Horocyclic harmonic Bergman spaces on homogeneous trees

TL;DR

This paper studies harmonic Bergman spaces on homogeneous trees, focusing on their structure, kernels, and boundedness properties of associated projections using harmonic analysis techniques.

Contribution

It introduces a new framework for harmonic Bergman spaces on homogeneous trees, including orthonormal bases, kernel formulas, and boundedness results for projections.

Findings

Established the structure of harmonic Bergman spaces on homogeneous trees.

Derived explicit formulas for reproducing kernels and orthonormal bases.

Proved boundedness of Bergman projections using Calderón-Zygmund theory.

Abstract

The main focus of this contribution is on the harmonic Bergman spaces $\mathcal{B}_{\alpha}^{p}$ on the $q$-homogeneous tree $\mathfrak{X}_q$ endowed with a family of measures $\sigma_\alpha$ that are constant on the horocycles tangent to a fixed boundary point and turn out to be doubling with respect to the corresponding horocyclic Gromov distance. A central role is played by the reproducing kernel Hilbert space $\mathcal{B}_{\alpha}^{2}$ for which we find a natural orthonormal basis and formulae for the kernel. We also consider the atomic Hardy space and the bounded mean oscillation space. Appealing to an adaptation of Calder\'on-Zygmund theory and to standard boundedness results for integral operators on $L^p_\alpha$ spaces with H\"ormander-type kernels, we determine the boundedness properties of the Bergman projection.