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Two kinds of Hp-classes of harmonic functions are defined on a general rank one symmetric space of noncompact type. The first one is introduced by using a nontangential maximal function. The second is related to the diffusion generated by the Laplace-Beltrami operator. The equivalence of the two classes is proven for 0 < p < oo. 0. Introduction. The aim of the present work is to prove an analogue, for a rank one symmetric space of noncompact type, of a result due to Burkholder, Gundy and Silverstein [2] for the Euclidean space R^_ and to Burkholder and Gundy [1] for RTM+1. Our results generalize and extend those of Debiard [5] who did his work on the generalized half plane equivalent to the hyperbolic ball of C". In §1 we start with some general background about symmetric spaces of noncompact type with special emphasis on those of rank one. The reader is referred to Helgason [8 or 9] and Korányi [11] for more details. We give then some basic definitions and illustrate them with an example. We end this section by stating the main result of this work (Theorem I) that consists in the equivalence of two Hpclasses of harmonic functions (//? and H^) previously defined. §2 contains several facts about Brownian motion in a Riemannian manifold that are used later on in the paper. A general reference for these is McKean [16]. §§3 and 4 are devoted to the proof of each of the two parts of Theorem I. In §3 we prove the inclusion Hft C HI following the techniques of [1 and 5] adapted to the symmetric space. It is remarkable that for this part of the proof we do not need to require harmonicity. In §4 the opposite inclusion is proven. Although we also follow here the path marked by [1 and 5] the problems that one has to face to extend the results are more serious. In particular, the techniques to prove Proposition 8 and Proposition 12 are new. Also for the part of Lemma 7 that deals with 0 < p < 1 we had to adapt to a symmetric space a technique due to De Giorgi [7] for the Euclidean space. The author is indebted to Dr. Choi of the University of Chicago who showed him De Giorgi's method. The results proven here are, in part, contained in the author's Ph.D. Dissertation presented at Washington University in St. Louis in May 1983. The author wishes to thank Adam Korányi for his help and encouragement. 1. Symmetric spaces. Definitions. A symmetric space of noncompact type is a Riemannian manifold of negative curvature that verifies the following property: Received by the editors November 22, 1983. 1980 Mathematics Subject Classification. Primary 22E30, 43A85; Secondary 60J60.

𝐻^{𝑝}-classes on rank one symmetric spaces of noncompact type. I. Nontangential and probabilistic maximal functions