Lusin area integral

Summary

The Lusin area integral is a square function built from the harmonic extension $u$ of a function $f$ on $\R^n$ to the upper half-space. At a point $x$ it integrates $\abs{\nabla u}^2$, suitably weighted, over the cone of points $(y, t)$ with $\abs{y - x} < t$. For $1 < p < \infty$ its $L^p$ norm is comparable to that of $f$, as for the Littlewood–Paley square function. It was introduced by Lusin for holomorphic functions on the disc, and it plays a key role in the real-variable theory of Hardy spaces.