Hardy-Littlewood maximal operators on trees with bounded geometry

TL;DR

This paper investigates the boundedness of Hardy-Littlewood maximal operators on trees with bounded geometry, establishing sharp $p$-ranges and extending results to certain graphs, advancing understanding in harmonic analysis on discrete structures.

Contribution

It provides the first sharp $p$-range for boundedness of maximal operators on trees with bounded geometry and extends results to roughly isometric graphs.

Findings

Sharp $p$-range for centered maximal operator boundedness.

Uncentered maximal operator bounded only at $p=\infty$ for some trees.

Extension of results to graphs roughly isometric to these trees.

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 $\Upsilon_{a,b}$.