BMO Spaces on Weighted Homogeneous Trees

Abstract

We consider an infinite homogeneous tree \({\mathcal {V}}\) endowed with the usual metric d defined on graphs and a weighted measure \(\mu \). The metric measure space \(({\mathcal {V}},d,\mu )\) is nondoubling and of exponential growth, hence the classical theory of Hardy and BMO spaces does not apply in this setting. We introduce a space \(BMO(\mu )\) on \(({\mathcal {V}},d,\mu )\) and investigate some of its properties. We prove in particular that \(BMO(\mu )\) can be identified with the dual of a Hardy space \(H^1(\mu )\) introduced in a previous work and we investigate the sharp maximal function related with \(BMO(\mu )\).

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Acknowledgements

Work partially supported by the MIUR project “Dipartimenti di Eccellenza 2018-2022” (CUP E11G18000350001). 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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Correspondence to Anita Tabacco.

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Dedicated to Guido Weiss on the occasion of his 90th birthday.

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Arditti, L., Tabacco, A. & Vallarino, M. BMO Spaces on Weighted Homogeneous Trees. J Geom Anal (2020). https://doi.org/10.1007/s12220-020-00435-w

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Keywords

  • Hardy spaces
  • BMO spaces
  • Homogeneous trees
  • Nondoubling measure
  • Sharp maximal function

Mathematics Subject Classification

  • 05C05
  • 30H35
  • 42B30