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

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.

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.

@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  - 
%0 Journal Article
%A Journé, Jean-Lin
%T Two problems of Calderón-Zygmund theory on product-spaces
%J Annales de l'Institut Fourier
%D 1988
%P 111-132
%V 38
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1125/
%R 10.5802/aif.1125
%G en
%F AIF_1988__38_1_111_0
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/

[1] R. Fefferman, Calderón-Zygmund theory for product domains-Hp spaces, P.N.S.A., vol. 83 (1986), 840-843. | MR | Zbl

[2] J. Pipher, Journé's covering lemma and its extension to higher dimensions, Duke Journal of Math, 53 n° 3 (1986), 683-690. | MR | Zbl

[3] J.-L. Journe, Calderón-Zygmund operators on product spaces, Mat. Iberoamerica, vol. 1, n° 3 (1985), 55-91. | MR | Zbl

[4] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math., (2), vol. 126 (1987), 109-130. | MR | Zbl

[5] R. Fefferman, Functions of bounded mean oscillation on the polydisc, Ann. of Math., (2), 10 (1979), 395-406. | MR | Zbl

[6] L. Carleson, A counterexample for measures bounded for Hp for the bi-disc, Mittag Leffer Report n° 7, 1974.

[7] S.-Y. A. Chang and R. Fefferman, A continuous version of the duality of H1 and BMO on the bi-disc, Ann. of Math., (2), 112 (1980), 179-201. | MR | Zbl

[8] J.-L. Rubio De Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, vol. 1, n° 2 (1985). | MR | Zbl

[9] P. Krikeles, Ph. D. dissertation, Yale University, 1982.

[10] M. Christ, and J.-L. Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80. | MR | Zbl

[11] S.-Y. A. Chang, Carleson measure on the bi-disc, Ann. of Math., (2), 109 (1979), 613-620. | MR | Zbl

[12] J.-L. Journé, Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, L.N. 994, Springer-Verlag. | MR | Zbl

[13] S.-Y. A. Chang and R. Fefferman, The Calderón-Zygmund decomposition on product domains, Am. J. of Math., vol. 104, 3, 455-468. | MR | Zbl

Cited by Sources: