0 TD[(n(d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)g;(3.2.15) andwesaythatasubsequenceN=fn10,whichmeansthatcontainsaconewithvertexat0.Theorem3.2.1a)

PAGE 42

If,inaddition,Gsatisestheconeconditionat0,then1)-2)arealsoequivalentto3) LetFbeacompactsetsuchthat0isontheboundaryoftheunboundedcomponentofRdnF,andletbdenotethebalayageofsomemeasureoutofRdn(F[ 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.09 TD[("

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B1+ F; Thelefthandsideiswhatisontheleftof(3.3.18),andsinceS8=S8=8d)]TJ/F8 7.97 Tf 6.59 0 TD[(2,and Foreverya2S8(3.1.9)withr=1andR=8showsthat 9d)]TJ/F8 7.97 Tf 6.58 0 TD[(1S1>1 9dS1; 9dS1;(3.3.20) andwehavetoestimatehowmuchofcS8goesontoF.SinceweassumedF2=F[ 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(2F2(F):(3.3.21) ThedistanceofthesetsF 1+2cap(F)cap( 3d)]TJ/F13 5.98 Tf 5.75 0 TD[(2(cap(F)+cap( 1+2 3d)]TJ/F13 5.98 Tf 5.75 0 TD[(2cap(F);

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3d)]TJ/F13 5.98 Tf 5.76 0 TD[(24"1+": 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(2" 8d)]TJ/F8 7.97 Tf 6.59 0 TD[(2S1)]TJ/F1 11.95 Tf 20.98 8.09 TD[(1 8d)]TJ/F8 7.97 Tf 6.59 0 TD[(2" 8d)]TJ/F8 7.97 Tf 6.59 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.09 TD[(" Weshalluse(3.3.18)inascaledform,namelyifEiscompact,0isontheboundaryoftheunboundedcomponentofRdnEandcap(E\( thenwehave[S2)]TJ/F12 5.98 Tf 5.76 0 TD[(n 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" wherenowbdenotesbalayageoutofRdn(E[ AfterthispreparationletusreturntothesetNE(")whichwasassumedtobeofpositivelowerdensity.Thenthereisan>0suchthatforlargeNthesetNE(")hasatleastNelementssmallerthanN.ForlargeNthenwecanselectasubsetKNE(")\f2;:::;N)]TJ/F1 11.95 Tf 11.95 0 TD[(2g38

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WesetEn=E[ Firstnotethat0 S2)]TJ/F12 5.98 Tf 5.75 0 TD[(n=c1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2S2)]TJ/F12 5.98 Tf 5.75 0 TD[(n)]TJ/F13 5.98 Tf 5.76 0 TD[(1:(3.3.24) Ontheotherhand,ifn=ni)]TJ/F1 11.95 Tf 10.82 0 TD[(1,withni2K,then(3.3.22)istrue,henceforsuchnwehave(3.3.23).Again,themeasuren+3isthebalayageofnontoEn+3,andintakingthisbalayagewesweepoutonlythepartofnthatissittingonS2)]TJ/F12 5.98 Tf 5.75 0 TD[(nnE.Thus,ifbdenotesthebalayageontoEn+3,thenn+3 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.08 TD[("

PAGE 46

8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2N)]TJ/F8 7.97 Tf 6.59 0 TD[(3kS2)]TJ/F12 5.98 Tf 5.76 0 TD[(N(S2)]TJ/F12 5.98 Tf 5.75 0 TD[(N)1 2N(d)]TJ/F8 7.97 Tf 6.59 0 TD[(2)1)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" 2N(d)]TJ/F8 7.97 Tf 6.59 0 TD[(2)1)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" 2N(d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)1)]TJ/F4 11.95 Tf 15.77 8.08 TD[(" andisdenedbytheequation2)]TJ/F6 7.97 Tf 6.58 0 TD[(=1)]TJ/F4 11.95 Tf 15.76 8.08 TD[("

