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2014 Optimal control of singular Fourier multipliers by maximal operators
Jonathan Bennett
Anal. PDE 7(6): 1317-1338 (2014). DOI: 10.2140/apde.2014.7.1317


We control a broad class of singular (or “rough”) Fourier multipliers by geometrically defined maximal operators via general weighted L2() norm inequalities. The multipliers involved are related to those of Coifman, Rubio de Francia and Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form ei|ξ|α for both α positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more novel “improper fractional averages” associated with “escape” regions. Some applications are given to the theory of LpLq multipliers, oscillatory integrals and dispersive PDE, along with natural extensions to higher dimensions.


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Jonathan Bennett. "Optimal control of singular Fourier multipliers by maximal operators." Anal. PDE 7 (6) 1317 - 1338, 2014.


Received: 7 June 2013; Accepted: 12 July 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1304.42028
MathSciNet: MR3270165
Digital Object Identifier: 10.2140/apde.2014.7.1317

Primary: 42B15 , 42B20 , 42B25
Secondary: 42B37

Keywords: Fourier multipliers , maximal operators , weighted inequalities

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 6 • 2014
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