Spiderwebs and sharp Lp bounds for the Hardy–Littlewood maximal operator on Gromov hyperbolic spaces

In this paper we prove that if 1<a≤b<a2 and X is a locally doubling δ-hyperbolic complete connected length metric measure space with (a,b)[jls-end-space/]-pinched exponential growth at infinity, then the centred Hardy–Littlewood maximal operator M is bounded on Lp(X) for all p>ρ[jls-end-space/], and it is of weak type (ρ,ρ)[jls-end-space/], where ρ:=loga⁡b[jls-end-space/]. A key step in the proof is a new structural theorem for Gromov hyperbolic spaces with (a,b)[jls-end-space/]-pinched exponential growth at infinity, consisting in a discretisation of X by means of certain graphs, introduced in this paper and called spiderwebs, with “good connectivity properties”. Our result applies to trees with bounded geometry, and Cartan–Hadamard manifolds of pinched negative curvature, providing new boundedness results in these settings. The index ρ is optimal in the sense that if p<ρ[jls-end-space/], then there exists X satisfying the assumptions above such that M is not of weak type (p,p)[jls-end-space/]. Furthermore, if b>a2[jls-end-space/], then there are examples of spaces X satisfying the assumptions above such that M bounded on Lp(X) if and only if p=∞[jls-end-space/].