English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 showthat 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 thedimension $N$. 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 sharpbound.  This exponential growth phenomenon extends to a class of graphs strictlylarger than trees, which we will call \emph{global antipode graphs}. Thisgrowth 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.

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