Horocyclic harmonic Bergman spaces on homogeneous trees

The main focus of this contribution is on the harmonic Bergman spaces [Formula: see text] on the [Formula: see text]-homogeneous tree [Formula: see text] endowed with a family of measures [Formula: see text] 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 [Formula: see text] 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 Calderón–Zygmund theory and to standard boundedness results for integral operators on [Formula: see text] spaces with Hörmander-type kernels, we determine the boundedness properties of the Bergman projection. This work was inspired by [J. M. Cohen, F. Colonna, M. A. Picardello and D. Singman, Bergman spaces and Carleson measures on homogeneous isotropic trees, Potential Anal. 44(4) (2016) 745–766, doi:10.1007/s11118-015-9529-7; F. De Mari, M. Monti and M. Vallarino, Harmonic Bergman projectors on homogeneous trees, Potential Anal. 61 (2024) 153–182].