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