The Spherical Maximal Operators on Hyperbolic Spaces

Abstract

In this article we investigate $L^p$ boundedness of the spherical maximal operator ${\mathfrak{m}}^{\hat{I}}\pm$ of (complex) order $\hat{I}\pm$ 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 $\hat{I}\pm \in \mathbb{R}$ and $1<p<\infty$, if ${\mathfrak{m}}^{\hat{I}}\pm$ is bounded on $L^p({\mathbb{H}}^n)$, then we must have $\hat{I}\pm >1-n+n/p$ for $1<p\le 2$; or $\hat{I}\pm \ge \max \{1/p-(n-1)/2,(1-n)/p\}$ for $2<p<\infty$. Furthermore, we improve El Kohen's result [J. Operator Theory 3 (1980)] on $L^p$ boundedness of ${\mathfrak{m}}^{\hat{I}}\pm$ by showing that ${\mathfrak{m}}^{\hat{I}}\pm$ is bounded on $L^p({\mathbb{H}}^n)$ provided that $\mathrm{Re}\hat{I}\pm >\max \{(2-n)/p-1/(p{p}_n),(2-n)/p-(p-2)/[p{p}_n(p_n-2)]\}$ for $2\le p\le \infty$, with $p_n=2(n+1)/(n-1)$ for $n\ge 3$ and $p_n=4$ for $n=2$.