Weak-type inequalities for Fourier multipliers with applications to the Beurling-Ahlfors transform

Abstract

The paper contains the study of weak-type constants of Fourier multipliers resulting from modulation of the jumps of Léevy processes. We exhibit a large class of functions $m: {\mathbb R}^d\to {\mathbb C}$, for which the corresponding multipliers $T_m$ satisfy the estimates

$ \|T_m f\|_{L^{p,\infty}({\mathbb R}^d)} \leq \bigg[ \frac{1}{2}\Gamma \bigg( \frac{2p-1}{p-1} \bigg) \bigg]^{(p-1)/p} \|f\|_{L^p({\mathbb R}^d)} $

for 1 < $p$ < 2, and

$ \|T_m f\|_{L^{p,\infty}({\mathbb R}^d)} \leq \bigg[ \frac{p^{p-1}}{2} \bigg]^{1/p} \|f\|_{L^p({\mathbb R}^d)} $

for $2 \leq p$ < $\infty$. The proof rests on a novel duality method and a new sharp inequality for differentially subordinated martingales. We also provide lower bounds for the weak-type constants by constructing appropriate examples for the Beurling-Ahlfors operator on ${\mathbb C}$.