The point-wise product of a function of bounded mean oscillation with a function of the Hardy space is not locally integrable in general. However, in view of the duality between and , 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 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 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 , . 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.
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.
Bonami, Aline 1; Iwaniec, Tadeusz 2; Jones, Peter 3; Zinsmeister, Michel 4
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TY - JOUR AU - Bonami, Aline AU - Iwaniec, Tadeusz AU - Jones, Peter AU - Zinsmeister, Michel TI - On the Product of Functions in BMO and H$^\text{1}$ JO - Annales de l'Institut Fourier PY - 2007 SP - 1405 EP - 1439 VL - 57 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2299/ DO - 10.5802/aif.2299 LA - en ID - AIF_2007__57_5_1405_0 ER -
%0 Journal Article %A Bonami, Aline %A Iwaniec, Tadeusz %A Jones, Peter %A Zinsmeister, Michel %T On the Product of Functions in BMO and H$^\text{1}$ %J Annales de l'Institut Fourier %D 2007 %P 1405-1439 %V 57 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2299/ %R 10.5802/aif.2299 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2299/
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