A note on discrete spherical averages over sparse sequences

Abstract

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