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Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. (Calderón-Zygmund operators, para-accretive functions and interpolation). (French) Zbl 0604.42014

This paper contains a generalization of boundedness criteria given by the first two authors, Ann. Math., II. Ser. 120 (1984; Zbl 0567.47025) and A. Mc Intosh and Y. Meyer, C. R. Acad. Sci., Paris, Ser. I 301, 395-397 (1985; Zbl 0584.47030). It concerns singular integral operators of Calderón-Zygmund type in \({\mathbb{R}}^ n\), i.e. operators that are associated to a kernel K(x,y) such that \(| K(x,y)| \leq C| x-y|^{-n}\) for \(x\neq y\); \(| K(x',y)-K(x,y)| \leq C| x'-x|^{\delta}| x-y|^{-n-\delta}\) for some \(0<\delta \leq 1\) and all x’,x,y such that \(| x'-x| <()| x- y|,\) and a similar condition for K(x,y).
A function \(b:{\mathbb{R}}^ n\to {\mathbb{C}}\) is said to be para-accretive if it is bounded, and if there exist constants \(C\geq 0\) and \(\epsilon >0\) such that, for each cube \(\tilde Q\) in \({\mathbb{R}}^ n\), one can find a cube \(\tilde Q\) such that \(C^{-1}| Q| \leq | \tilde Q| \leq C| Q|\), such that the distance from Q to \(\tilde Q\) is less than \(C| Q|^{1/n}\), and such that \(| 1/| \tilde Q| \int_{\tilde Q}b(y)dy| \geq \epsilon.\) For instance, any bounded function by such that Re b(x)\(\geq \epsilon >0\) is para-accretive.
The result is the following: if \(b_ 1\) and \(b_ 2\) are two para-accretive functions, and T is a singular integral operator with a kernel as above, then T is bounded on \(L^ 2({\mathbb{R}}^ n)\) if and only if the image of \(b_ 1\) by T is in BMO, the image of \(b_ 2\) by the transpose of T is in BMO, and some reasonable easy-to-check additional property (”weak boundedness”) is satisfied. The proof uses Cotlar-type almost-orthogonal decompositions of operators, and Littlewood-Paley decompositions in the framework of spaces of homogeneous type.

MSC:

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