%0 Journal Article
%T Subdyadic square functions and applications to weighted harmonic analysis
%A David Beltran
%A Jonathan Bennett
%J Mathematics
%D 2015
%I arXiv
%X 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 $L^2$-weighted inequalities, in the spirit of a well-known conjecture of E. M. 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.
%U http://arxiv.org/abs/1510.01897v1