On Astala's theorem for martingales and Fourier multipliers

We exhibit a large class of symbols $m$ on ${\mathbb{R}}^d$, $d\ge 2$, for which the corresponding Fourier multipliers $T_m$ satisfy the following inequality. If $D$, $E$ are measurable subsets of ${\mathbb{R}}^d$ with $E\subseteq D$ and $\mathrm{\mid }D\mathrm{\mid }<\mathrm{\infty }$, then $${\int }_{D\setminus E}\mathrm{\mid }{T}_m{\chi }_E(x)\mathrm{\mid }dx\le \{\begin{array}{lll}\mathrm{\mid }E\mathrm{\mid }+\mathrm{\mid }E\mathrm{\mid }\ln \left(\frac{\mathrm{\mid }D\mathrm{\mid }}{2\mathrm{\mid }E\mathrm{\mid }}\right),& if\mathrm{\mid }E\mathrm{\mid }<\mathrm{\mid }D\mathrm{\mid }\mathrm{/}2,\mathrm{\mid }D\setminus E\mathrm{\mid }+\frac{1}{2}\mathrm{\mid }D\setminus E\mathrm{\mid }\ln \left(\frac{\mathrm{\mid }E\mathrm{\mid }}{\mathrm{\mid }D\setminus E\mathrm{\mid }}\right),& if\mathrm{\mid }E\mathrm{\mid }\ge \mathrm{\mid }D\mathrm{\mid }\mathrm{/}2.\end{array}\mathrm{.}$$ Here $\mathrm{\mid }\cdot \mathrm{\mid }$ denotes the Lebesgue measure on ${\text{bR}}^d$. 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.