The Spherical Maximal Operators on Hyperbolic Spaces

TL;DR

This work analyzes the spherical maximal operators of complex order $\alpha$ on the hyperbolic space $\mathbb{H}^n$ and provides a detailed $L^p$ boundedness theory. By leveraging the Helgason Fourier transform, Harish-Chandra function asymptotics, and analytic continuation, the authors establish sharp necessary conditions on $\alpha$ and $p$ in Theorem 1.1, and derive improved sufficient conditions in Theorem 1.2 through local smoothing estimates for Fourier integral operators in Iwasawa coordinates. The results extend El Kohen's earlier hyperbolic bounds and enhance the understanding of how geometry and oscillatory analysis govern boundedness ranges for $\mathfrak m^\alpha$. The approach showcases a synthesis of hyperbolic harmonic analysis with modern local smoothing and decoupling techniques, yielding precise $(\alpha,p)$-regions that govern boundedness and informing potential extensions to related geometric settings.

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 $1<p<\infty$, if $\mathfrak{m}^α$ is bounded on $L^p(\mathbb{H}^n)$, then we must have $α>1-n+n/p$ for $1<p\leq 2$; or $α\geq \max\{1/p-(n-1)/2,(1-n)/p\}$ for $2<p<\infty$. Furthermore, we improve El Kohen's result [J. Operator Theory 3 (1980)] on $L^p$ boundedness of $\mathfrak{m}^α$ by showing that $\mathfrak{m}^α$ is bounded on $L^p(\mathbb{H}^n)$ provided that $\mathop{\mathrm{Re}}α> \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 $n=2$.

Figures (1)

• Figure 1.1: The $(1/p,\alpha)$-plot for $n\geq 3$. El Kohen's result Ko indicates that $\mathfrak{m}^\alpha$ is bounded on $L^p(\mathbb{H}^n)$ if $(1/p,\alpha)$ is strictly above the dashed folded segments $OBA$. We extend the range of $(1/p,\alpha)$ in El Kohen's result for $2<p<\infty$ to the range strictly above the dotted folded segments $OCB$. In addition, we showed that $\mathfrak{m}^\alpha$ is bounded on $L^p(\mathbb{H}^n)$ only if $(1/p,\alpha)$ is on or above the solid folded segments $ODB$ for $2<p<\infty$, or strictly above the dashed segment $BA$ for $1<p\leq 2$.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5