Abstract
In this paper we investigate some properties of the harmonic Bergman spaces \(\mathcal A^p(\sigma )\) on a q-homogeneous tree, where \(q\ge 2\), \(1\le p<\infty \), 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 Hörmander’s condition.
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Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures for analytic Besov spaces. Rev. Mat. Iberoamericana 18(2), 443–510 (2002)
Arcozzi, N., Rochberg, R., Sawyer, E., Wick, B.: Potential theory on trees, graphs and Ahlfors-regular metric spaces. Potential Anal. 41(2), 317–366 (2014)
Békollé, D.: Inégalité à poids pour le projecteur de Bergman dans la boule unité de \(\mathbb{C} ^{n}\). Studia Math. 71(3), 305–323 (1982)
Békollé, D., Bonami, A.: Inégalités à poids pour le noyau de Bergman. C. R. Acad. Sci. Paris Sér. A-B 286(18), A775–A778 (1978)
Békollé, D., Bonami, A., Garrigós, G., Nana, C., Peloso, M.M., Ricci, F.: Lecture notes on Bergman projectors in tube domains over cones: an analytic and geometric viewpoint. IMHOTEP J. Afr. Math. Pures Appl. 5, (2004)
Boiko, T., Woess, W.: Moments of Riesz measures on Poincaré disk and homogeneous tree-a comparative study. Expo. Math. 33(3), 353–374 (2015)
Cartier, P.: Harmonic analysis on trees. In: Proceedings of Symposia in Pure Mathematics, vol. 26, pp. 419–424 (1973)
Cohen, J.M., Colonna, F.: Embeddings of trees in the hyperbolic disk. Complex Variables Theory Appl. 24(3–4), 311–335 (1994)
Cohen, J.M., Colonna, F., Picardello, M., Singman, D.: Bergman spaces and Carleson measures on homogeneous isotropic trees. Potential Anal. 44, 05 (2016)
Cohen, J.M., Colonna, F., Picardello, M., Singman, D.: Fractal functions with no radial limits in Bergman spaces on trees. Hokkaido Math. J. 47, 269–289 (2018)
Cohen, J.M., Colonna, F., Picardello, M., Singman, D.: Carleson measures for non-negative subharmonic functions on homogeneous trees. Potential Anal. 52(1), 41–67 (2020)
Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83(4), 569–645 (1977)
Cowling, M., Meda, S., Setti, A.G.: An overview of harmonic analysis on the group of isometries of a homogeneous tree. Expo. Math. 16(5), 385–423 (1998)
Cowling, M., Meda, S., Setti, A.G.: Estimates for functions of the Laplace operator on homogeneous trees. Trans. Amer. Math. Soc. 352, 01 (2000)
Deng, Y., Huang, L., Zhao, T., Zheng, D.: Bergman projection and Bergman spaces. J. Operator Theory 46(1), 3–24 (2001)
Figà-Talamanca, A., Nebbia, C.: Harmonic analysis and representation theory for groups acting on homogenous trees, vol. 162. Cambridge University Press, (1991)
Forelli, F., Rudin, W.: Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J. 24(6), 593–602 (1974)
Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Modern Birkhäuser Classics. Birkhäuser Boston Inc, Boston, MA (2007)
Hörmander, L.: Estimates for translation invariant operators in \(L^{p}\) spaces. Acta Math. 104, 93–140 (1960)
Stein, E.M.: Singular integrals and estimates for the Cauchy-Riemann equations. Bull. Amer. Math. Soc. 79(2), 440–445 (1973)
Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton, NJ (1993)
Zhu, K.: Operator Theory in Function Spaces. American Mathematical Society, Mathematical surveys and monographs (2007)
Acknowledgements
The authors are grateful to Marco M. Peloso for suggesting references on the weak type (1,1) boundedness of the Bergman projector in the case of the hyperbolic disk, and to Matteo Levi for useful comments.
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Filippo De Mari, Matteo Monti, and Maria Vallarino wrote the main manuscript text. All authors reviewed the manuscript.
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This work is partially supported by the project “Harmonic analysis on continuous and discrete structures” funded by Compagnia di San Paolo (Cup E13C21000270007). Furthermore, the authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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De Mari, F., Monti, M. & Vallarino, M. Harmonic Bergman Projectors on Homogeneous Trees. Potential Anal 61, 153–182 (2024). https://doi.org/10.1007/s11118-023-10106-4
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DOI: https://doi.org/10.1007/s11118-023-10106-4