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arXiv · 2205.13859

Harmonic Bergman projectors on homogeneous trees

Filippo De Mari·Matteo Monti·Maria Vallarino
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Abstract

In this paper we investigate some properties of the harmonic Bergman spaces $\mathcal A^p(\sigma)$ on a $q$-homogeneous tree, where $q\geq 2$, $1\leq p<\infty$, and $\sigma$ is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J.~Cohen, F.~Colonna, M.~Picardello and D.~Singman. When $p=2$ they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on $L^p(\sigma)$ for $1<p<\infty$ and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral H\"ormander's condition.

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Filippo De Mari, Matteo Monti, Maria Vallarino. 2022-05-27. Harmonic Bergman projectors on homogeneous trees. https://arxiv.org/abs/2205.13859

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