Optimal control of singular Fourier multipliers by maximal operators

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Colleges, School and Institutes

Abstract

We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form $e^{i|\xi|^\alpha}$ for both $\alpha$ 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 $L^p-L^q$ multipliers, oscillatory integrals and dispersive PDE, along with natural extensions to higher dimensions.

Bibliographic note

22 pages

Details

Original languageEnglish
Pages (from-to)1317–1338
Number of pages22
JournalAnalysis and PDE
Volume7
Issue number6
Publication statusPublished - 18 Oct 2014

Keywords

  • math.CA, 44B20, 42B25, Fourier multipliers, maximal operators, weighted inequalities