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 El Kohen. We prove that when $n\ge 2$, for $\alpha \in \mathbb{R}$ and $1<p<\mathrm{\infty }$, if ${\mathfrak{m}}^{\alpha }$ is bounded on $L^p({\mathbb{H}}^n)$, then we must have $\alpha >1-n+n\mathrm{/}p$ for $1<p\le 2$; or $\alpha \ge \max \{1\mathrm{/}p-(n-1)\mathrm{/}2,(1-n)\mathrm{/}p\}$ for $2<p<\mathrm{\infty }$. Furthermore, we improve El Kohen’s result [J. Operator Theory 3 (1980)] on $L^p$ boundedness of ${\mathfrak{m}}^{\alpha }$ by showing that ${\mathfrak{m}}^{\alpha }$ is bounded on $L^p({\mathbb{H}}^n)$ provided that $\alpha >\max \{(2-n)\mathrm{/}p-1\mathrm{/}(p{p}_n),(2-n)\mathrm{/}p-(p-2)\mathrm{/}[p{p}_n(p_n-2)]\}$ for $2\le p\le \mathrm{\infty }$, with $p_n=2(n+1)\mathrm{/}(n-1)$ for $n\ge 3$ and $p_n=4$ for $n=2$.