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 StrichartzBourgain 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

Prepares students for research in partial differential equations Guides readers through the proofs of significant theorems Includes challenging exercises covering important results, with detailed solutions to some

Autorentext

Albert Ai is a postdoctoral researcher at the University of Wisconsin-Madison. He received his PhD from the University of California, Berkeley in 2019. His primary research interests include the analysis of nonlinear dispersive PDEs and harmonic analysis. In particular, he has worked on low regularity solutions of fluid models and wave equations. Thomas Alazard is Director of Research at the CNRS and Associate Professor at the École normale supérieure Paris-Saclay. He received his PhD from the University of Bordeaux, France, in 2005. Previously, he worked at the Orsay mathematics department and at the École normale supérieure in Paris. His research focuses on the analysis of partial differential equations, a subject on which he has written several books. Mihaela Ifrim is currently a Professor of Mathematics at the University of Wisconsin, Madison. She studied mathematics in Bucharest and at the University of California, Davis, where she received her Ph.D. in 2012. After a Simons postdoctoral fellowship at UC Berkeley, in 2017 she moved Madison, where she was a Sloan Research Fellow and a CAREER grant recipient. Her work spans many directions in nonlinear partial differential equations, including fluid dynamics and nonlinear dispersive flows, with an emphasis on free boundary problems and on the study of low regularity local and global dynamics of the solutions. Daniel Tataru is a Distinguished Professor in Mathematics at the University of California, Berkeley. He is well known for his substantial contributions to dispersive pde's, also in connection to fluid dynamics, harmonic analysis, geometry, general relativity and free boundary problems. He studied mathematics at the University of Iasi, in Romania, and then at University of Virginia, where he received his Ph.D. in 1992. He spent almost a decade at Northwestern university, before moving to Berkeley. Among other honors, he is a Fellow of both the American Academy of Arts and Sciences and the European Academy of Sciences, as well as a Simons Investigator since 2013.

Inhalt

Part I Functional Analysis.- 1 Topological Vector Spaces.- 2 Fixed Point Theorems.- 3 Hilbertian Analysis, Duality and Convexity.- Part II Harmonic Analysis.- 4 Fourier Series.- 5 Fourier Transform.- 6 Convolution.- 7 Sobolev Spaces.- 8 Harmonic Functions.- Part III Microlocal Analysis.- 9 Pseudo-Differential Operators.- 10 Symbolic Calculus.- 11 Hyperbolic Equations.- 12 Microlocal Singularities.- Part IV Analysis of Partial Differential Equations.- 13 The Calderón Problem.- 14 De Giorgi's Theorem.- 15 Schauder's Theorem.- 16 Dispersive Estimates.- Part V Recap and Solutions to the Exercises.- 17 Recap on General Topology.- 18 Inequalities in Lebesgue Spaces.- 19 Solutions.