The Spherical Maximal Operators on Hyperbolic Spaces

Abstract

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^α$ of (complex) order $α$ 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 $α\in\mathbb{R}$ and $1 1-n+n/p$ 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 $n=2$.