Modern Fourier analysis. 3rd ed. (English) Zbl 1304.42002

Grafakos, Loukas

Graduate Texts in Mathematics 250. New York, NY: Springer (ISBN 978-1-4939-1229-2/hbk; 978-1-4939-1230-8/ebook). xvi, 624 p. (2014).

Publisher’s description: This text is addressed to graduate students in mathematics and to interested researchers who wish to acquire an in depth understanding of Euclidean Harmonic analysis. The text covers modern topics and techniques in function spaces, atomic decompositions, singular integrals of nonconvolution type, and the boundedness and convergence of Fourier series and integrals. The exposition and style are designed to stimulate further study and promote research. Historical information and references are included at the end of each chapter.
This third edition includes a new chapter entitled “Multilinear Harmonic Analysis” which focuses on topics related to multilinear operators and their applications. Sections 1.1 and 1.2 are also new in this edition. Numerous corrections have been made to the text from the previous editions and several improvements have been incorporated, such as the adoption of clear and elegant statements. A few more exercises have been added with relevant hints when necessary.
Editorial remark: See Zbl 1158.42001 for a review of the second edition.

MSC:

42-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
42B35	Function spaces arising in harmonic analysis
42B30	\(H^p\)-spaces
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25	Maximal functions, Littlewood-Paley theory
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

smoothness; Hardy spaces; Sobolev spaces; Besov spaces; Triebel-Lizorkin spaces; atomic decomposition; BMO; Carleson measures; weighted inequalities; time-frequencey analysis; singular integrals