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\(H^ p\)-and \(L^ p\)-variants of multiparameter Calderón-Zygmund theory. (English) Zbl 0770.42010

The authors consider Calderón-Zygmund operators on product domains. In particular, they show that under weak conditions on the kernel of the operator and assuming that the behavior of the operator is known in \(L^ 2\) and for certain “rectangle atoms” the boundedness of the operator on product \(H^ p\) spaces can be proven. This gives a multiparameter, but necessarily weaker extension of R. Fefferman’s two parameter theory [cf. R. Fefferman, Ann. Math., II. Ser. 126, 109-130 (1987; Zbl 0644.42017) and J.-L. Journé, Rev. Mat. Iberoam. 1, No. 3, 55-91 (1985; Zbl 0634.42015)].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B15 Multipliers for harmonic analysis in several variables
42B30 \(H^p\)-spaces
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