  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.
