Littlewood–Paley square function

Summary

The Littlewood–Paley square function of a function $f$ on $\R^n$ is $Sf = \left( \sum_{j \in \Z} \abs{\Delta_j f}^2 \right)^{1/2}$, where the $\Delta_j f$ are the pieces of the Littlewood–Paley decomposition. By Plancherel's theorem its $L^2$ norm is comparable to that of $f$, and by the Littlewood–Paley inequality the same holds in $L^p$ for $1 < p < \infty$. It turns the frequency information of $f$ into a pointwise quantity that can be estimated with real-variable tools. It is a model for many other square functions, such as the Lusin area integral.