On Astala's theorem for martingales and Fourier multipliers

TL;DR

This paper establishes sharp inequalities for Fourier multipliers related to martingales, including the Beurling-Ahlfors operator, using probabilistic methods inspired by quasiconformal map distortion results.

Contribution

It introduces a broad class of Fourier multipliers satisfying new sharp inequalities, connecting harmonic analysis with martingale inequalities and quasiconformal map theory.

Findings

Inequalities hold for Fourier multipliers including the Beurling-Ahlfors operator.

The inequalities are sharp for the real and imaginary parts of the Beurling-Ahlfors operator.

Probabilistic methods and martingale inequalities are used to prove these results.

Abstract

We exhibit a large class of symbols $m$ on $\R^d$, $d\geq 2$, for which the corresponding Fourier multipliers $T_m$ satisfy the following inequality. If $D$, $E$ are measurable subsets of $\R^d$ with $E\subseteq D$ and $|D|<\infty$, then $$ \int_{D\setminus E} |T_{m}\chi_E(x)|\mbox{d}x\leq \begin{cases} |E|+|E|\ln\left(\frac{|D|}{2|E|}\right), & \mbox{if}|E|<|D|/2, |D\setminus E|+\frac{1}{2}|D \setminus E|\ln \left(\frac{|E|}{|D\setminus E|}\right), & \mbox{if}|E|\geq |D|/2. \end{cases}. $$ Here $|\cdot|$ denotes the Lebesgue measure on $\bR^d$. When $d=2$, these multipliers include the real and imaginary parts of the Beurling-Ahlfors operator $B$ and hence the inequality is also valid for $B$ with the right-hand side multiplied by $\sqrt{2}$. The inequality is sharp for the real and imaginary parts of $B$. This work is motivated by K. Astala's celebrated results on the Gehring-Reich conjecture concerning the distortion of area by quasiconformal maps. The proof rests on probabilistic methods and exploits a family of appropriate novel sharp inequalities for differentially subordinate martingales. These martingale bounds are of interest on their own right.