Year: 2026
Type: preprint
Abstract: Let $Λ$ denote the von Mangoldt function; we prove that for each $r > 2$, there exist constants \[ r' < \mathbf{c}(r) < 2 < \mathbf{C}(r), \qquad \lim_{r \to \infty} \mathbf{c}(r) = 1, \ \lim_{r \to \infty} \mathbf{C}(r) = \infty \] so that the discrete variational Carleson operator along the primes \begin{align} \mathcal{V}^r \Big( \sum_{n \neq 0} f(x-n) Λ(|n|) \frac{e^{2πi λn}}{n} : λ\in \mathbb{T} \Big) \end{align} is bounded on $\ell^p$ for all $\mathbf{c}(r) < p < \mathbf{C}(r)$, while the variation is unbounded when $p \leq r'$. At the non-variational endpoint, the same argument gives the sharp maximal result: the prime Carleson operator \[ \sup_{λ\in\mathbb T} \Big|\sum_{n\neq0} f(x-n)Λ(|n|)\frac{e^{2πiλn}}{n}\Big| \] is bounded on \(\ell^p(\mathbb Z)\) for the full expected range \(1
Source: arXiv (Cornell University)
Institutions: King's College London, University of Bristol
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167665164
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