Lecture Notes
Cours d'Intégration en ligne. Cette page (en français) contient un cours sur la théorie de l'intégration et des exercices corrigés.
Sa lecture pourra être complétée par le livre
A Course on Integration Theory
These lecture notes contain some basic elements of Functional Analysis, Distribution Theory and Fourier Analysis.
Un cours d'introduction aux fonctions spéciales.
These notes contain an introduction to Partial Differential Equations.
These notes describe first some elements of Fourier Analysis and
some Basic Convolution Inequalities (Young, Hardy-Littlewood-Sobolev).
Next are given the Marcinkiewicz Interpolation Theorem, the Lebesgue Differentiation Theorem,
the Gagliardo-Nirenberg inequality and the Sobolev Injection Theorems.
The last chapter is devoted to an introduction to Pseudodifferential Operators.
These notes contain an introduction to Carleman Inequalities.
The first chapter describes the basic tools related to Carleman inequalities:
notions of well-posed, ill-posed problems, and the basic Carleman idea is given a detailed treatment via an example.
This chapter gives a proof of Calderón's uniqueness Theorem on simple characteristics as well as Hörmander's Theorem under the pseudo-convexity assumption.
The second chapter is devoted to second-order Elliptic Operators with jumps and to proving Carleman inequalities for this type of operators.
The third chapter deals with the notion of conditional-pseudo-convexity, useful in Lorentzian geometry.