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

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.

17 pages, no figures