Sharp Localized Inequalities for Fourier Multipliers

In this paper we study sharp localized L-q -> L-P estimates for Fourier multipliers resulting from modulation of the jumps of Levy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on 2 x 2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator.