On the Product of Functions in BMO and H 1
Annales de l'Institut Fourier, Volume 57 (2007) no. 5, p. 1405-1439

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 BMO, 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.

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.

DOI : https://doi.org/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.
@article{AIF_2007__57_5_1405_0,
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     pages = {1405-1439},
     doi = {10.5802/aif.2299},
     zbl = {1132.42010},
     language = {en},
     url = {http://www.numdam.org/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, Volume 57 (2007) no. 5, pp. 1405-1439. doi : 10.5802/aif.2299. http://www.numdam.org/item/AIF_2007__57_5_1405_0/

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