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

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 $$Mf(x):=\underset{\begin{array}{c}r\ge 0\\ {\mathbb{S}}_r^N(x)\mathrm{\ne }\mathrm{\varnothing }\end{array}}{\sup }\frac{1}{\mathrm{\mid }{\mathbb{S}}_r^N(x)\mathrm{\mid }}\mathrm{\mid }\sum _{{\mathbb{S}}_r^N(x)}f(y)\mathrm{\mid }\mathrm{.}$$ We will show that for any fixed $1\le p\le \mathrm{\infty }$, the $L^p$ operator norm of $M$, i.e. $$\mathrm{\parallel }M{\mathrm{\parallel }}_p:=\underset{\mathrm{\parallel }f{\mathrm{\parallel }}_p=1}{\sup }\mathrm{\parallel }Mf{\mathrm{\parallel }}_p,$$ grows exponentially in the dimension $N$. In particular, if $r$ is the probability that a random vertex of $T$ is a leaf, then $\mathrm{\parallel }M{\mathrm{\parallel }}_p\ge r^{-N\mathrm{/}p}$, 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.