Hardy-Littlewood maximal operator on spaces of exponential volume growth

TL;DR

This work studies the Hardy-Littlewood maximal operator for ball averages on spaces with exponential volume growth, focusing on finitely generated groups equipped with invariant length functions and radial structure. The authors develop a robust framework based on spherical shells, radial rapid decay, and the spherical coarse median inequality to obtain weak-type maximal inequalities, including $\mathcal{L}(\log\mathcal{L})^{\bf c}$ bounds with an explicit ${\bf c}$ dependent on geometric parameters. They apply the theory to broad classes such as lattices in connected semisimple Lie groups, right-angled Artin groups, Coxeter and braid groups, and non-elementary word-hyperbolic groups, obtaining optimal $(1,1)$-type results for the hyperbolic case and concrete bounds in other settings. The paper also analyzes rational growth and almost exact polynomial-exponential growth, extends results to lattice subgroups and product groups, and discusses the sharpness and potential optimality of the exponent ${\bf c}$, highlighting open questions and directions for further study.

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.

Figures (2)

• Figure 1: $\delta$-thin triangle. $1$ is the identity element $e$.
• Figure 2: Schematic pictures for the $\delta$-thin triangles

Theorems & Definitions (44)

• Definition 1
• Remark 2
• Definition 3
• Definition 4: rapid decay of spherical shell correlations
• Proposition 5
• proof
• Proposition 6
• proof
• Proposition 7
• proof