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Journé’s covering lemma and its extension to higher dimensions. (English) Zbl 0645.42018

The author extends Journé’s covering lemma concerning dyadic coverings which are maximal in one coordinate direction, to the case of product domains \({\mathbb{R}}^{n_ 1}\times...\times {\mathbb{R}}^{n_ m}\) (m\(\geq 3)\) [originally \(m=2\), J. Journé, Proc. Am. Math. Soc. 96, 593-598 (1986; Zbl 0594.42015)]. Using it the author establishes the boundedness of Calderón-Zygmund singular integrals (of product type) from the product Hardy space \(H^ p({\mathbb{R}}^{n_ 1}\times...\times {\mathbb{R}}^{n_ m})\) to \(L^ p({\mathbb{R}}^{n_ 1}\times...\times {\mathbb{R}}^{n_ m})\), which extends R. Fefferman’s result in the case \(m=2\) [Proc. Nat. Acad. Sci. 83, 840-843 (1986; Zbl 0602.42023)]. It is pointed out that several differences occur in two cases \(m=2\) and \(m\geq 3\).
Reviewer: K.Yabuta

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
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[1] S.-Y. A. Chang and R. Fefferman, A continuous version of duality of \(H^1\) with BMO on the bidisc , Ann. of Math. (2) 112 (1980), no. 1, 179-201. JSTOR: · Zbl 0451.42014
[2] R. Fefferman, Calderón-Zygmund theory for product domains-\(H^p\) spaces , to appear in Proc. Nat. Acad. Sci. · Zbl 0602.42023
[3] R. F. Gundy and E. M. Stein, \(H\spp\) theory for the poly-disc , Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1026-1029. JSTOR: · Zbl 0405.32002
[4] J.-L. Journé, Calderón-Zygmund operators on product spaces , Rev. Mat. Iberoamericana 1 (1985), no. 3, 55-91. · Zbl 0634.42015
[5] J.-L. Journé, A covering lemma for product spaces , · Zbl 0594.42015
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