Dimension dependence and dimension-free $\ell^2$ estimates for variation seminorms of spherical means on the hypercube

TL;DR

This paper investigates the behavior of variation seminorms of spherical means on the hypercube, revealing dimension dependence for all variation ranges and dimension-free bounds under specific radius restrictions.

Contribution

It establishes the dimension dependence of $oldsymbol{ ext{ell}}^2$ bounds for all radii and provides conditions for dimension-free estimates when radii are parity-restricted.

Findings

No dimension-free bounds for all radii and all $r$ in $[1, \, \infty)$.

Dimension-free estimates hold when radii are restricted to a fixed parity for $r \, ext{in} \, (2, \, \infty)$.

The results delineate the precise conditions under which dimension-free bounds are achievable.

Abstract

We prove that the $\ell^2 \to \ell^2$ norm of the $r$-variation seminorm of spherical means on the hypercube admits no dimension-free bounds for any $r \in [1, \infty)$ when the variation is taken over all possible radii. Furthermore, we establish that if the radii are restricted to a fixed parity, dimension-free estimates hold for all $r \in (2, \infty)$.