This book is a collection of lecture notes for the LIASFMA School and Workshop on 'Harmonic Analysis and Wave Equations' which was held on May 8-18, 2017 at Fudan University, in Shanghai, China. The aim of the LIASFMA School and Workshop is to bring together Chinese and French experts to discuss and dissect recent progress in these related fields; and to disseminate state of art, new knowledge and new concepts, to graduate students and junior researchers.The book provides the readers with a unique and valuable opportunity to learn from and communicate with leading experts in nonlinear wave-type equations. The readers will witness the major development with the introduction of modern harmonic analysis and related techniques.

​The theory of distributions constitutes an essential tool in the study of partial differential equations. This textbook would offer, in a concise, largely self-contained form, a rapid introduction to the theory of distributions and its applications to partial differential equations, including computing fundamental solutions for the most basic differential operators: the Laplace, heat, wave, Lam\'e and Schrodinger operators.​

This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.

The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.

Alberto P. Calderón (1920-1998) was one of this century's leading mathematical analysts. His contributions, characterized by great originality and depth, have changed the way researchers approach and think about everything from harmonic analysis to partial differential equations and from signal processing to tomography. In addition, he helped define the "Chicago school" of analysis, which remains influential to this day. In 1996, more than 300 mathematicians from around the world gathered in Chicago for a conference on harmonic analysis and partial differential equations held in Calderón's honor. This volume originated in papers given there and presents timely syntheses of several major fields of mathematics as well as original research articles contributed by some of the finest scholars working in these areas. An important addition to the literature, this book is essential reading for researchers in these and other related fields.

This self-contained text provides an introduction to modern harmonic analysis in the context in which it is actually applied, in particular, through complex function theory and partial differential equations. It takes the novice mathematical reader from the rudiments of harmonic analysis (Fourier series) to the Fourier transform, pseudodifferential operators, and finally to Heisenberg analysis.

Presents an account of the basic facts concerning the linear wave equation and the methods from harmonic analysis that are necessary when studying nonlinear hyperbolic differential equations. This book examines quasilinear equations with small data where the Klainerman-Sobolev inequalities and weighted space-time estimates are introduced.