A Discrete Carleson Theorem Along the Primes with a Restricted Supremum

Abstract

Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_Λf(n) := \sup_{λ\in Λ} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2πi λp}}{p} |,\] where $\pm \mathbb{P}:=\{\pm p:p\text{ is a prime}\}$, and $\Lambda \subset [0,1]$. We give sufficient conditions on $\Lambda$, met by (finite unions of) lacunary sets, for this to be a bounded sublinear operator on ${\ell }^p(\mathbb{Z})$ for $\frac{3}{2}<p<4$.