Two problems of Calderón-Zygmund theory on product-spaces
Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 111-132.

R. Fefferman a montré que sur un espace-produit à deux facteurs un opérateur T borné sur L2 est également borné de L∞ dans BMO du produit si l’oscillation moyenne sur un rectangle R de l’image d’une fonction bornée supportée en dehors d’un multiple R′ de R est dominée par C|R|s|R′|-s pour un s>0. Nous montrons ici que ce résultat n’est plus vrai en général pour un produit E de trois facteurs ou plus mais s’étend à ce cas lorsque l’opérateur T est un opérateur de convolution et s>s0(E). Également nous montrons que les bicommutateurs de Calderón-Coifman, obtenus à partir des commutateurs de Calderón par produit tensoriel multilinéaire, sont bornés sur L2 avec une croissance de norme polynomiale.

R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on L2 maps L∞ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by C|R|s|R|-s, for some s>0. We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided s>s0(E). We also show that the Calderon-Coifman bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on L2 with norms growing polynomially.

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     title = {Two problems of {Calder\'on-Zygmund} theory on product-spaces},
     journal = {Annales de l'Institut Fourier},
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Journé, Jean-Lin. Two problems of Calderón-Zygmund theory on product-spaces. Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 111-132. doi : 10.5802/aif.1125. https://www.numdam.org/articles/10.5802/aif.1125/

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