Singular Integrals and Differentiability Properties of Functions. (PMS-30)

The printing is not clear. For example, there were several places where I could not tell if what was on the page was a >= or <= symbol. This was not just on one page. There were a lot of instances of this. If I had to, I could consult other sources or determine for myself what it should probably say, but for the price of this book, this is completely unacceptable.

This appreciated book constitutes since its first printing one of the finest references on advanced harmonic analysis and some related topics. The author, one of the leading experts in the field, exposes clearly most of the general background as well as recent results, orienting the reader directly to the current trends in research.The book is valuable not only for harmonic analysis speciallists, but for every mathematician who wants to get well trained in some important and subtle topics of analysis which are shown by this approach as being closely related, leading the reader to a deep and thorough understanding.The contents of the book are: Some fundamental notions of real-variable theory; Singular integrals; Riesz transforms, Poisson integrals, and spherical harmonics; The Littlewood-Paley theory and multipliers; Differentiability properties in terms of function spaces; Extensions and restrictions; Return to the theory of harmonic functions; Differentiation of functions; Appendices: Some inequalities; The Marcinkiewicz interpolation theorem; Some elementary properties of harmonic functions; inequalities for Rademacher functions.Includes motivation and detailed explanations for each topic, excercises for each chapter, called "further results", which are small research projects on their own, and extensive references. The printing and the clothbound are exquisite.This kind of material should be included in every graduate mathematics program. Should read companion "Introduction to Fourier Analysis on Euclidean Spaces" (another jewel) by Stein and Weiss, and later the recent volume "Harmonic Analysis" also by Stein, both reviewed by myself.Please take a look at the rest of my reviews (just click on my name above).

Being an algebraic and differential topologist by calling and by education, I used to think that the analysis is mostly an art of combining inequalities into long strings, depending on good luck much more than on any insight. Few very conceptual analytic theories, like the measure theory or the theory of distributions were exceptions. Definitely, the required courses in analysis were to a big extent responsible for this point of view.Early in the graduate school, tired after a series of meaningless exams, but still willing to at least read something mathematical, I had come across this book by E. Stein. I started to read, and was astonished by the elegance of theorems and proofs. Estimates of integrals can be interesting, beautiful, and based on deep and to a big extent geometric insights! This changed my perception of the analysis forever. I did not get very far, but used the book later as a reference.So, my opinion is: if you are really interested in analysis, you should read this book from cover to cover. If not, at least give it a try. It will be fun, and it will give you a more balanced view of mathematics. Some basic (nowadays usually graduate) courses in analysis is a sufficient preparation.

EL libro se deshojó la primera vez que se abrió. Pedí este libro usado, intentando conseguir una impresión que fuese de una calidad decente; dicen que se perdió en el camino...Volví a pedir este libro "nuevo" me enviaron esta basura de print on demand que se deshojó al abrirlo por primera vez. Lo estoy regresando y lo que sugiero es que no lo compren.