On the Product of Functions in BMO and H$^\text{1}$

Bonami, Aline; Iwaniec, Tadeusz; Jones, Peter; Zinsmeister, Michel

On the Product of Functions in BMO and H

^{1}

[Produits de fonctions de

H^{1}

B M O

]

Bonami, Aline ; Iwaniec, Tadeusz ; Jones, Peter ; Zinsmeister, Michel

Annales de l'Institut Fourier, Tome 57 (2007), p. 1405-1439 / Harvested from Numdam

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Résumé
Abstract

Le produit d’une fonction à oscillation moyenne bornée avec une fonction de l’espace de Hardy $H^{1}$ n’est pas intégrable en général. Nous montrons toutefois qu’on peut lui donner un sens en tant que distribution tempérée, ceci grâce à la dualité $H^{1}$ , $B M O$ . Cette distribution peut de plus s’écrire comme la somme d’une fonction intégrable et d’une distribution appartenant à un espace de Hardy-Orlicz adapté. Lorsqu’on considère un tel produit pour les fonctions holomorphes du disque unité, cet énoncé possède une réciproque : toute fonction holomorphe de l’espace de Hardy-Orlicz considéré peut s’écrire comme un tel produit.

The point-wise product of a function of bounded mean oscillation with a function of the Hardy space $H^{1}$ is not locally integrable in general. However, in view of the duality between $H^{1}$ and $B M O$ , we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic of the corresponding Hardy-Orlicz space can be written as a product of a function in the holomorphic Hardy space $H^{1}$ and a holomorphic function with boundary values of bounded mean oscillation.

Publié le : 2007-01-01

Zbl 1132.42010

DOI : https://doi.org/10.5802/aif.2299
Classification: 42B25, 42B30, 30H
Mots clés: Espaces de Hardy, fonctions à oscillation moyenne bornée, lemme du Jacobien, équation du Jacobien, espaces de hardy-Orlicz, lemme div-curl, factorisation dans les classes de hardy, Jacobien faible.

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     author = {Bonami, Aline and Iwaniec, Tadeusz and Jones, Peter and Zinsmeister, Michel},
     title = {On the Product of Functions in BMO and H$^\text{1}$},
     journal = {Annales de l'Institut Fourier},
     volume = {57},
     year = {2007},
     pages = {1405-1439},
     doi = {10.5802/aif.2299},
     zbl = {1132.42010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2007__57_5_1405_0}
}

Bonami, Aline; Iwaniec, Tadeusz; Jones, Peter; Zinsmeister, Michel. On the Product of Functions in BMO and H$^\text{1}$. Annales de l'Institut Fourier, Tome 57 (2007) pp. 1405-1439. doi : 10.5802/aif.2299. http://gdmltest.u-ga.fr/item/AIF_2007__57_5_1405_0/

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