Variational estimates for averages and truncated singular integrals along the prime numbers

TL;DR

This paper establishes unified r-variational estimates for averages and truncated singular integrals along prime numbers on bcl^{s}(bZ) spaces, extending harmonic analysis tools to prime number settings.

Contribution

It provides the first unified proof of r-variational estimates for prime-based averages and singular integrals, for all s in (1, bInfinity) and r > 2.

Findings

Proves r-variational bounds for prime averages

Establishes bounds for truncated singular integrals along primes

Extends harmonic analysis techniques to prime number sequences

Abstract

We prove, in a unified way, $r$-variational estimates, $r>2$, on $\ell^{s}(\mathbb{Z})$ spaces, $s \in (1, \infty)$, for averages and truncated singular integrals along the set of prime numbers.