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DJam Comptes Rendus de l'Academies des Sciences Series I Mathematiques Vol No

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Niveau: Supérieur, Doctorat, Bac+8
[DJam] Comptes Rendus de l'Academies des Sciences - Series I Mathematiques, Vol 332, No 12 (2001) 1053–1058. Estimations du noyau de Green, propriete de valeur moyenne et geometrie des boules hyperboliques Cyrille DOMENICHINO? & Philippe JAMING ? LAPT (UMR 6632), Universite de Provence, 39, rue Joliot-Curie, 13453 Marseille cedex 13, FRANCE Resume : Dans cette note, nous obtenons des estimes du noyau de Green dans les boules hyperboliques reelles, complexes et quaternioniques. Celles-ci nous permettent ensuite de montrer que, dans ces boules, les seuls domaines de classe C1+?, ? > 0 pour lesquels l'egalite de la moyenne surfacique est vraie pour toutes les fonctions harmoniques sont les boules geodesiques. English title : Green kernel estimates, mean value properties and geometry of classical rank one balls. English abstract : In this Note, we obtain estimates of the Green kernel of real, complex and quaternionic hyperbolic balls. We then apply these to show that in such balls the only domains of class C1+?, ? > 0 for which the spherical mean value identity holds for every harmonic function are the geodesic balls. English Abridged Version In this Note, we denote by F = R,C or H and n ≥ 2 an integer (n ≥ 3 if F = R). Let Bn be the Euclidean unit ball of Fn.

  • bn de bord ∂? de classe c1

  • green kernel

  • boules hyperboliques

  • boules geodesiques

  • estimation du noyau de green complexe

  • ?z ?

  • ∂?

  • operateur de laplace-beltrami sur bn


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[DJam]ComptesRendusdelAcade´miesdesSciences-SeriesIMathe´matiques,Vol332,No12(2001)10531058. EstimationsdunoyaudeGreen,proprie´te´devaleurmoyenne etg´eom´etriedesbouleshyperboliques
Cyrille DOMENICHINO & Philippe JAMING
LAPT(UMR6632),Universit´edeProvence,39,rueJoliot-Curie,13453Marseillecedex13, FRANCE
R´esume´:´mitseseayonudsesoou,ntesdonenbtDnacsteetonreenudeG danslesbouleshyperboliquesr´eelles,complexesetquaternioniques.Celles-ci nous permettent ensuite de montrer que, dans ces boules, les seuls domaines 1+α de classeC,α >cifauresvrsteequeiaesq0upourl´egelsle´edlatieynnalom pourtouteslesfonctionsharmoniquessontlesboulesg´eod´esiques. English title :Green kernel estimates, mean value properties and geometry of classical rank one balls. English abstract :In this Note, we obtain estimates of the Green kernel of real, complex and quaternionic hyperbolic balls. We then apply these to show 1+α that in such balls the only domains of classC,α >0 for which the spherical mean value identity holds for every harmonic function are the geodesic balls.
English Abridged Version In this Note, we denote byF=R,CorHandn2 an integer (n3 ifF=R). Let n Bnbe the Euclidean unit ball ofF. DefineGasF=R:G=SO0(n,1),F=C:G= SU(n,1),F=H:G=Sp(n,1); and letG=KANIt isbe its Iwasawa decomposition. well known thatBncan be identified withG/K, in particularGacts onBnandBncan be endowed with aGWe will refer to this metric as the hyperbolic metric on-invariant metric. Bnlet. Further, d= dimRFand definem1=d(n1) andm2=d1 the multiplicities of the roots ofG. IfzBn, there existsgGsuch thatg.z= 0 andg.0 =z. ForζBnwe will then write ϕz(ζ) =g.ζso that 2 zPzζ1− kzkQzζ ϕz(ζ) = 1− hζ, zi hζ, zi withPz(ζ) =zandQz(ζ) =ζPz(ζ). hz, zi Denote byDFthe Laplace operator onBnthat is invariant underG, bytheG-invariant 1+α measure onBn. Let Ω be a relatively compact domain inBnof classCsuch thatΩ =Ω. Denote bygthe contraction ofwith regard to the outward normal vector toΩ with respect to the hyperbolic metric. The Green function Γ forDFis then given by Z m 1 1 +m21 (1t) 2 Γ(ζ, z) =cn1+m+mdt. 2 t kϕz(ζ)k2 65