# Raphaël Danchin: A primer on Littlewood-Paley decomposition

Raphaël Danchin (Université Paris-Est Créteil)
### Abstract

This lecture is a very basic and short introduction to Fourier analysis and paradifferential calculus.
After motivating the introduction of a device allowing for splitting
any tempered distribution into a countable sum of spectrally localized functions,
we introduce the so-called Littlewood-Paley decomposition.
This enables us to define the standard Besov spaces (that may be seen as a natural
generalization of more classical spaces like e.g. the Holder or Sobolev ones)
and establish various embeddings.
Then we introduce the paraproduct and remainder operators that allow
to decompose any product of tempered distributions (whenever it makes sense)
into a sum of three terms corresponding to low frequencies / high frequencies
multiplication (and symmetric term)
and multiplication of frequencies of comparable size.
As in application, we give examples of quadratic estimates in various spaces.
Finally, we explain how Littlewood-Paley decomposition allows
to pinpoint optimal regularity estimates for the heat flow.
As an application, we provide an elementary proof of the global well-posedness of the incompressible
Navier-Stokes equations supplemented with small data in critical regularity spaces.
References
[1] H. Bahouri, J.-Y. Chemin and R.~Danchin:
*Fourier analysis and nonlinear partial differential equations*,
Grundlehren der mathematischen Wissenschaften, 343, Springer (2011).