  Through the study of novel variants of the classical Littlewood-Paley-Stein
$g$-functions, we obtain pointwise estimates for broad classes of
highly-singular Fourier multipliers on $\mathbb{R}^d$ satisfying regularity
hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to
efficiently bound such multipliers by geometrically-defined maximal operators
via general weighted $L^2$ inequalities, in the spirit of a well-known
conjecture of Stein. Our framework applies to solution operators for dispersive
PDE, such as the time-dependent free Schr\"odinger equation, and other highly
oscillatory convolution operators that fall well beyond the scope of the
Calder\'on-Zygmund theory.
