Subdyadic square functions and applications to weighted harmonic analysis

TL;DR

This paper introduces new subdyadic square functions to analyze highly-singular Fourier multipliers, enabling weighted $L^2$ bounds and applications to dispersive PDEs beyond classical Calderón-Zygmund methods.

Contribution

It develops a novel framework of subdyadic square functions for pointwise estimates of singular multipliers, extending harmonic analysis tools to broader classes of operators.

Findings

Established pointwise bounds for singular Fourier multipliers.

Bounded multipliers by geometric maximal operators in weighted $L^2$ spaces.

Applied results to dispersive PDE solution operators.

Abstract

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.