PAGE 47

Fornotationalconveniencelet1=B2)]TJ/F13 5.98 Tf 5.75 0 TD[(1,L=B2)]TJ/F12 5.98 Tf 5.75 0 TD[(L,L+1=B2)]TJ/F12 5.98 Tf 5.76 0 TD[(L)]TJ/F13 5.98 Tf 5.76 0 TD[(1,3=2=B32)]TJ/F12 5.98 Tf 5.75 0 TD[(L)]TJ/F13 5.98 Tf 5.75 0 TD[(2,andlettheboundingsurfacesoftheseballsbedenotedbyT1,TL,TL+1andT3=2,respectively.SetalsoF3=2=F\ isthebalayageoutofRdn WestartfromtherepresentationF=L+1[j=2F\( Nowleta21n WithoutlossofgeneralitywemayassumethatFisofpositivecapacity(otherwise41

PAGE 48

F=ZU Weshallneedasimilarreasoningforthebalayage=a:=eaofaoutof(Rdn Thus,ifK:=a:UF(a)1 thenfora62Kwehave(3.3.29),andifK3=2:=a:UF3=2(a)1 thenfora2TLnK3=2wehave(3.3.30).ForthecapacityofKwegetfrom(3.1.3){42

PAGE 49

IfLdenotesthe(d)]TJ/F1 11.95 Tf 10.85 0 TD[(1)-dimensionalnormalizedsurfaceareameasureonTL,thenby(3.1.5)wehaveL(K\TL)cp AnidenticalinequalityistrueforK3=2(c.f.(3.3.34)):L(K3=2\TL)c2L=2p Leta;b2TL,andletea,ebbethebalayageofa;boutofthedomainRdn(F3=2[ 1)]TJ/F1 11.95 Tf 11.95 0 TD[(3=4db=7db;

PAGE 50

Chooseandxab2TLnK3=2(see(3.3.32)).By(3.3.36)if"issucientlysmallcomparedtoL,thenthereissuchab.Inthiscase(3.3.30)giveseb(F3=2)1=L,hencethebalayage:= thenL+1(H)1 foralla2TLandallx2TL+1nH. Nextconsiderthebalayagea:= c 2L+1d)]TJ/F8 7.97 Tf 6.58 0 TD[(23=2L+2+1=2L+1 c

PAGE 51

Inasimilarfashionweobtainfora:= (2L)]TJ/F1 11.95 Tf 11.95 0 TD[(1)d)]TJ/F8 7.97 Tf 6.58 0 TD[(1ba(T1)c0(2L+1) (2L)]TJ/F1 11.95 Tf 11.96 0 TD[(1)d)]TJ/F8 7.97 Tf 6.59 0 TD[(1:(3.3.41) Nownotethata+a+a+ a(y) (2L)]TJ/F1 11.95 Tf 11.95 0 TD[(1)d)]TJ/F8 7.97 Tf 6.59 0 TD[(1; 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjd)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 Thisderivationusedtheexistenceofb2TLnK3=2,anditisvalidif"issucientlysmallcomparedtoL(see(3.3.36)).StepII Let>0.SupposethatisameasureonTLsuchthatd(y) whereHTLisof(normalizedsurfacearea)measureatmost.Letbbethebalayageofoutof(1n

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2d)]TJ/F8 7.97 Tf 6.59 0 TD[(2(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)fory2TL+1nH;(3.3.44)whereHisasetof(normalizedsurfacearea)measureatmostandc2>0isaconstantdependingonlyond. Firstofallnotethatwehave(3.3.42)fora2TLnKandy2TL+1nH,whereHisthexedsetdenedin(3.3.38),andalsonotethattheintegraloverTL+1ofthersttermontherightof(3.3.42)withrespecttodLisZTL3 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjddL(a)=ZTLd a(y) L(y) TL(y) 2(L+1)(d)]TJ/F8 7.97 Tf 6.59 0 TD[(2)dTL+1=1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2; 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjd)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)]TJ/F4 11.95 Tf 17.22 8.09 TD[(c1 Wewritewitha2TLandy2TL+1db(y) 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjddL(a):

