Muckenhoupt weights for the Hardy-Littlewood maximal operator on hyperbolic spaces

On the real hyperbolic space ${\mathbb{H}}^n$, the classical Muckenhoupt condition does not describe the weights $w$ for which the Hardy-Littlewood maximal operator $M$ is bounded on $L^p(w)$. We show that, for $1<p<\mathrm{\infty }$, $M$ is bounded on $L^p(w)$ if and only if $w$ is a local $A_p$ weight and satisfies a two-layer condition, which compares $w$ and its dual weight on two layers of the shadow of a point. We also treat the weak type case, radial weights and the range $p\le 1$, and we prove a weighted Herz principle for radial convolution operators.