Hardy–Littlewood maximal operators on trees with bounded geometry
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- by Matteo Levi, Stefano Meda, Federico Santagati and Maria Vallarino;
- Trans. Amer. Math. Soc. 378 (2025), 3951-3979
- DOI: https://doi.org/10.1090/tran/9229
- Published electronically: March 19, 2025
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Abstract:
In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy–Littlewood maximal operators on the class $\Upsilon _{a,b}$, $2\leq a\leq b$, of trees with $(a,b)$-bounded geometry. We find the sharp range of $p$, depending on $a$ and $b$, where the centred maximal operator is bounded on $L^p(\mathfrak {T})$ for all $\mathfrak {T}$ in $\Upsilon _{a,b}$. Precisely, the lower endpoint is $\log _a b$ if $b \leq a^2$ and $\infty$ otherwise. In particular, we show that if $b>a^2$, then there exists a tree in $\Upsilon _{a,b}$ for which the uncentred maximal function is bounded on $L^p$ if and only if $p=\infty$. We also extend these results to graphs which are strictly roughly isometric, in the sense of Kanai, to trees in the class $\Upsilon _{a,b}$.References
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Bibliographic Information
- Matteo Levi
- Affiliation: Fac. de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain; and Dept. Matemàtica i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
- MR Author ID: 1267296
- ORCID: 0000-0002-6242-1770
- Email: matteo.levi@ub.edu
- Stefano Meda
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via R. Cozzi 53, I-20125 Milano, Italy
- MR Author ID: 122835
- Email: stefano.meda@unimib.it
- Federico Santagati
- Affiliation: Dipartimento di Matematica, Dipartimento di Eccellenza 2023–2027; and Malga Center, Università di Genova, via Dodecaneso 35, Genova, Italy
- MR Author ID: 1486817
- ORCID: 0000-0003-4685-9956
- Email: fedirico.santagati@edu.unige.it
- Maria Vallarino
- Affiliation: Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
- MR Author ID: 787152
- ORCID: 0000-0002-0023-9121
- Email: maria.vallarino@polito.it
- Received by editor(s): September 28, 2023
- Received by editor(s) in revised form: April 18, 2024, and May 7, 2024
- Published electronically: March 19, 2025
- Additional Notes: The first, third, and fourth 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). The first, third, and fourth authors were supported in part by the Project “Harmonic analysis on continuous and discrete structures” (bando Trapezio Compagnia di San Paolo CUP E13C21000270007). The first author was supported in part by the Generalitat de Catalunya (grant 2021 SGR 00071), the Spanish Ministerio de Ciencia e Innóvacion (project PID2021-12315NB-100).
- Dedicated: This paper is dedicated to the memory of A. M.
- © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 378 (2025), 3951-3979
- MSC (2020): Primary 05C05, 43A99
- DOI: https://doi.org/10.1090/tran/9229