The spherical maximal operators on hyperbolic spaces

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by Kohen [13]. We prove that when $n\geq 2$, for $\alpha\in\mathbb{R}$ and $11-n+n/p$ for $1 \max \{{(2-n)/p}-{1/(p p_n)}, \ {(2-n)/p} - (p-2)/ [p p_n(p_n-2) ] \} $ for $2\leq p\leq \infty$, with $p_n=2(n+1)/(n-1)$ for $n\geq 3$ and $p_n=4$ for $n=2$.