The main focus of this contribution is on the harmonic Bergman spaces ℬαp on the q-homogeneous tree q 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, 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].
Horocyclic harmonic Bergman spaces on homogeneous trees
De Mari F.;Monti M.;
2024-01-01
Abstract
The main focus of this contribution is on the harmonic Bergman spaces ℬαp on the q-homogeneous tree q 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, 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].
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