We control a broad class of singular (or “rough”) Fourier multipliers
by geometrically defined maximal operators via general weighted
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
for
both
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
–
multipliers, oscillatory integrals and dispersive PDE, along with natural extensions to
higher dimensions.
Dedicated to the memory of Adela
Moyua, 1956–2013.
