This is a very good book for graduate students and for mathematicians interested in Fourier analysis and PDEs. Based on the title, one might expect that the subject matter would be closer to that of a traditional PDEs textbook, either oriented more towards Functional Analysis or dedicated mostly to applications. This book is different.
The first comparison that springs to mind is probably Lars Hörmander’s treatise on linear partial differential operators, even though the book under review is much smaller in scale. The author presents the bases of the theory of distributions and Fourier transform and then goes on to discuss quite advanced topics such as inverse backscattering, Fourier integral operators, the Atiyah-Singer Index Theorem, and the oblique derivative problem. Some of the results presented are relatively recent and closely related to the author’s own research.
The Fourier transform is a constant presence throughout these “lectures.” The choice of topics is progressive and the presentation is clear, rigorous and inviting. The reader feels encouraged to learn what follows next. At the end of each of the eight chapters there are problem sets, consisting mostly in exercises of theoretical nature which complement the text. The exercises are suitably chosen, and most contain hints or references to the original research.
The book is very well written. While it is too sophisticated for a first course in PDEs, it can definitely be used as a textbook for an advanced graduate class. I would recommend this book without reservations to anyone who wants an unambiguous and fast introduction to an eclectic selection of topics in linear PDEs.
Florin Catrina is Assistant Professor of Mathematics at St. John's University in Queens, New York.