paraproduct

In mathematics, a paraproduct is a non-commutative bilinear operator acting on functions that in some sense is like the product of the two functions it acts on. According to Svante Janson and Jaak Peetre, in an article from 1988,Svante Janson and Jaak Peetre, [https://www.jstor.org/stable/2000875 "Paracommutators-Boundedness and Schatten-Von Neumann Properties"], Transactions of the American Mathematical Society, Vol. 305, No. 2 (Feb., 1988), pp. 467–504. "the name 'paraproduct' denotes an idea rather than a unique definition; several versions exist and can be used for the same purposes." The concept emerged in J.-M. Bony’s theory

of paradifferential operators.{{cite journal |last1=Bony |first1=J.-M. |title=Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires |journal=Ann. Sci. Éc. Norm. Supér. |date=1981 |volume=14 |issue=2 |pages=209–246|doi=10.24033/asens.1404 |doi-access=free }}

This said, for a given operator $\backslash Lambda$ to be defined as a paraproduct, it is normally required to satisfy the following properties:

• It should "reconstruct the product" in the sense that for any pair of functions $(f,\; g)$ in its domain,

:: $fg\; =\; \backslash Lambda(f,\; g)\; +\; \backslash Lambda(g,\; f).$

• For any appropriate functions $f$ and $h$ with $h(0)=0$, it is the case that $h(f)\; =\; \backslash Lambda(f,\; h\text{'}(f))$.
• It should satisfy some form of the Leibniz rule.

A paraproduct may also be required to satisfy some form of Hölder's inequality.