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Jam6 Bollettino della Unione Matematica Italiana pp

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Niveau: Supérieur, Doctorat, Bac+8
[Jam6] Bollettino della Unione Matematica Italiana (8) 4 (2001) pp 685–702. Harmonic functions on classical rank one balls Philippe JAMING Abstract : In this paper, we study the links between harmonic functions on the hyperbolic balls (real, complex or quaternionic), the euclidean harmonic functions on these balls and pluriharmonic functions under growth conditions. In particular, we extend results by A. Bonami, J. Bruna and S. Grellier (com- plex case) and the author (real case) to the quaternionic case. Keywords : rank one ball, harmonic functions, boundary values. AMS subject class : 48A85, 58G35. 1. Introduction. In this paper, we study the links between harmonic functions on the hyperbolic balls (real, complex or quaternionic), the euclidean harmonic functions on these balls and pluriharmonic functions. In particular we investigate whether growth conditions may separate these classes. More precisely, let F = R,C or H (the quaternions) and let n be an integer, n ≥ 2 (n ≥ 3 if F = R). Let Bn be the euclidean ball in Fn, let ∆ be the euclidean laplacian operator on Bn and let N = r ∂∂r be the normal derivation operator. For k ? N? a function u of class C2k is said to be k-hamonic if ∆ku = 0, in particular for k = 1 this are the euclidean harmonic functions.

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  • invariant laplacian

  • harmonic functions

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  • laplace-beltrami operator

  • df-harmonic function

  • operator can


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[Jam6]BollettinodellaUnioneMatematicaItaliana(8)4(2001)pp685–702.HarmonicfunctionsonclassicalrankoneballsPhilippeJAMINGAbstract:Inthispaper,westudythelinksbetweenharmonicfunctionsonthehyperbolicballs(real,complexorquaternionic),theeuclideanharmonicfunctionsontheseballsandpluriharmonicfunctionsundergrowthconditions.Inparticular,weextendresultsbyA.Bonami,J.BrunaandS.Grellier(com-plexcase)andtheauthor(realcase)tothequaternioniccase.Keywords:rankoneball,harmonicfunctions,boundaryvalues.AMSsubjectclass:48A85,58G35.1.Introduction.Inthispaper,westudythelinksbetweenharmonicfunctionsonthehyperbolicballs(real,complexorquaternionic),theeuclideanharmonicfunctionsontheseballsandpluriharmonicfunctions.Inparticularweinvestigatewhethergrowthconditionsmayseparatetheseclasses.Moreprecisely,letF=R,CorH(thequaternions)andletnbeaninteger,n2(n3ifF=R).LetBnbetheeuclideanballinFn,letΔbetheeuclideanlaplacianoperatoronBnandletN=rrbethenormalderivationoperator.ForkNafunctionuofclassC2kissaidtobek-hamonicifΔku=0,inparticularfork=1thisaretheeuclideanharmonicfunctions.TheballBncanalsobeendowedwiththehyperbolicgeometry.LetDFbetheassociatedLaplace-Beltramioperator.Letρ=n21,n,2n+1accordingtoF=R,CorH.Itiswellknownthatifuiseuclideanharmonicormoregenerallyk-harmonicforkNwithaboundarydistribution,theneverynormalderivativeofu,Nku,hasalsoaboundarydistribution.WewillshowthatifuisaDF-harmonicfunctionwithaboundarydistribution,thenforeveryintegerk<ρ,Nkuhasalsoaboundarydistribution.Next,wedefineapluriharmonicfunctionasafunctionthatiseuclideanharmonicovereveryF-linewhereFisseenasRdwithd=dimRF.ThisextendsaclassicaldefinitionfromthecaseF=Ctothetwoothercasesandseemstobethemostpertinentdefinitionforourstudy.Itisshownin[Jam5]forF=Randnoddandin[BBG]forF=C,thatifuisDF-harmonicwithaboundarydistribution,thenNρuhasaboundarydistributionifandonlyuisalsoeuclideanharmonic.NotethatforF=R,ρisanintegerifnisodd,whereasforneven,ρisahalf-integer.Inthislastcase,althoughonemightgiveameaningtoNρ,theaboveresultisnolongertrue.Actually,ifF=Randniseven,wewillshowthatifuisDR-harmonicthenuisalso2n-harmonic(uptoachangeofvariables),implyingthatubehavesmorealiketheeuclidean-harmonicfunctions.Inparticular,ashasalreadybeenshownin[Jam5]bydifferentmethods,ifuhasaboundarydistribution,thenNkuhasalsoaboundarydistributionforeveryk.So,inevendimension,DR-harmonicfunctionsbehavelikeeuclideanharmonicfuntions.Further,inthecaseF=R,theonlyfunctionsthatarebothDR-harmonicandeuclideanharmonic(andmoregenerallyk-harmonicwithk1)aretheconstants.InthecaseF=C,15