Harmonic Analysis

Rating:
Summary: Nothing better than learning from the experts.
Review: If you get involved in analysis sooner or later you will have to read Elias Stein. Why? Because the core of analysis is harmonic analysis, and this man has been one of the leading experts in the field over at least 35 years, so, whatever branch of analysis you choose, Dr. Stein will be there.

This thick book (695 pages) includes most of the topics in harmonic analysis which have been researched extensively during the last 20 years. It should be taken as a complement to Stein's "Singular Integrals and Differentiability Properties of Functions" (1970) and Stein & Weiss' "Introduction to Fourier Analysis on Euclidean Spaces" (1971).

Its contents are: Real variable theory (covering lemmas, maximal function, generalized Calderón-Zygmund decomposition, etc.), maximal functions (vector-valued max. functions, Carlesson measures, etc.), Hardy spaces (characterization, atomic decomposition, singular integrals, etc.), H^1 and BMO (sharp function, interpolation, etc.), weighted inequalities (the class A_p, etc.), Fourier transform, almost orthogonality, oscillatory integrals, maximal operators, maximal averages, the Heisenberg group.

The style is concise and presents motivation for each topic. Includes excercises - called "further results" - which are short research topics on their own, and an extensive list of references. The cloth bound is lovely.

Superb book; a must in every analyst's library.

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