## 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.)

### Citations:

Zbl 0567.47025; Zbl 0584.47030
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