Some recent developments in Fourier analysis and \(H^ p\)-theory on product domains. (English) Zbl 0557.42007

Chang, Sun-Yung A.; Fefferman, Robert

Bull. Am. Math. Soc., New Ser. 12, 1-43 (1985).

This article is a survey of results in ”multi-parameter” harmonic analysis, equivalently, harmonic analysis on product spaces. The prototypical such space is the bidisc, and the authors concentrate on this example (equivalently, on the product of half-planes).
The article begins with a brief overview of the ”classical theory” of the 1970’s which was initiated by papers of Stein, Stein/Weiss, and C. Fefferman/Stein; the theory was developed by others too numerous to mention. The article then surveys in more detail the basic developments in the multi-parameter theory: differentiation theory, maximal functions, covering lemmas, singular integrals, \(H^ p\) theory, functions of bounded mean oscillation, the atomic decomposition of \(H^ p\) functions, unconditional bases for \(H^ 1\), the corona problem, and interpolation of operators.
The authors are two of the principal workers in the multi-parameter theory. Thus they are very well qualified to write this survey. They are careful to point out the differences between the one-parameter and multi- parameter theories; this is a real blessing for the novice. The paper is well written. It is a welcome guide to the often difficult and technical literature connected with this subject.

Reviewer: S.Krantz

MSC:

42B30	\(H^p\)-spaces
32A35	\(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25	Maximal functions, Littlewood-Paley theory

Keywords:

product spaces; maximal functions; singular integrals; \(H^ p\); bounded mean oscillation; corona problem; interpolation of operators

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