Two problems of Calderón-Zygmund theory on product-spaces
Annales de l'Institut Fourier, Tome 38 (1988) no. 1, p. 111-132
R. Fefferman a montré que sur un espace-produit à deux facteurs un opérateur T borné sur L 2 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>s 0 (E). Également nous montrons que les bicommutateurs de Calderón-Coifman, obtenus à partir des commutateurs de Calderón par produit tensoriel multilinéaire, sont bornés sur L 2 avec une croissance de norme polynomiale.
R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on L 2 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>s 0 (E). We also show that the Calderon-Coifman bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on L 2 with norms growing polynomially.
@article{AIF_1988__38_1_111_0,
     author = {Journ\'e, Jean-Lin},
     title = {Two problems of Calder\'on-Zygmund theory on product-spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {38},
     number = {1},
     year = {1988},
     pages = {111-132},
     doi = {10.5802/aif.1125},
     zbl = {0638.47026},
     mrnumber = {90b:42031},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1988__38_1_111_0}
}
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/item/AIF_1988__38_1_111_0/

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