## 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

### MSC:

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