These are notes from the book “Pseudo-differential Operators and the Nash-Moser Theorem” by S. Alinhac and P. Gérard. I have to introduce myself (or remember the memories of the course I took in my M2) into the theory of pseudo-differential operators and semi-classical analysis. For this purpose I found pedagogical and simplified the introduction to this subject given by Alinhac and Gérard.

The first charper present the main elements of the theory of pseudo-differential operators: symbols, operator symbolic calculus, action in Sobolev spaces, and invariance under change of coordinates.

Introduction

Let $(a(\xi)\in \mathcal{C}^\infty$ be a slowly increasing function, we denote $a(D)$ the operator defined on $\mathcal{S}’(\mathbb{R}^n)$ by \(\mathcal{F}(a(D)u)(\xi):= a(\xi)\hat{u}(\xi), \qquad \forall u\in\mathcal{S}(\mathbb{R}^n).\) The function $a(\xi)$ is called the symbol of the operator $a(D)$.

When we compare the formula \((a(D)u)(x) = (2\pi)^{-n}\int e^{ix\xi}a(\xi)\hat u(\xi)d\xi\) with the Fourier inversion formula of $u$, it shows that $a(D)$ acts on $u$ only modifying the fragment $e^{ix\xi}\hat u$ of frequency $\xi$ by multiplying its ‘amplitude’ $\hat u(\xi)$ by $a(\xi)$. Morover, \(\mathcal{F}(a(D)b(D)u)(\xi) = a(\xi)b(\xi)\hat u(\xi).\) We can write this like \(a(D)b(D) = (ab)(D).\) This is a first example of “symbolic calculus”. In this trivial case, $a(D)$ is a p.d.o with constants coefficients, thereby underlying the fact that its symbol does not depend on $x$.

In this chapter, we associate operators $a(x,D)$ with general symbols $a(x, \xi)$ such that the “amplitude modulating” nature of the action of $a(x,D)$ emphasized above is preserved, and we obtain symbolic calculus of the type: \(a(x, D)b(x, D) = (a \# b)(x, D),\) where $a#b$ is a certain symbol which we know how to calculate, recovering $ab$ if $b$ has constant coefficients.

Example of interest (Laplacian)

Consider the constant coefficients operator $P = \Delta = \partial_1^2 + … + \partial_n^2$ with associated symbol $a(\xi) = -|\xi|^2$. Let us find $E\in D’(\mathbb R^n)$ satisfying $PE = \delta + \omega$ where $\omega\in \mathcal S$ is a smooth “correction”. Formally, taking Fourier transform we see that it suffices to consider \(\hat E(\xi) = -\frac{1 - \chi(\xi)}{|\xi|^2}\) since \(\widehat{PE}(\xi) = -|\xi|^2\hat E(\xi) = 1 - \chi(\xi).\) Such a distribution $E$ is $\mathcal C^\infty$ away from 0, since $\widehat{(ix_j)^k E} = \partial_{\xi_j}^k\hat E \sim \xi_j^{-(2+k)}$ is homogeneous of degree -(2+k) for $|\xi|$ large, and $\widehat{D_j^k E}= \xi_j^k \hat E \sim -\xi_j^k/|\xi|^2$.