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Sharp Localized Inequalities for Fourier Multipliers

Published online by Cambridge University Press:  20 November 2018

Adam Osękowski
Adam Osękowski*
Affiliation:
Department of Mathematics, Informatics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland. e-mail: ados@mimuw.edu.pl
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Abstract

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In this paper we study sharp localized ${{L}^{q}}\,\to \,{{L}^{p}}$ 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 2×2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling–Ahlfors operator.

Keywords

42B1560G4442B20Fourier multipliermartingalelaminate
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 6 , 01 December 2014 , pp. 1358 - 1381
Copyright
Copyright © Canadian Mathematical Society 2014

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