Harmonic functions on Hermitian hyperbolic space

Abstract

We say that a function F is harmonic in 9 if AF=0, and we define the classes .X/P(9) of harmonic functions in analogy with the classes HP(9) of holomorphic functions. The Poisson kernel corresponding to A is explicitly known [4]; since 9 is a symmetric space of rank one, '(9) is exactly the set of all Poisson integrals of L-Q), X denoting the boundary of 9 [3]. As we show by an easy reduction to the case of p = oo, the same statement is true for every p ? 1 (with a slight modification if p = 1). Our main concern is the generalization of the classical Fatou theorem, and of its local version due to Privalov and Calderon ([1], [7]). For this purpose we define the notion of admissible convergence in ?3. We show that in the case of n= 1 admissible convergence coincides with nontangential convergence, while for n > 1 it is stronger. It is a notion invariant under the group of holomorphic automorphisms of q; nontangential convergence in the case n> 1 is not. We prove Fatou's theorem for admissible convergence by some explicit estimates on the Poisson kernel and by using an extension of the Hardy-Littlewood Maximal Theorem due to Edwards and Hewitt [2]. It is perhaps worth mentioning that this is a new result even for holomorphic functions since previous investigations, being based on the euclidean Poisson integral, yielded only radial or nontangential convergence [1], [6]. The generalized Cayley transformation carries 9 onto a generalized halfplane D. In analogy with [5] one can again define the spaces of harmonic functions JVPP(D). The results described above all have their analogues in this situation, and the proofs are parallel to those for 9. It should be noted, however, that these