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Aline Bonami; Tadeusz Iwaniec; Peter Jones; Michel Zinsmeister
On the Product of Functions in BMO and H$^\text{1}$
(Produits de fonctions de $H^1$ et $BMO$)
Annales de l'institut Fourier, 57 no. 5 (2007), p. 1405-1439, doi: 10.5802/aif.2299
Article PDF | Analyses Zbl 1132.42010
Class. Math.: 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.

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$, $BMO$. 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.

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