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Poincaré and Hardy Inequalities on Homogeneous Trees

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Geometric Properties for Parabolic and Elliptic PDE's

Part of the book series: Springer INdAM Series ((SINDAMS,volume 47))

Abstract

We study Hardy-type inequalities on infinite homogeneous trees. More precisely, we derive optimal Hardy weights for the combinatorial Laplacian in this setting and we obtain, as a consequence, optimal improvements for the Poincaré inequality.

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Acknowledgements

We thank Y. Pinchover for let us know that the property “optimality near infinity” can be derived from the criticality of H − W and the null-criticality of the operator H − W with respect to the Hardy weight W. This remarkable fact has been recently proved in [25] in the continuous setting.

The first author is partially supported by the INdAM-GNAMPA 2019 grant “Analisi spettrale per operatori ellittici con condizioni di Steklov o parzialmente incernierate” and by the PRIN project “Direct and inverse problems for partial differential equations: theoretical aspects and applications” (Italy). 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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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Dipartimento di Eccellenza 2018–2022, Politecnico di Torino, Torino, Italy

    Elvise Berchio, Federico Santagati & Maria Vallarino

Authors
  1. Elvise Berchio
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  2. Federico Santagati
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  3. Maria Vallarino
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Corresponding author

Correspondence to Elvise Berchio .

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Editors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Napoli, Italy

    Vincenzo Ferone

  2. Applied Mathematics and Informatics Course, Faculty of Advanced Science and Technology, Ryukoku University, Otsu, Shiga, Japan

    Tatsuki Kawakami

  3. DIMAI Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Firenze, Italy

    Paolo Salani

  4. Department of Mathematics, Osaka City University, Osaka, Japan

    Futoshi Takahashi

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Berchio, E., Santagati, F., Vallarino, M. (2021). Poincaré and Hardy Inequalities on Homogeneous Trees. In: Ferone, V., Kawakami, T., Salani, P., Takahashi, F. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-030-73363-6_1

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