Two-parameter version of Bourgain's inequality: Rational frequencies

TL;DR

This paper extends Bourgain's maximal logarithmic inequality to two parameters for rational frequencies in $L^2( eal^2)$, using a novel two-parameter Rademacher--Menschov inequality to control oscillations.

Contribution

It introduces the first two-parameter version of Bourgain's inequality for rational frequencies, advancing harmonic analysis techniques.

Findings

Established a two-parameter maximal inequality for rational frequencies.

Developed a new two-parameter Rademacher--Menschov inequality.

Controlled oscillation seminorms in the two-parameter setting.

Abstract

Our aim is to establish the first two-parameter version of Bourgain's maximal logarithmic inequality on $L^2(\mathbb R^2)$ for the rational frequencies. We achieve this by introducing a variant of a two-parameter Rademacher--Menschov inequality. The method allows us to control an oscillation seminorm as well.