Littlewood–Paley theory

Summary

Littlewood–Paley theory splits a function on $\R^n$ into pieces whose Fourier transforms live in dyadic annuli $\set{2^{j-1} < \abs{\xi} < 2^{j+1}}$, by the Littlewood–Paley decomposition. Its main result is the Littlewood–Paley inequality. It says that, for $1 < p < \infty$, the $L^p$ norm of $f$ is comparable to that of its square function $\left( \sum_j \abs{\Delta_j f}^2 \right)^{1/2}$. So the dyadic pieces behave in $L^p$ almost like orthogonal pieces in $L^2$. The theory underlies multiplier theorems, the description of function spaces such as Sobolev, Besov and Triebel–Lizorkin spaces, and paraproduct decompositions of products.