Hardy-Littlewood maximal operator on spaces of exponential volume growth

TL;DR

This paper investigates the Hardy-Littlewood maximal operator on spaces with exponential volume growth, establishing weak-type inequalities under various geometric and algebraic conditions, including for groups like lattices in Lie groups and hyperbolic groups.

Contribution

It proves weak-type maximal inequalities for the Hardy-Littlewood operator on spaces with exponential growth, extending results to diverse groups and geometric structures.

Findings

Weak-type $( ext{log} ext{L})^c$ inequality for certain groups

Weak-type $(1,1)$ inequality for hyperbolic groups

Applicability to lattices in Lie groups and Artin, Coxeter, braid groups

Abstract

We consider the Hardy-Littlewood maximal function associated with ball averages on spaces with exponential volume growth. We focus on discrete groups with balls defined by invariant metrics associated with a variety of length functions. Under natural assumptions on the rough radial structure of the group in question, we establish a weak-type $\mathcal{L}\left(\log \mathcal{L}\right)^{\bf c}$ maximal inequality for the Hardy-Littlewood maximal function. We give a variety of examples where the rough radial structure assumptions hold, based on considerations from geometric group theory, or on analytic considerations related to the regular representation of the group. We elucidate the connections of these assumptions to a spherical coarse median inequality, to almost exact polynomial-exponential growth of balls, and to the radial rapid decay property. In particular, the weak-type maximal inequality in $\mathcal{L}\left(\log \mathcal{L}\right)^{\bf c}$ is established for any lattice in a connected semisimple Lie group with finite center, with respect to the distance function restricted from the Riemannian distance on symmetric space to an orbit of the lattice. It is also established for right-angled Artin groups, Coxeter groups and braid groups, for a suitable choice of word metric. For non-elementary word-hyperbolic group we establish that the Hardy-Littlewood maximal operator with respect to balls defined by a word length satisfies the weak-type $(1,1)$ maximal inequality, which is the optimal result.