PAGE 53

2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2(1)]TJ/F1 11.95 Tf 11.96 0 TD[(2d);(3.3.46) providedc1 ThisconditionshouldbeunderstoodinthesensethatrstwechooseLlargeenough,thenforxedLchoose">0smalltosatisfy(3.3.47).Furthermore,assumingthiscondition,HhasmeasureatmostL+1(H)1 Thus,withsuchachoiceforLand"theestimate(3.3.44)holdswithc2=2d.StepIII Letc3=4>>0,wherec3isaconstanttobechosenlater,andsupposethatisameasureonTLsuchthatd(y) whereH0isof(normalizedsurfacearea)measureatleastc3)]TJ/F4 11.95 Tf 12.19 0 TD[(.Thus,weconsider47

PAGE 54

Let,asbefore,bbethebalayageofoutof(1n whereHisasetof(normalizedsurfacearea)measureatmostandc4dependsonlyond. ForaproofjustfollowtheproofinstepII.Wehave(3.3.42)fora2TLnKwithHgivenin(3.3.38),andnotethat3 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjd3 22d: 22d)]TJ/F4 11.95 Tf 17.23 8.08 TD[(c1 NotethatbothoftheseconditionsaresatisedifLissucientlylargeand"issucientlysmall. 48

PAGE 55

kxk1)]TJ/F4 11.95 Tf 11.96 0 TD[(g:(3.3.51) Considerthedomain(B2n Let0insuchawaythatalltheestimatesinstepsII{IVhold. LetEn=E[

PAGE 56

WewanttocompareAn+1withAnfornL.Let Inacompletelysimilarmanner,ifn21,thenAnisofthesecondtypewhileAn+1isofthersttype,i.e.vn(y)Anforally2S2)]TJ/F12 5.98 Tf 5.75 0 TD[(n\Cwiththeexceptionofasetofmeasureandvn+1(y)An+1forally2S2)]TJ/F12 5.98 Tf 5.76 0 TD[(n)]TJ/F13 5.98 Tf 5.75 0 TD[(1withtheexceptionofasetofmeasure.Nowinsteadof(3.3.44)weapply(3.3.50)toconcludethatAn+1c4An. Finally,ifn22,nL,thenAn+1isdenitelyofthesecondtype,butAn50

PAGE 57

Insummary,wehaveAn+1(1=2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)(1)]TJ/F4 11.95 Tf 12.44 0 TD[(c2)Anforn20,An+1c4Anforn21andAn+1(c3c=2)Anforn22.Ifs=sNdenotesthenumberofelementsofNE(")[f0gnotlargerthanN,thenthereareatmostselementsof1andatmostsLelementsof2notlargerthanN.Thus,wecanconcludeAN+11 2d)]TJ/F6 7.97 Tf 6.59 0 TD[(sN(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)N(c4)sNc3c ForallsuchNwecanconcludethatAN+1(2=c3)(1=2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)N(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)2N; 2d)]TJ/F8 7.97 Tf 6.59 0 TD[(2N(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)2N(3.3.54) because,independentlyifANisoftherstorsecondtype,wehavevN(y)ANonasetofmeasureatleastc3)]TJ/F4 11.95 Tf 11.95 0 TD[(c3=2. Nowwecaneasilycompletetheproof.LetN+1betheunboundedcomponentofRdnEN+1.ConsiderGreen'sfunctionwithpoleaty02N+1andintegrateitoverthesphereSrwithr=rN=2)]TJ/F6 7.97 Tf 6.58 0 TD[(N.ZSrgN+1(x;y0)dSr(x)=ZSr)]TJ/F4 11.95 Tf 5.48 -9.68 TD[(gN+1(x;y0))]TJ/F4 11.95 Tf 11.95 0 TD[(gN+1(0;y0)dSr(x)=ZSr)]TJ/F4 11.95 Tf 5.48 -9.68 TD[(Uy0(x))]TJ/F4 11.95 Tf 11.95 0 TD[(Uy0(0)dSr(x)51

