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= r} be the sphere of radius r centered at x and M be the spherical maximal averaging operator on T N given byf (y) .We will show that for any fixed 1 ≤ p ≤ ∞, the L p operator norm of M , i.e.M p := supgrows exponentially in the dimension N . In particular, if r is the probability that a random vertex of T is a leaf, then M p ≥ 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 global antipode graphs. It stands in contrast to the results of [2], [3], and [1] which together prove that the spherical maximal L p bounds (for p > 1) are dimension-independent for finite cliques.