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

摘要

The paper contains the study of weak-type constants of Fourier multipliers resulting from modulation of the jumps of Levy processes. We exhibit a large class of functions m : R~d →C, for which the corresponding multipliers T_m satisfy the estimates ‖T_mf‖_(LP,∞(R~d))≤[1/2 Γ(2p-1/p-1)]~((p-1)/p)‖f‖_(LP(R~d)) for 1 ＜ p ＜ 2, and ‖T_mf‖_(LP,∞(R~d))≤[p~(p-1)/2]~(1/p)‖f‖_(LP(R~d)) 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.