On restricted weak-type constants of Fourier multipliers

Abstract

We exhibit a large class of symbols $m\colon \mathbb{R}^d\to \mathbb{C}$ for which the corresponding Fourier multipliers $T_m$ satisfy the following restricted weak-type estimates: if $A\subset \mathbb{R}^d$ has finite Lebesgue measure, then

$$||T_m\chi_A||_{p,\infty}\leq \frac{p}{2}e^{(2-p)/p}||\chi_A||_p,\quad p\geq 2.$$

In particular, this leads to novel sharp estimates for the real and imaginary part of the Beurling-Ahlfors operator on $\mathbb{C}$. 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.