Stability in Burkholder's differentially subordinate martingales inequalities and applications to Fourier multipliers

TL;DR

This paper investigates stability estimates for extremal functions related to the Beurling-Ahlfors operator's $L^p$ bounds, using probabilistic methods and martingale inequalities to extend results to broader Fourier multipliers.

Contribution

It introduces new stability inequalities for Fourier multipliers based on martingale inequalities, expanding the class of operators analyzed.

Findings

Established stability estimates for extremal functions of the Beurling-Ahlfors operator.

Extended stability results to a larger class of Fourier multipliers.

Demonstrated the effectiveness of probabilistic methods in harmonic analysis.

Abstract

We study stability estimates for the almost extremal functions associated with the $L^p$-bound for the real and imaginary parts of the Beurling-Ahlfors operator. The proof exploits probabilistic methods and rests on analogous results for differentially subordinate martingales which are of independent interest. This allows us to obtain stability inequalities for a larger class of Fourier multipliers.