Harmonic Bergman spaces on locally finite trees
Alessandro Ottazzi, Federico Santagati
TL;DR
This work develops a harmonic Bergman theory on locally finite trees using a forward Laplacian and flow-weighted measures. It derives an explicit Bergman kernel and proves that the associated Bergman projection is bounded on $L^p$ for every $p>1$ and has weak type $(1,1)$, employing nondoubling Calderón–Zygmund methods. It further analyzes exponential-weight settings to obtain sharp kernel estimates and necessary and sufficient conditions for the $L^p$-boundedness of Toeplitz-type operators built from the Bergman kernel, with an equivalence of conditions in homogeneous trees. The results bridge discrete harmonic analysis on graphs with classical Bergman theory and provide tools for anisotropic operator analysis on trees.
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.