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Rating: 
Summary: Must-have advanced text on harmonic analysis.
Review: 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).
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