Hardy-Littlewood maximal operators on trees with bounded geometry

In this paper we study the Lp boundedness of the centred and the uncentred Hardy–Littlewood maximal operators on the class Υa,b, 2 ≤ a ≤ 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 Lp(T) for all T in Υa,b. Precisely, the lower endpoint is loga b if b ≤ a2 and ∞ otherwise. In particular, we show that if b > a2, then there exists a tree in Υa,b for which the uncentred maximal function is bounded on Lp if and only if p = ∞. We also extend these results to graphs which are strictly roughly isometric, in the sense of Kanai, to trees in the class Υa,b