Horocyclic harmonic Bergman spaces on homogeneous trees

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