Harmonic Bergman Projectors on Homogeneous Trees

In this paper we investigate some properties of the harmonic Bergman spaces A(p)(sigma) on a q-homogeneous tree, where q >= 2, 1 <= p < infinity, and sigma is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J. Cohen, F. Colonna, M. Picardello and D. Singman. When p=2 they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on L-p(sigma) for 1 < p < infinity and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral Hormander's condition.