Harmonic Bergman projectors on homogeneous trees

TL;DR

This paper explores harmonic Bergman spaces on homogeneous trees, analyzing their properties, reproducing kernels, and the boundedness of associated projectors, especially under non-doubling measures with exponential decay.

Contribution

It provides explicit formulas for reproducing kernels and establishes boundedness results for Bergman projectors on these non-doubling harmonic Bergman spaces.

Findings

Explicit reproducing kernel for $p=2$ spaces.

Boundedness of Bergman projector on $L^p$ spaces.

Weak type (1,1) boundedness under exponential decay measures.

Abstract

In this paper we investigate some properties of the harmonic Bergman spaces $\mathcal A^p(\sigma)$ on a $q$-homogeneous tree, where $q\geq 2$, $1\leq p<\infty$, 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 H\"ormander's condition.