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

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.

Dedicated to the memory of Adela Moyua, 1956–2013.

Fourier multipliers, maximal operators, weighted inequalities
Mathematical Subject Classification 2010
Primary: 42B15, 42B25, 42B20
Secondary: 42B37
Received: 7 June 2013
Accepted: 12 July 2014
Published: 18 October 2014
Jonathan Bennett
School of Mathematics
The University of Birmingham
The Watson Building
B15 2TT
United Kingdom