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In this article we present the Littlewood-Paley theory and illustrate the effectiveness of this microlocal analysis tool in the study of partial differential equations, in a context which is the least technical possible. As we shall see below, the Littlewood-Paley theory provides a robust approach not only to the separate study of the various regimes of solutions to nonlinear partial differential equations, but also to the fine study of functional inequalities, and to make them accurate. 1. The Littlewood-Paley theory : a tool that has become indispensable The Littlewood-Paley theory is a localization procedure in the frequency space that, since about three decades ago, has established itself as a very powerful tool in harmonic analysis. The first goal of this text is to present it in a way as simple as possible 1. Its basic idea is contained in two fundamental inequalities known as Bernstein's inequalities, that describe some properties of functions whose Fourier transform have compact support. The first inequality says that, for a tempered distribution 2 in R d whose Fourier transform is supported in a annulus of size λ, to differentiate first and then take the L p norm is the same as to apply a homothety of ratio λ on the L p norm. In the L 2 setting this remarkable property is an easy consequence of the action of the Fourier transform on derivatives and of the Fourier-Plancherel formula. The proof in the case of general L p spaces uses Young's inequalities and the fact that the Fourier transform of a convolution is the product of the Fourier transforms. In the other hand, the second inequality tells us that, for such a distribution, the change from the L p norm to the L q norm, with q ≥ p ≥ 1, costs λ d 1 p − 1 q , which must be understood as a Sobolev embedding. It is proved like the first inequality, using Young's inequalities and the relation between the Fourier transform and the convolution product. Fourier Analysis is at the heart of the Littlewood-Paley theory, which has inspired a large number of my works. It was in conducting experiments on the propagation of heat that Joseph Fourier at the end of the 18th century opened the door to that theory, which was hugely expanded on the 20th century and intervenes in the majority of branches of Physics. In this theory having the name of its creator, one performs the frequency analyis of a function f of L 1 (R d) by the formula : f (ξ) = R d e −ix·ξ f (x) dx .

The Littlewood–Paley Theory: A Common Thread of Many Works in Nonlinear Analysis