Subdyadic square functions and applications to weighted harmonic analysis

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 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 L² 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 Schrödinger equation, and other highly oscillatory convolution operators that fall well beyond the scope of the Calderón–Zygmund theory.