This book offers a rigorous and self contained introduction to the analytical methods used in nonlinear partial differential equations, with a special emphasis on dispersive equations.

The book addresses a central difficulty of the field: the essential tools of modern PDE theory are fundamental, but they are often scattered across different sources, developed with incompatible notation, or presented without the needs of dispersive analysis in mind. In this book, they are organized into a coherent framework, with complete proofs and a unified point of view. Readers are guided from foundational topics such as Lebesgue and Lorentz spaces, convolution, weak convergence, and Fourier transforms to more advanced techniques including oscillatory integrals, maximal functions, singular integrals, dyadic methods, and spaces of regular functions. This provides the language and techniques needed to study regularity, decay, dispersive estimates, and the behavior of solutions to evolution equations. The final part of the book shows how this analytical toolbox is applied to the Schrödinger and wave equations, highlighting the core ideas of Euclidean dispersive analysis in a concrete and mathematically rigorous way.

By combining breadth, detail, and an explicit focus on applications, the book serves both as a reference work and as a pathway into a demanding area of research. It is intended for graduate students, doctoral students, and early career researchers in analysis, partial differential equations, harmonic analysis, and mathematical physics.

Piero D'Ancona has been a full professor of mathematical analysis at Sapienza Università di Roma since 1997. He earned his PhD from the Scuola Normale Superiore di Pisa and has held extended visiting positions at several prestigious international institutions, including various institutes in Paris, the Courant Institute in New York, and Imperial College in London. Prior to his current role, he served as an assistant professor at the University Tor Vergata (1989–1991), an associate professor at the University of Pisa (1992–1993), and a full professor at the University of L'Aquila (1994–1997).