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

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

doi:10.5802/aif.2299

On the Product of Functions in BMO and H

^{1}

[Produits de fonctions de

H^{1}

B M O

]

Bonami, Aline ¹ ; Iwaniec, Tadeusz ² ; Jones, Peter ³ ; Zinsmeister, Michel ⁴

¹ Université d’Orléans MAPMO BP 6759 45067 Orléans cedex
² Syracuse University 215 Carnegie Hall Syracuse NY 13244-1150 (USA)
³ Yale University Mathematics Dept. PO Box 208 283 New Haven CT 06520-8283 (USA)
⁴ Université d’Orléans MAPMO BP 6759 45067 Orléans cedex et Ecole Polytechnique PMC 91128 Palaiseau

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1405-1439.

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.

Zbl

DOI : 10.5802/aif.2299

Classification : 42B25, 42B30, 30H
Keywords: Hardy spaces, bounded mean oscillation, Jacobian lemma, Jacobian equation, Hardy-Orlicz spaces, div-curl lemma, factorization in Hardy spaces, weak Jacobian.
Mot 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.

Affiliations des auteurs :

Bonami, Aline ¹ ; Iwaniec, Tadeusz ² ; Jones, Peter ³ ; Zinsmeister, Michel ⁴

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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) no. 5, pp. 1405-1439. doi : 10.5802/aif.2299. https://www.numdam.org/articles/10.5802/aif.2299/

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