Harmonic Functions on the Real Hyperbolic Ball II : Hardy and Lipschitz Spaces

Sandrine Grellier (1) , Philippe Jaming (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started in \cite{Ja3}. Our focus here is on the theory of Hardy-Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman-Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball. In doing so, we exhibit differences of behaviour of derivatives of harmonic functions depending on the parity of the dimension of the ball and on the parity of the order of derivation.

Mots clés: espaces de Hardy; espaces de Hardy-Sobolev; espaces de Lipschitz; classes de Zygmund; théorie de Fefferman-Stein; fonctions maximales; intégrales d'aire; fonction g de Littlewood-Paley; boule hyperbolique réelle; fonctions harmoniques

Domaines: Analyse classique [math.CA]

Sandrine Grellier, Philippe Jaming. Harmonic Functions on the Real Hyperbolic Ball II : Hardy and Lipschitz Spaces. Mathematical News / Mathematische Nachrichten, 2004, 268, pp.50-73. ⟨hal-00005815⟩