Jam6 Bollettino della Unione Matematica Italiana pp

Philippe Jaming

Description

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.

• then


• invariant laplacian


• harmonic functions


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• laplace-beltrami operator


• df-harmonic function


• operator can

Sujets

• Harmonic function
• Laplace?Beltrami operator

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 6
• Langue: English

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[Jam6]BollettinodellaUnioneMatematicaItaliana(8) 4 (2001)pp685–702. Harmonicfunctionsonclassicalrankoneballs PhilippeJAMING Abstract:Inthispaper,westudythelinksbetweenharmonicfunctionson thehyperbolicballs(real,complexorquaternionic),theeuclideanharmonic functionsontheseballsandpluriharmonicfunctionsundergrowthconditions. Inparticular,weextendresultsbyA.Bonami,J.BrunaandS.Grellier(com- plexcase)andtheauthor(realcase)tothequaternioniccase. Keywords:rankoneball,harmonicfunctions,boundaryvalues. AMSsubjectclass:48A85,58G35. 1. Introduction. Inthispaper,westudythelinksbetweenharmonicfunctionsonthehyperbolicballs(real, complexorquaternionic),theeuclideanharmonicfunctionsontheseballsandpluriharmonic functions.Inparticularweinvestigatewhethergrowthconditionsmayseparatetheseclasses. Moreprecisely,let F = R , C or H (thequaternions)andletnbeaninteger,n≥2(n≥3 if F = R ).Let B n betheeuclideanballin F n ,letΔbetheeuclideanlaplacianoperatoron B n andletN=r ∂ ∂ r bethenormalderivationoperator.Fork∈ N ∗ afunctionuofclassC 2k issaidtobek-hamonicifΔ k u=0,inparticularfork=1thisaretheeuclideanharmonic functions. Theball B n canalsobeendowedwiththehyperbolicgeometry.LetD F betheassociated Laplace-Beltramioperator.Letρ= n 2 −1 ,n,2n+1accordingto F = R , C or H . Itiswellknownthatifuiseuclideanharmonicormoregenerallyk-harmonicfork∈ N ∗ withaboundarydistribution,theneverynormalderivativeofu,N k u,hasalsoaboundary distribution.WewillshowthatifuisaD F -harmonicfunctionwithaboundarydistribution, thenforeveryintegerk<ρ,N k uhasalsoaboundarydistribution. Next,wedefineapluriharmonicfunctionasafunctionthatiseuclideanharmonicover every F -linewhere F isseenas R d withd=dim R F .Thisextendsaclassicaldefinitionfrom thecase F = C tothetwoothercasesandseemstobethemostpertinentdefinitionforour study. Itisshownin[Jam5]for F = R andnoddandin[BBG]for F = C ,thatifuisD F -harmonic withaboundarydistribution,thenN ρ uhasaboundarydistributionifandonlyuisalso euclideanharmonic.Notethatfor F = R ,ρisanintegerifnisodd,whereasforneven, ρisahalf-integer.Inthislastcase,althoughonemightgiveameaningtoN ρ ,theabove resultisnolongertrue.Actually,if F = R andniseven,wewillshowthatifuisD R - harmonicthenuisalso 2 n -harmonic(uptoachangeofvariables),implyingthatubehaves morealiketheeuclidean-harmonicfunctions.Inparticular,ashasalreadybeenshownin [Jam5]bydifferentmethods,ifuhasaboundarydistribution,thenN k uhasalsoaboundary distributionforeveryk.So,inevendimension,D R -harmonicfunctionsbehavelikeeuclidean harmonicfuntions. Further,inthecase F = R ,theonlyfunctionsthatarebothD R -harmonicandeuclidean harmonic(andmoregenerallyk-harmonicwithk ≥ 1)aretheconstants.Inthecase F = C , 15