1. Introduction

Throughout the paper n is a positive integer greater than 1. We will be working with functions defined on all or part of ℝn. Let Dj denote the partial derivative with respect to the j-th coordinate variable. Recall that ▽u(x) = (Diu(x), • • •, Dnu(x)). The Laplacian of u is. A real- or complex-valued function u is harmonic on an open subset Q of ℝn if △u = 0 on Ω The purpose of this article is to present an elementary treatment of some known results for the harmonic Bergman spaces consisting of all harmonic functions on the unit ball in ℝn that are p-integrable with respect to volume measure. Several properties of these spaces are analogous to those of the Bergman spaces of analytic functions on the unit ball in ℂn. As in the analytic case, there is a reproducing kernel and associated projection. Duality results follow once we know that the projection is Lp-bounded. Coifman and Rochberg [1980] used deep results from harmonic analysis to establish Lp-boundedness of the harmonic Bergman projection. An explicit formula for the harmonic Bergman reproducing kernel has only been determined recently; see [Axler et al. 1992]. We give a simple derivation for such a formula in Section 2.