On restricted weak-type constants of Fourier multipliers

Abstract

We exhibit a large class of symbols m: Rd ! C for which the corresponding Fourier multipliers Tm satisfy the following restricted weak-type estimates: if A Rd has finite Lebesgue measure, then... In particular, this leads to novel sharp estimates for the real and imaginary part of the Beurling-Ahlfors operator on C. The proof rests on probabilistic methods: we exploit a stochastic representation of the multipliers in terms of Lévy processes and appropriate sharp inequalities for differentially subordinated martingales.