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 Lévy processes. We exhibit a large class of functions m: ℝ^d → ℂ, for which the corresponding multipliers T_m satisfy the estimatesfor 1 < p < 2, and for 2 ≤ p < ∞. 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 ℂ.