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

TL;DR

This paper investigates the boundedness of Hardy--Littlewood maximal operators on Riemannian manifolds with bounded geometry, revealing stability under conformal changes and establishing sharp estimates on models with negative curvature.

Contribution

It provides new stability results under conformal metric changes, sharp $L^p$ bounds on negatively curved models, and insights into boundedness properties on complex geometric structures.

Findings

Centered operator is weak type (1,1) on connected sums of negative curvature spaces.

Uncentered operator is only bounded on $L^ abla$ for the space of essentially bounded functions.

Shared boundedness properties for quasi-isometric spaces with doubling measures.

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 Str\"omberg, providing new interesting results.