On Astala's theorem for martingales and Fourier multipliers

We exhibit a large class of symbols m   on RdRd, d≥2d≥2, for which the corresponding Fourier multipliers TmTm satisfy the following inequality. If D, E   are measurable subsets of RdRd with E⊆DE⊆D and |D|<∞|D|<∞, then∫D∖E|TmχE(x)|dx≤{|E|+|E|ln⁡(|D|2|E|),if |E|<|D|/2,|D∖E|+12|D∖E|ln⁡(|E||D∖E|),if |E|≥|D|/2. Here |⋅||⋅| denotes the Lebesgue measure on IRd. When d=2d=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 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.