Two-dimensional Hardy-Littlewood theorem for functions with general monotone Fourier coefficients

TL;DR

This paper extends the Hardy-Littlewood theorem to two-dimensional functions with Fourier coefficients that are generally monotone and not necessarily positive, establishing sharp conditions for the theorem's validity.

Contribution

It introduces a two-dimensional Hardy-Littlewood theorem for functions with general monotone Fourier coefficients, including non-positive coefficients, and demonstrates the theorem's sharpness with a counterexample.

Findings

The theorem holds for a broad class of Fourier coefficients with general monotonicity.

Counterexample shows the theorem fails if the class of coefficients is slightly extended.

The result clarifies the limits of Hardy-Littlewood type inequalities in two dimensions.

Abstract

We prove the Hardy-Littlewood theorem in two dimensions for functions whose Fourier coefficients obey general monotonicity conditions and, importantly, are not necessarily positive. The sharpness of the result is given by a counterexample, which shows that if one slightly extends the considered class of coefficients, the Hardy-Littlewood relation fails.