A Discrete Carleson Theorem Along the Primes with a Restricted Supremum

TL;DR

This paper establishes boundedness of a discrete maximal function along primes with a restricted supremum over certain sets, extending Carleson-type theorems to a prime-based discrete setting.

Contribution

It introduces new conditions on the set \\Lambda, including lacunary sets, ensuring boundedness of a prime-based discrete maximal function on \\ell^p for specific p ranges.

Findings

Boundedness of the maximal function for \\Lambda with lacunary structure

Valid for \\ell^p with 1.5 < p < 4

Extends Carleson theorem to prime-based discrete operators

Abstract

Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_\Lambda f(n) := \sup_{\lambda \in \Lambda} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2\pi i \lambda p}}{p} |,\] 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 $\frac{3}{2} < p < 4$.