Spider's webs and sharp $L^p$ bounds for the Hardy--Littlewood maximal operator on Gromov hyperbolic spaces

TL;DR

The paper addresses sharp $L^p$ bounds for the centred Hardy--Littlewood maximal operator on δ-hyperbolic spaces with $(a,b)$-pinched exponential growth. It introduces spider's webs as a novel discretisation that preserves large-scale geometry via strict rough isometries, and proves that the maximal operator is bounded on $L^p(X)$ for all $p>\tau$ and of weak type $(\tau,\tau)$ with $\tau=\log_a b$ under $1<a\le b< a^2$. The threshold $\tau$ is shown to be optimal, with counterexamples for $p<\tau$ and breakdown to boundedness only at $p=\infty$ when $b>a^2$, and the results extend to Cartan--Hadamard manifolds with pinched negative curvature as well as to Damek--Ricci spaces and trees. The approach combines geometric discretisation, hyperbolic-geometry techniques, and Nazarov–Naor–Tao–type weak-type control to transfer $L^p$ bounds from spider's webs to the ambient spaces, yielding robust, large-scale bounds that are stable under rough isometries.

Abstract

In this paper we prove that if $1<a\leq b<a^2$ and $X$ is a locally doubling $δ$-hyperbolic complete connected length metric measure space with $(a,b)$-pinched exponential growth at infinity, then the centred Hardy--Littlewood maximal operator $\mathcal M$ is bounded on $L^p(X)$ for all $p>τ$, and it is of weak type $(τ,τ)$, where $τ:= \log_ab$. A key step in the proof is a new structural theorem for Gromov hyperbolic spaces with $(a,b)$-pinched exponential growth at infinity, consisting in a discretisation of $X$ by means of certain graphs, introduced in this paper and called spider's webs, with ``good connectivity properties". Our result applies to trees with bounded geometry, and Cartan--Hadamard manifolds of pinched negative curvature, providing new boundedness results in these settings. The index $τ$ is optimal in the sense that if $p<τ$, then there exists $X$ satisfying the assumptions above such that $\mathcal M$ is not of weak type $(p,p)$. Furthermore, if $b>a^2$, then there are examples of spaces $X$ satisfying the assumptions above such that $\mathcal M$ bounded on $L^p(X)$ if and only if $p=\infty$.

Figures (3)

• Figure 1: The red line represents a standard geodesic connecting $x$ to $y$, while the blue line depicts an alternative path between them.
• Figure 2: The figure represents the points $o$, $x$, $x'$, $\lfloor x\rfloor$, $y, y', [y]$, and $p$, as well as the geodesics in $\Gamma_0$ connecting $o$ to $x$ (orange), $o$ to $y$ (blue), and $x$ to $y$ (red). The geodesic in $X$ connecting $x$ to $y$ is represented by a green dashed curve.
• Figure :

Theorems & Definitions (32)

• Theorem 1.1
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Lemma 2.7
• Theorem 2.8
• Definition 2.9