Monotone extremal functions and the weighted Hilbert's inequality

TL;DR

This paper constructs optimal exponential majorants for the signum function under monotonicity constraints and applies them to provide a straightforward Fourier analysis proof of a weighted Hilbert inequality.

Contribution

It introduces new monotone extremal functions and uses them to simplify the proof of a classical weighted Hilbert inequality.

Findings

Optimal one-sided majorants of exponential type for the signum function are found.

A simple Fourier analysis proof of the weighted Hilbert-Montgomery-Vaughan inequality is provided.

The results improve understanding of extremal functions and their applications in harmonic analysis.

Abstract

In this note we find optimal one-sided majorants of exponential type for the signum function subject to certain monotonicity conditions. As an application, we use these special functions to obtain a simple Fourier analysis proof of the (non-sharp) weighted Hilbert-Montgomery-Vaughan inequality.