PAGE 58

whereedenotesthebalayageoutofN+1. Heretherstintegrandis1 wherec6dependsonlyony0andd.ForthesecondintegralwehaveZSrUfy0(0))]TJ/F4 11.95 Tf 11.95 0 TD[(Ufy0(x)dSr(x)=ZEN+11 SinceZSr1 provided<(1)]TJ 11.95 9.89 TD[(p

PAGE 59

log2log1 1)]TJ/F4 11.95 Tf 11.96 0 TD[(c2: TheproofaboveusedthatEiscontainedintheunitballandomitstheconeC2.Thegeneralcasecanbesimilarlyhandled.Infact,letcontaintheconeC=C(x0;2;`).SelectasphereSr0,r0
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Conversely,suppose2).Letjxj=rbesmall,andE=EnB2r.Let a(y)):

PAGE 61

Nextweshowthat3)implies1).LetRbesolargethat@GBRandconstructadomainTinthefollowingway.Ifa62 Itislefttoshow1))3undertheconecondition.Undertheconecondition1)impliesthepositivelowerdensityofNE(")forsome">0,i.e.thereisanandanN1suchthatjNE(")\f0;1;:::;Ngj4NforNN1.ThenforlargeN,sayforNN2,wealsohavejNE(")\f[(2)N]+1;[(2)N]+2;:::;MgjM

PAGE 62

56

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L.V.Ahlfors(1973):ConformalInvariants.NewYork:McGraw-Hill.[2] V.V.Andrievskii,ThehighestsmoothnessoftheGreen'sfunctionimpliesthehighestdensityoftheset(manuscript)[3] D.H.Armitage,S.J.Gardiner,ClassicalPotentialTheory,Springermonographsinmathematics,Springer-Verlag,London,2001.[4] L.Carleson,V.Totik,HoldercontinuityofGreen'sfunctions,ActaSci.Math.,Szeged,toappear[5] L.L.Helms,IntroductiontoPotentialTheory,Pureandappliedmathematics,XXII,Wiley-Interscience,NewYork,1969.[6] N.S.Landkof,FoundationsofModernPotentialTheory,Grundlehrendermath-ematischenWissenschaften,180,SpringerVerlag,Berlin,1972.[7] V.G.Maz'ja,Regularityattheboundaryofsolutionsofellipticequations,andcomformalmappings,Dokl.Akad.Nauk.SSSR,152,1297-1300(Russian)1963.[8] V.G.Maz'ja,OnthemodulusofcontinuityofthesolutionsoftheDirichletproblemnearirregularboundary,in:Problemsinmathematicalanalysis;Bound-aryproblemsandintegralequations,Izdat.LeningradUniv.,Leningrad,45-48.(Russian)1966.[9] V.G.Maz'ja,Ontheregularityofboundarypointsforellipticequations,in:Investigationsonlinearoperatorsandfunctiontheory;99unsolvedproblems57

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V.G.Maz'ja,Onthemodulusofcontinuityofaharmonicfunctionataboundarypoint,Zap.Nauchn.Sem.Leningrad.Otdel.Mat.Inst.Steklov.(LOMI),135,87-95.(Russian)1984.[11] M.E.Munroe,MeasureandIntegration,Addison-Wesleymathematicsseries,Addison-Wesley,Reading,1959.[12] T.Ransford:PotentialTheoryintheComplexplane.Cambridge:CambridgeUniversityPress,1995.[13] E.B.Sa,V.Totik,LogarithmicPotentialswithExternalFields,GrundlehrendermathematischenWissenschaften,316,Springer-Verlag,NewYork/Berlin,1997.58

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FerencreceivedhisM.S.degreeinMathematicsattheUniversityofSzeged,Hun-garyin2000.HeenteredthePh.D.programinMathematicsattheUniversityofSouthFloridain2001.HiseldisPotentialTheoryandApproximationTheory,hisadvisorwasDr.VilmosTotik.WhileinthePh.D.program,FerenctaughtseveralundergraduatecoursesasaTeachingAssistant. Ferenclikesallkindsofsports,especiallytennis,pingpongandsoccer.