Spider's webs and sharp $L^p$ bounds for the Hardy--Littlewood maximal operator on Gromov hyperbolic spaces

In this paper we prove that if $1<a\le b<a^2$ and $X$ is a locally doubling $\delta$-hyperbolic complete connected length metric measure space with $(a,b)$-pinched exponential growth at infinity, then the centred Hardy–Littlewood maximal operator $\mathcal{M}$ is bounded on $L^p(X)$ for all $p>\tau$, and it is of weak type $(\tau ,\tau )$, where $\tau :={\log }_{a}b$. A key step in the proof is a new structural theorem for Gromov hyperbolic spaces with $(a,b)$-pinched exponential growth at infinity, consisting in a discretisation of $X$ by means of certain graphs, introduced in this paper and called spider’s webs, 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 $\tau$ is optimal in the sense that if $p<\tau$, then there exists $X$ satisfying the assumptions above such that $\mathcal{M}$ is not of weak type $(p,p)$. Furthermore, if $b>a^2$, then there are examples of spaces $X$ satisfying the assumptions above such that $\mathcal{M}$ bounded on $L^p(X)$ if and only if $p=\mathrm{\infty }$.