Vol. 7, No. 6, 2014

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Optimal control of singular Fourier multipliers by maximal operators

Jonathan Bennett

Vol. 7 (2014), No. 6, 1317–1338
Abstract

We control a broad class of singular (or “rough”) Fourier multipliers by geometrically defined maximal operators via general weighted ${L}^{2}\left(ℝ\right)$ 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 ${e}^{i|\xi {|}^{\alpha }}$ 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.

 Dedicated to the memory of Adela Moyua, 1956–2013.
Keywords
Fourier multipliers, maximal operators, weighted inequalities
Mathematical Subject Classification 2010
Primary: 42B15, 42B25, 42B20
Secondary: 42B37