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Calderón-Zygmund operators on product spaces. (English) Zbl 0634.42015

The author develops and to a certain degree unifies some results both of Calderon-type theory for singular integral operators in the non convolutional case [see e.g. R. R. Coifman, Y. Meyer, Astérisque 57 (1978; Zbl 0483.35082); G. David and the author, Ann. Math. 120, 371-397 (1984; Zbl 0567.47025); the author, Lect. Notes Math. 994 (1983; Zbl 0508.42021) and of the Fefferman-Stein theory for singular operators on product spaces. He gives the corresponding setting of the operators under consideration, one of the typical examples being the operator associated to the kernel \[ K_ a(x,y)=[\prod^{n}_{i=1}(x_ i-y_ i)+\int^{y_ 1}_{x_ 1}...\int^{y_ n}_{x''}a(u_ 1,...,u_ n)du]^{-1} \] (“nth Cauchy operator on Lipschitz curve”), \(\| a\|_{\infty}<1\). The author investigates the BMO-boundedness of the operators on product spaces, reduced “L\({}^ 2\)-boundedness” to \(``L^{\infty}\to BMO\) boundedness” by means of a certain geometrical lemma and considers some other aspects of the Calderón-Zygmund-type theory for product spaces.
Reviewer: S.G.Samko

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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