Subdyadic square functions and applications to weighted harmonic analysis

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

To appear in Advances in Mathematics

References in corpus (3)

• Sharp bounds for $t$-Haar multipliers on $L^2$
• Some Problems in Harmonic Analysis
• Optimal control of singular Fourier multipliers by maximal operators

Cited by in corpus (3)

• Multi-scale sparse domination
• Two-weight extrapolation on function spaces and applications
• Endpoint sparse domination for classes of multiplier transformations