On Astala's theorem for martingales and Fourier multipliers

We exhibit a large class of symbols m on R-d, d >= 2, for which the corresponding Fourier multipliers T-m satisfy the following inequality. If D, E are measurable subsets of R-d with E subset of D and vertical bar D vertical bar < infinity, then integral(D\E) vertical bar T-m chi E(x)vertical bar dx <= { vertical bar E vertical bar + vertical bar E vertical bar ln (vertical bar D vertical bar/2 vertical bar E vertical bar), if vertical bar E vertical bar < vertical bar D vertical bar/2, vertical bar D\E vertical bar+1/2 vertical bar D\E vertical bar ln (vertical bar E vertical bar/vertical bar D\E vertical bar), if vertical bar E vertical bar >= vertical bar D vertical bar/2. Here vertical bar center dot vertical bar denotes the Lebesgue measure on R-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 root 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. (C) 2015 Elsevier Inc. All rights reserved.