A modification of the Chang-Wilson-Wolff Inequality via the Bellman Function

TL;DR

This paper introduces a Bellman function approach to establish sharp inequalities in harmonic analysis, demonstrating its application through a new result related to the Chang-Wilson-Wolff Inequality, inspired by Burkholder's work.

Contribution

It presents a novel application of the Bellman function technique to prove a new inequality connected to the Chang-Wilson-Wolff Inequality, expanding its use in harmonic analysis.

Findings

Bellman function technique effectively proves sharp inequalities.

New inequality related to Chang-Wilson-Wolff established.

Method builds on Burkholder's approach to martingale transforms.

Abstract

We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we present how it was originally used by Donald Burkholder to prove $L^p$ boundedness of the $\pm 1$ martingale transform. Finally, with Burkholder's result as a blueprint, we use the Bellman function to prove a new result related to the Chang-Wilson-Wolff Inequality.