The Spherical Maximal Operators on Hyperbolic Spaces

TL;DR

This paper studies the boundedness of spherical maximal operators of complex order on hyperbolic spaces, establishing new conditions on the order parameter for different p-values and improving previous results.

Contribution

It provides new necessary and sufficient conditions for the $L^p$ boundedness of spherical maximal operators on hyperbolic spaces, extending and refining earlier work by El Kohen.

Findings

Derived bounds for $ ext{Re}\alpha$ ensuring $L^p$ boundedness.

Extended the range of $p$ for which boundedness holds.

Improved previous results by El Kohen on the operator.

Abstract

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 $n=2$.