Horocyclic harmonic Bergman spaces on homogeneous trees.

The main focus of this contribution is on the harmonic Bergman spaces ℬαp on the q-homogeneous tree 픛q endowed with a family of measures σα 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 ℬα2 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 Lαp 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, <italic>Potential Anal.</italic> <bold>44</bold>(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, <italic>Potential Anal.</italic> <bold>61</bold> (2024) 153–182]. [ABSTRACT FROM AUTHOR]