Lecture Notes

Lectures on Integration

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 à paraître, A Course on Integration Theory.

Lecture Notes on Real Analysis.

These lecture notes contain some basic elements of Functional Analysis, Distribution Theory and Fourier Analysis.

Fonctions spéciales.

Un cours d'introduction aux fonctions spéciales.

Lecture Notes on Partial Differential Equations

These notes contain an introduction to Partial Differential Equations.

Elements of Graduate Analysis

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.

Carleman Inequalities

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.

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