Lacunary Spherical Maximal Operators on Hyperbolic Spaces

We prove that the lacunary spherical maximal operator, defined on the $n$-dimensional real hyperbolic space, is bounded on L^p(\H^n) for all $n\ge 2$ and $1<p\le \mathrm{\infty }$. In particular, the lacunary set is significantly larger than its Euclidean counterpart, reflecting the influence of the geometry at infinity of the hyperbolic space.