A note on discrete spherical averages over sparse sequences

Abstract

This note presents an example of an increasing sequence $({\lambda }_l{)}_{l=1}^{\infty }$ such that the maximal operators associated to normalized discrete spherical convolution averages \[ \sup_{l\geq 1}\frac{1}{r(λ_l)}\left|\sum_{|x|^2=λ_l}f(y-x)\right|\] for functions $f:{\mathbb{Z}}^n\to {\mathbb{C}}^n$ are bounded on ${\ell }^p$ for all $p>1$ when the ambient dimension $n$ is at least five.