Key Takeaway: The Littlewood-Paley operator can characterize H-classes of harmonic functions on rank one symmetric spaces of noncompact type, proving L-boundedness if the area integral of 2 that function is in L.

Abstract

A characterization of the Hardy class H2 on a rank one symmetric space of noncompact type by a Littlewood-Paley type operator defined through the Green potential of the norm square of the invariant gradient. 0. INTRODUCTION In the present article we characterize H -classes of harmonic functions on rank one symmetric spaces of noncompact type using a Littlewood-Paley type operator that is defined as the Green potential of the square of the gradient. It seems that this type of operator was first suggested by R. Gundy (cf. GetoorSharpe [5] and Meyer [10]) to characterize BMO of Rn+1 . The idea of using this operator to characterize HP is due to Debiard (cf. Debiard [4]). The method that we follow here is to show that the Littlewood-Paley operator of harmonic function verifies a L -boundedness if and only if the area integral of 2 that function is in L In ? 1 we state several facts about symmetric spaces and give the basic definitions. The reader is referred to Helgason [6] and Koranyi [9] for more details. In ?2 we state and prove our results. 1. GENERALITIES Let X be a rank one symmetric space of noncompact type. Let G be the identity component of the group of isometries of X. Fix a point o E X as the base point for the space and let K be the isotropy subgroup of G at o. The group G is a semisimple Lie group with finite center, and K is a compact subgroup. The space X can be identified with the homogeneous space G/K. The Lie algebras of G and K are denoted as g and k. Let 0 be the Cartan involution in g relative to k and let p be the subspace of g with eigenvalue1 relative to 0: g = k (E p. Let a be a fixed maximal subalgebra of p, the dimension of a is the same as the rank of X, in our case it is 1 , therefore Received by the editors March 1, 1988 and, in revised form, June 15, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 22E30, 43A85. ( 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page