Dimensionally Exponential Lower Bounds on the $L^p$ Norms of the Spherical Maximal Operator for Cartesian Powers of Finite Trees and Related Graphs

TL;DR

This paper proves that the $L^p$ norms of the spherical maximal operator on Cartesian powers of finite trees grow exponentially with dimension, contrasting with dimension-independent bounds on finite cliques.

Contribution

It establishes the first exponential lower bounds on the $L^p$ norms of spherical maximal operators for Cartesian powers of finite trees and related graphs.

Findings

$L^p$ norms grow exponentially with dimension $N$

Lower bounds depend on leaf probability $r$

Extension to a class of graphs called global antipode graphs

Abstract

Let $T$ be a finite tree graph, $T^N$ be the Cartesian power graph of $T$, and $d^N$ be the graph distance metric on $T^N$. Also let \[ \mathbb S_r^N(x) := \{v \in T^N: d^N(x,v) = r\} \] be the sphere of radius $r$ centered at $x$ and $M$ be the spherical maximal averaging operator on $T^N$ given by \[ Mf(x) := \sup_{\substack{r \geq 0 \\ \mathbb S_r^N(x) \neq \emptyset}} \frac{1}{|\mathbb S_r^N(x)|} |\sum_{\mathbb S_r^N(x)} f(y)|. \] We will show that for any fixed $1 \leq p \leq \infty$, the $L^p$ operator norm of $M$, i.e. \[ \|M\|_p := \sup_{\|f\|_p = 1} \|Mf\|_p, \] grows exponentially in the dimension $N$. In particular, if $r$ is the probability that a random vertex of $T$ is a leaf, then $\|M\|_p \geq r^{-N/p}$, although this is not a sharp bound. This exponential growth phenomenon extends to a class of graphs strictly larger than trees, which we will call \emph{global antipode graphs}. This growth result stands in contrast to the work of Greenblatt, Harrow, Kolla, Krause, and Schulman that proved that the spherical maximal $L^p$ bounds (for $p > 1$) are dimension-independent for finite cliques.