---
tag: H4E1PW67Z1
kind: concept
name: Littlewood–Paley square function
msc: [42B25]
sources:
  - url: https://en.wikipedia.org/wiki/Littlewood%E2%80%93Paley_theory
---

## 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|D1PAT0K446]].
By Plancherel's theorem its $L^2$ norm is comparable to that of $f$, and by the [[Littlewood–Paley inequality|6GYQNB2862]] 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|KHY35P4AKF]].

## Explanations

### L1 human

Split $f$ into octaves of frequency, look at the size of each octave at a point $x$, and combine these sizes by a square root of a sum of squares.
The result is a new nonnegative function $Sf$, which ignores the phases of the pieces and only remembers their sizes.
Changing the signs of the pieces does not change $Sf$, which explains why such sign changes do not change the $L^p$ norm much.
The square root of a sum of squares is used because it matches the $L^2$ norm exactly when the pieces are orthogonal.