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Hardy spaces on the Euclidean space. (English) Zbl 0984.42015

Springer Monographs in Mathematics. Berlin: Springer. xiii, 305 p. (2001).
From the book-cover: “This monograph is based on a draft that the author had completed by 1990. It deals with the theory of real Hardy spaces on the \(n\)-dimensional Euclidean space. The author explains in detail some important results on Hardy-spaces using real-variable methods. In particular, the atomic decomposition of elements in Hardy spaces and the author’s own constructive proof of the Fefferman-Stein decomposition of BMO functions into the sum of a bounded function and Riesz transforms of bounded functions is presented.”
In a forward to this book, Peter W. Jones provides an informative and well-written essay about Uchiyama’s personality and his mathematical work. The book is divided into 28 paragraphs and contains at the end a symbol index and a ten-page reference list. Here, the editor should have taken care for a better optical presentation. Each paragraph concludes with a note containing information about the contributed results in the text. The book, written in a very unusual style, contains a wealth of deep results and is recommended to every mathematician interested in the field.

MSC:

42B30 \(H^p\)-spaces
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42B35 Function spaces arising in harmonic analysis
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