---
tag: 640A1MTM4Z
kind: definition
name: Paraproduct
msc: [42B25, 42B20, 35S50]
defines: 9RCFS9W90H
uses: [D1PAT0K446, SASP14MA1Z, DPVM34ADZV]
sources:
  - url: https://en.wikipedia.org/wiki/Paraproduct
---

## Definition

Let $n \ge 1$, and let $\chi$, $\psi$ and $\Delta_j$, $j \in \Z$, be as in the [[Littlewood–Paley decomposition|D1PAT0K446]] on $\R^n$.
For $m \in \Z$ and a bounded function $g$, let $S_m g$ be defined by $\widehat{S_m g}(\xi) = \chi(2^{-m}\xi) \hat g(\xi)$, where $\hat{\ }$ is the [[Fourier transform|DPVM34ADZV]] of tempered distributions.
For $f$ in the [[Lp space|SASP14MA1Z]] $L^2(\R^n)$ and $g \in L^\infty(\R^n)$, the *paraproduct* of $f$ by $g$ is
$$\Pi_g f = \sum_{j \in \Z} (S_{j-3} g) \, (\Delta_j f),$$
where the series converges in $L^2(\R^n)$.

## Remarks

The function $S_m g$ is the convolution of $g$ with $2^{mn} \check\chi(2^m \cdot)$, a smooth average of $g$ at scale $2^{-m}$, so $\abs{S_m g} \le C \norm{g}_\infty$.
The Fourier transform of the $j$-th term is supported in $\set{2^{j-2} \le \abs{\xi} \le 2^{j+2}}$, because $\Delta_j f$ has frequencies between $2^{j-1}$ and $2^{j+1}$ and $S_{j-3} g$ has frequencies below $2^{j-2}$.
So the terms are almost orthogonal, which gives the convergence and the bound $\norm{\Pi_g f}_2 \le C \norm{g}_\infty \norm{f}_2$.
The shift $3$ is a convention; other authors use a different shift or a different cut-off.

Grouping the terms of $fg = \sum_{j, k} \Delta_j f \, \Delta_k g$ according to $k \le j - 3$, $j \le k - 3$ or $\abs{j - k} \le 2$ gives Bony's decomposition
$$fg = \Pi_g f + \Pi_f g + \sum_{\abs{j - k} \le 2} \Delta_j f \, \Delta_k g$$
for $f$ and $g$ both in $L^2(\R^n) \cap L^\infty(\R^n)$, since $\sum_{k \le m} \Delta_k g = S_m g$.

Example: if $g = c$ is a constant, then $\hat g$ is a multiple of the point mass at $0$ and $\chi(0) = 1$, so $S_m g = c$ for every $m$.
Hence $\Pi_g f = c \sum_j \Delta_j f = c f$.

Non-example: the term $\Delta_j f \, \Delta_j g$, a product of two pieces at the same frequency, is not part of a paraproduct.
Its Fourier transform can be supported near $0$, for instance when $\widehat{\Delta_j g}$ is the reflection of $\widehat{\Delta_j f}$, so high frequencies combine into low ones; such terms go into the remainder.