Harmonic analysis on product spaces. (English) Zbl 0644.42017

It is well-known that, in the product space \({\mathbb{R}}^ n\times {\mathbb{R}}^ m\), direct analogues of the notions in the case \({\mathbb{R}}^ n\) are not always appropriate. The author discusses several substitutes in several cases. Firstly he introduces the notion of product Marcinkiewicz integral and investigates its relation to the Hardy-Littlewood maximal operators in the coordinate directions. Secondly he shows that atoms supported in rectangles (rectangle atoms) still play important roles to check the boundedness of \(L^ 2\)-bounded operators from \(H^ p({\mathbb{R}}\times {\mathbb{R}})\) to \(L^ p({\mathbb{R}}\times {\mathbb{R}})\), although \(H^ p({\mathbb{R}}\times {\mathbb{R}})\) functions cannot be decomposed into rectangle atoms. Similar considerations are given to the notion of mean oscillation over rectangles discussed by Carleson as a direct analogue of the one in the \({\mathbb{R}}^ n\) case. Thirdly he defines a substitute “sharp operator” for Fefferman-Stein’s sharp maximal function. For \(L^ 2\) bounded operators T with “sharp operators” \(T^{\#}\), he gives a sort of weighted \(L^ 2\) inequality involving generalized Lusin’s area integral and his sharp operator. Finally he gives a large class of singular integrals for which \(T^{\#}f=M_ s(| f|^ 2)^{1/2},\) and hence to which his theorem can be applied.
Reviewer: K.Yabuta


42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
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