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Calderón-Zygmund theory for product domains: \(H^ p\) spaces. (English) Zbl 0602.42023

This paper proves a theorem about singular integral operators on Hardy spaces. The theorem is that if T is a linear operator on \(L^ 2\) and if T maps p-atoms into functions which decay (in a very mild sense) at infinity, then T satisfies a weak ype (1,1) estimate.
There are two surprising features of this paper. One is that, even though the atomic decomposition fails for \(H^ p\), atoms can be used to test operators. The other is that the proof is so clean. The paper is written in a leisurely style and is a pleasure to read.
Reviewer: S.G.Krantz

MSC:

42B30 \(H^p\)-spaces
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
31B35 Connections of harmonic functions with differential equations in higher dimensions
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