Let $\operatorname S_\omega f=\int_\omega \widehat f(\xi)e^{ix\xi}\; d\xi$ be the Fourier projection operator to an interval $\omega$ in the real line. Rubio de Francia's Littlewood--Paley inequality (Rubio de Francia, 1985) states that for any collection of disjoint intervals $\Omega$, we have \begin{equation*}\notag \NORM \Biggl[ \sum_{\omega\in\Omega} \abs{\operatorname S_\omega f}^2\Biggr]^{1/2} .p.\lesssim{}\norm f.p.,\qquad 2\le{}p<\infty. \end{equation*} We survey developments related to this inequality, including the higher dimensional case, and consequences for multipliers.

Research supported in part by a National Science Foundation Grant. The author is a Guggenheim Fellow.