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arXiv · 2308.07128

Hardy-Littlewood maximal operators on trees with bounded geometry

Matteo Levi·Stefano Meda·Federico Santagati·Maria Vallarino
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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 $\Upsilon_{a,b}$. We show that 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 $\Upsilon_{a,b}$.

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Matteo Levi, Stefano Meda, Federico Santagati, Maria Vallarino. 2023-08-14. Hardy-Littlewood maximal operators on trees with bounded geometry. https://arxiv.org/abs/2308.07128

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