Horocyclic harmonic Bergman spaces on homogeneous trees

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 Calderón-Zygmund theory and to standard boundedness results for integral operators on $L_{\alpha }^p$ spaces with Hörmander-type kernels, we determine the boundedness properties of the Bergman projection.