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Sharp Localized Inequalities for Fourier Multipliers

Published:2014-04-05

Department of Mathematics, Informatics and Mechanics , Warsaw University , Banacha 2, 02-097 Warsaw , Poland
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Abstract

In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of Lévy 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 $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator .
 Keywords: Fourier multiplier, martingale, laminate
 MSC Classifications: 42B15 - Multipliers 60G44 - Martingales with continuous parameter 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.)