Harmonic Bergman projectors on homogeneous trees

Abstract

In this paper we investigate some properties of the harmonic Bergman spaces ${\mathcal{A}}^p(\sigma )$ on a $q$-homogeneous tree, where $q\ge 2$, $1\le p<\infty$, 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 $1<p<\infty$ 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 Hörmander's condition.