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 \[ \sup_{l\geq 1}\frac{1}{r(\lambda_l)}\left|\sum_{|x|^2=\lambda_l}f(y-x)\right|\] 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.