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