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 $n\geq 2$, for $\alpha\in\mathbb{R}$ and $1<p<\infty$, if $\mathfrak{m}^\alpha$ is bounded on $L^p(\mathbb{H}^n)$, then we must have $\alpha>1-n+n/p$ for $1<p\leq 2$; or $\alpha\geq \max\{1/p-(n-1)/2,(1-n)/p\}$ for $2<p<\infty$. 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 $\mathop{\mathrm{Re}}\alpha> \max \{{(2-n)/p}-{1/(p p_n)},{(2-n)/p}- (p-2)/[p p_n(p_n-2)]\}$ for $2\leq p\leq \infty$, with $p_n=2(n+1)/(n-1)$ for $n\geq 3$ and $p_n=4$ for $n=2$.