Harmonic Bergman spaces on locally finite trees

TL;DR

This paper introduces harmonic Bergman spaces on locally finite trees, characterizes the Bergman projection, and establishes boundedness properties of associated operators, advancing the understanding of harmonic analysis in discrete structures.

Contribution

It defines harmonic Bergman spaces on trees with probabilistic Laplacians and characterizes the boundedness of the Bergman projection and Toeplitz-type operators.

Findings

Bergman projection is bounded on L^p for p > 1

Projection is of weak type (1,1)

Necessary and sufficient conditions for Toeplitz operator boundedness

Abstract

We define the harmonic Bergman space on locally finite trees with respect to a suitable probabilistic Laplacian and a class of weighted flow measures. We characterise the corresponding Bergman projection and prove that it is bounded on $L^p$ for every $p>1$, and of weak type $(1,1)$. We also prove necessary and sufficient conditions for the $L^p$-boundedness of the extension of a class of Toeplitz-type operators.