Directional maximal function along the primes

TL;DR

This paper investigates the behavior of a two-dimensional discrete directional maximal operator along primes, demonstrating that for certain vector sets, the operator's norm can grow significantly with the number of vectors.

Contribution

It introduces a new analysis of the maximal operator along primes, extending previous work on integer directions and revealing growth properties in specific vector configurations.

Findings

Existence of vector sets causing operator norm growth

Growth of the maximal operator's norm with epsilon power

Extension of prior integer-based results to prime directions

Abstract

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.