Hardy--Littlewood maximal operators on certain manifolds with bounded geometry

TL;DR

The paper investigates the $L^p$ boundedness of the centred $\mathscr{M}$ and uncentred $\mathscr{N}$ Hardy–Littlewood maximal operators on Riemannian manifolds with bounded geometry and exponential volume growth. It develops a framework based on local doubling, the uniform ball size condition, and strict rough isometries, and examines conformal perturbations, rotationally symmetric models with pinched negative scalar curvature, connected sums, and Strömberg-type counterexamples. Key contributions include the $L^p$ stability of $\mathscr{M}$ under conformal changes, sharp $L^p$ bounds on rotational models, and a sharp dichotomy on connected sums where $\mathscr{M}$ remains well-behaved while $\mathscr{N}$ can fail except on $L^\infty$; plus, robustness results under strict quasi-isometries and detailed Strömberg-type analyses. These results extend Hardy–Littlewood maximal operator theory to broad geometric contexts and clarify how curvature, rough isometries, and manifold constructions influence harmonic analysis on noncompact spaces.

Abstract

In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under mild assumptions, $L^p$ estimates for the centred operator are ``stable'' under conformal changes of the metric, and prove sharp~$L^p$ estimates for the centred operator on Riemannian models with pinched negative scalar curvature. Furthermore, we prove that the centred operator is of weak type $(1,1)$ on the connected sum of two space forms with negative curvature, whereas the uncentred operator is, perhaps surprisingly, bounded only on $L^\infty$. We also prove that if two locally doubling geodesic metric measure spaces enjoying the uniform ball size condition are strictly quasi-isometric, then they share the same boundedness properties for both the centred and the uncentred maximal operator. Finally, we discuss some $L^p$ mapping properties for the centred operator on a specific Riemannian surface introduced by Strömberg, providing new interesting results.

Figures (1)

• Figure 1: Examples of functions $\phi_\tau$ for increasing values of $\tau$ and $T$ and constant value of $\delta$.

Theorems & Definitions (30)

• Proposition 2.1
• proof
• Theorem 3.1
• Remark 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4
• Theorem 4.1
• proof