Harmonic Functions on the Real Hyperbolic Ball I : Boundary Values and Atomic Decompositions of Hardy Spaces

Philippe Jaming (1)

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

In this article we study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic space $\B_n$. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution. In doing so, we put forward different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball $\B_n$. We then study Hardy spaces $H^p(\B_n)$, $0< p <\infty$, whose elements appear as the hyperbolic harmonic extensions of distributions belonging to the Hardy spaces of the sphere $H^p(\S^{n-1})$. In particular, we obtain an atomic decomposition of this spaces.

Mots clés: real hyperbolic ball; harmonic functions; boundary values; Hardy spaces; atomic decomposition

Domaines: Analyse classique [math.CA]

Philippe Jaming. Harmonic Functions on the Real Hyperbolic Ball I : Boundary Values and Atomic Decompositions of Hardy Spaces. Colloquium Mathematicum, 1999, 80, pp.63-82. ⟨hal-00005820⟩