R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on maps 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 , for some . 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 . We also show that the Calderon-Coifman bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on with norms growing polynomially.
R. Fefferman a montré que sur un espace-produit à deux facteurs un opérateur borné sur est également borné de dans BMO du produit si l’oscillation moyenne sur un rectangle de l’image d’une fonction bornée supportée en dehors d’un multiple de est dominée par pour un . Nous montrons ici que ce résultat n’est plus vrai en général pour un produit de trois facteurs ou plus mais s’étend à ce cas lorsque l’opérateur est un opérateur de convolution et . Également nous montrons que les bicommutateurs de Calderón-Coifman, obtenus à partir des commutateurs de Calderón par produit tensoriel multilinéaire, sont bornés sur avec une croissance de norme polynomiale.
@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}, pages = {111--132}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {38}, number = {1}, year = {1988}, doi = {10.5802/aif.1125}, mrnumber = {90b:42031}, zbl = {0638.47026}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1125/} }
TY - JOUR AU - Journé, Jean-Lin TI - Two problems of Calderón-Zygmund theory on product-spaces JO - Annales de l'Institut Fourier PY - 1988 SP - 111 EP - 132 VL - 38 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.1125/ DO - 10.5802/aif.1125 LA - en ID - AIF_1988__38_1_111_0 ER -
Journé, Jean-Lin. Two problems of Calderón-Zygmund theory on product-spaces. Annales de l'Institut Fourier, Volume 38 (1988) no. 1, pp. 111-132. doi : 10.5802/aif.1125. http://www.numdam.org/articles/10.5802/aif.1125/
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