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

Subdyadic square functions and applications to weighted harmonic analysis