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We control a broad class of singular (or "rough") Fourier multipliers bygeometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$norm inequalities. The multipliers involved are related to those ofCoifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-typeconditions that permit highly oscillatory factors of the form$e^{i|\xi|^\alpha}$ for both $\alpha$ positive and negative. The maximalfunctions that arise are of some independent interest, involving fractionalaverages associated with tangential approach regions (related to those of Nageland 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 naturalextensions to higher dimensions.

Optimal control of singular Fourier multipliers by maximal operators