; TeX output 2003.08.26:1356 hGS 9K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen28,2003,279{302$Fq7"V cmbx10NONLINEARHARMONICMEASURESONTREES5B"V cmbx10RobQertTKaufman,Jos«}WeG.Lloren9te,andJang-MeiW uɍdX+UnivĽersityofIllinois,DepartmenĽtofMathematics +1409WN8estGreenStreetUrbana,Illinois61801,U.S.A.;rpk{aufma@math.uiuc.edu DUnivĽersitatAutUonomadeBarcelona,DepartmentdeMatemUatiquesPDES-08193Bellaterra,Barcelona,Spain;gonzalez@mat.uab.esdX+UnivĽersityofIllinois,DepartmenĽtofMathematics51409WN8estGreenStreet,Urbana,Illinois61801,U.S.A.;wu@math.uiuc.edu}D"V UU cmbx10DAbstract.}WN8e%shoĽwthatnonlinearharmonicmeasuresontreeslackmanydesirableprop;Ber- ties ofsetfunctionsencounĽteredinclassicalanalysis.Let <b> UU cmmi10Flb;Beanaveragingop;Beratoron DR=0er U cmmi7and!?F b;Be_Rthe_RF(L-harmonicmeasureona -regularforwĽardbranchingtree.|Unless_RFistheusualaĽverage,N!?F is3nnotaCho;BquetcapacitĽy;iunionofsetsof3n!?FmeasurezerocanhaĽve3np;Bositive3n!?Fmeasure;when;Fisp;BermĽutationinv{ariant;andthereexistsetsoffull;!?F /measurehaving\small"dimension.LetԺAb;BeԺamonotoneoperatoronԺDRf,"thenA -harmonicԺfunctionsontreesneednotob;BeythestrongmaximĽumprincipleunlesstheratiooftheellipticityconstantsiscloseto1 .v썑 K`y cmr10W*exshowthatnonlinearharmonicmeasuresontreeslackmanydesirableprop- ertiesUUofsetfunctionsencounteredinclassicalanalysis.ɍLetb> cmmi10T&bGeadirectedtreewithregular -branching;/inthispapGerwecontinueearlierworkonp -harmonicfunctionsontreesin[CFPR]Eand[KW].W*etreatthewp -LaplacianwasaspGecialcaseofanonlinearaveragingwoperatorwFc:UPR^ 0er cmmi7.!", cmsy10! 3R^ٓR cmr71studiedFbyAlvqarez,RoGdrqguezandY*akubovichFin[AR*Y].ThentheFFc-potentialtheoryonT^%isthediscreteversionofthenonlinearpGotentialtheoryintheEu-clideanspacestructuredonthenonlinearEulerequationofthevqariationalintegralu cmex10R052}>y rsfs10F](x;ru)dxUUwithF(x;h)jhj^pR;UUsee[GLM]and[HKM].Each[JaveragingopGerator[JF٫leadstoaharmonicmeasureontheboundary@ 8TR_ofthetree. >7ExceptwhenбFistheusualaverage,U.thereexistsanumbGerd(;Fc)ostrictlyolessthanthedimensionof@ 8Tc,sothateverycompactseton@ 8Tofedimensione<d(;Fc)musthavezeroeFc-harmonicmeasure,#andthereexistsetsofdimensionsd(;Fc)havingfullFc-harmonicmeasure.IfwecouldshowthateveryBorelseton@ 8T⮫ofdimension<fd(;Fc)haszeroFc-harmonicmeasure,thenִthestatement\thedimensionofִFc-harmonicmeasureisd(;Fc) "wouldE ff xs22000rMathematicsSubvjectClassication:Primary31C05,31C45;Secondary31C20,28A12.RTheKsecondauthorwĽaspartiallysupp;BortedbyagrantfromtheMinisteriodeEducaciUon, Spain.dPĽarttofthisresearchwasstartedwhenthesecondauthorwasvisitingtheUniversityofIllinoisinFN8ebruary2002.cHewishestothanktheDepartmenĽtofMathematicsforhospitalityN8.cTheЍthirdauthorwĽaspartiallysupp;BortedbytheNationalScienceFN8oundationDMS-0070312. * 280Mx!p0J cmsl10R.UUKaufman,J.G.Llorente,andJ.-M.W*uhGfollow;here1dimensionof1Fc-harmonicmeasureisdenedtobGetheinmumofthe dimensionsTofthoseBorelsetsonT@ 8T1havingfullFc-harmonicmeasure.oWhenFisthep -LaplacianopGerator, d(;Fc)isintroGducedin[KW]andusedtostudythe>"sizesofF*atousetsandsetsofniteradialvqariationsforbGounded>"p -harmonicfunctionsontrees.iHereagaind(;Fc)isthecriticaldimensionofF*atousetsforbGoundedUUFc-harmonicUUfunctions.淍W*efialsoprovefithatwhenfiFispGermutationinvqariantandisnottheusualaverage,5thereexisttwosetsonű@ 8Tc,ofzeroűFc-harmonicmeasure,whoseunionhasҸpGositiveҸFc-harmonicmeasure.ThisanswersaquestionraisedbyMartioontrees([AR*Y],[M]).Ourworkismotivqatedby[AR*Y],inwhichitisprovedthatforcertainձFc's,thereexistcongruentsetsB1|s;B2;:::;B \ofarbitrarilysmallpGositiveFc-harmonicmeasure,Twhoseunionis@ 8T.8OurconstructionstartsfromtheroGotofUUthetreeandtheirsstartsfromthebGoundary*.W*ealsoshowthatwhenF/ispGermutationinvqariantandisnottheusualaverage, Fc-harmonic*measureisnotaChoGquetcapacity;!consequently*, it*isnotleftUUcontinuousonincreasingsequencesofsets.WhileOtheseresultsshowthatOFc-harmonicmeasurelacksmanydesirableprop-erties%ofsetfunctionsinthelineartheory*,manyproblemsremain:fsomeconcerninnerJapproximationofBorelsetsbycompactsets,M othersareabGoutthebehaviorofUUmonotonesequencesUUAn ӫsuchthatSލO! cmsy71%19An8=@ 8Tc.Inanotherdirection,westudytheanalogueofthequasilinearellipticequationdiv (Aru)=0O([HKM])#onOtrees.6W*eexaminethenotionofA -harmonicfunctionsontrees,#^whenAisamonotoneopGeratoronR^..ƩW*endthatA -harmonicfunctions3donotalways3satisfythestrongmaximumprincipledenedinSection1;this-showsthatsomecautionisnecessaryinarguingfromellipticopGeratorsintheEuclidean;9spacetopGotentialtheoryontrees.iWhenan;9A -operatorisclosetothep -Laplacian,Gthe]JstrongmaximumprincipleholdsandaF*atoutypGetheoremisvqalid;8our*msucientconditionissharpwhen*mp=2 .czW*e*malsoshowthatthecriticaldimension<d(;A)for<theA -opGeratorapproachesthecriticaldimensiond(;p)forlthelp -LaplacianuniformlyastheellipticityconstantsoflAapproachlthatoftheUUp -Laplacian.Finally*,[we'Ycommentonthemeaningsof'Y1 -Laplacianand1 -Laplacianontrees.Our opGeratorsontreesmaybeconsideredassimpleanaloguesofthe p -Laplacian=div (jruj^p 2Ʒru),w1JE