Hardy–Littlewood maximal operators on trees with bounded geometry

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 $\Upsilon _{a,b}$.