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

Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself.

Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance.

Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.

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).