Analysis and Partial Differential Equations (Universitext)

This textbook provides a modern introduction to advanced concepts and methods of mathematical analysis. The first three parts of the book cover functional analysis, harmonic analysis, and microlocal analysis. Each chapter is designed to provide readers with a solid understanding of fundamental concepts while guiding them through detailed proofs of significant theorems. These include the universal approximation property for artificial neural networks, Brouwer's domain invariance theorem, Nash's implicit function theorem, Calderón's reconstruction formula and wavelets, Wiener's Tauberian theorem, Hörmander's theorem of propagation of singularities, and proofs of many inequalities centered around the works of Hardy, Littlewood, and Sobolev. The final part of the book offers an overview of the analysis of partial differential equations. This vast subject is approached through a selection of major theorems such as the solution to Calderón's problem, De Giorgi's regularity theorem for elliptic equations, and the proof of a Strichartz–Bourgain estimate. Several renowned results are included in the numerous examples. Based on courses given successively at the École Normale Supérieure in France (ENS Paris and ENS Paris-Saclay) and at Tsinghua University, the book is ideally suited for graduate courses in analysis and PDE. The prerequisites in topology and real analysis are conveniently recalled in the appendix Preface Part I Functional Analysis Part II Harmonic Analysis Part III Microlocal Analysis Part IV Partial Differential Equations Part V Recap and Solutions to the exercises Acknowledgments Contents Part I Functional Analysis Chapter 1 Topological Vector Spaces Normed Vector Spaces Convex, Bounded, Balanced Sets Finite-Dimensional Spaces Semi-Norms Banach Spaces The Space of Continuous Functions Applications to Neural Networks Exercises Chapter 2 Fixed Point Theorems Differential Calculus Reminders Banach's Fixed Point Theorem The Local Inversion Theorem The Cauchy–Lipschitz Theorem The Caristi and Ekeland Theorems Brouwer's Fixed Point Theorem Invariance of Domain Nash's Theorem Exercises Chapter 3 Hilbertian Analysis, Duality and Convexity Introduction to Hilbert Spaces Hilbert Basis The Hahn–Banach Theorem Lebesgue Spaces Weak Convergence, Weak-* Convergence The Banach–Alaoglu Theorem Exercises Part II Harmonic Analysis Chapter 4 Fourier Series Introduction Square-Integrable Functions Pointwise Convergence and Uniform Convergence Applications of the Plancherel Formula Wiener's Theorem Exercises Chapter 5 Fourier Transform Introduction The Schwartz space Tempered Distributions The Littlewood–Paley Decomposition Exercises Chapter 6 Convolution Definition of the Convolution Product Approximations of the Identity The Hardy–Littlewood Maximal Function Pointwise Convergence of an Approximation of the Identity The Hardy–Littlewood–Sobolev Inequality Sobolev Inequalities Calderón's Reconstruction Formula and Wavelets Wiener's Tauberian Theorem Exercises Chapter 7 Sobolev Spaces Introduction The Poincaré Inequality and the Poisson Problem Sobolev Spaces Defined on any Open Set Fourier Analysis and Sobolev Spaces Sobolev Embeddings Dyadic Characterization of Sobolev Spaces Exercises Chapter 8 Harmonic Functions The mean value property The Fundamental Solution of the Laplacian Regularity of Harmonic Functions Exercises Part III Microlocal Analysis Chapter 9 Pseudo-Differential Operators Symbols Continuity of Pseudo-Differential Operators Generalizations Exercises Chapter 10 Symbolic Calculus Introduction to Symbolic Calculus Oscillatory Integrals Adjoint and Composition Applications of Symbolic Calculus Exercises Chapter 11 Hyperbolic Equations Transport Equations Pseudo-Differential Hyperbolic Equations Exercises Chapter 12 Microlocal Singularities Local Properties The Wave Front Set The Theorem of Propagation of Singularities Nonlinear Problems Exercises Part IV Analysis of Partial Differential Equations Chapter 13 The Calderón Problem Introduction Density of the Products of Harmonic Functions Equations with Variable Coefficients The Sylvester–Uhlmann Theorem An Exercise Chapter 14 De Giorgi's Theorem Introduction Subsolutions and Nonlinear Transformations Moser Iterations The Harnack Inequality Hölder Regularity Exercises Chapter 15 Schauder's Theorem Local Averages and Elliptic Equations Local Averages and Hölder Spaces Campanato's Theorem The Schauder–Campanato Theorem H2 Regularity Regularity of Minimal Surfaces Exercises Chapter 16 Dispersive Estimates The Schrödinger Equation A Strichartz–Bourgain Estimate for KdV Exercises Part V Recap and Solutions to the Exercises Chapter 17 Recap on General Topology Topological Spaces Separability, Compactness and Completeness Baire's Theorem Regular Functions with Compact Support Exercises Chapter 18 Inequalities in Lebesgue Spaces The Lp Spaces Hölder, Minkowski, and Hardy Inequalities Distribution Functions and Marcinkiewicz's Theorem Exercises Chapter 19 Solutions References Notation Index