# Paraproducts and Products of functions in $BMO(\mathbb R^n)$ and $H^1(\mathbb R^n)$ through wavelets

Abstract : In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $\BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times \BMO(\bR^n)$ into $L^1(\bR^n)$, the other one from $\H^1(\bR^n)\times \BMO(\bR^n)$ into a new kind of Hardy-Orlicz space denoted by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies $\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{ \log(e+|x|) +\log (e+ |\mathcal Mf(x)|)}dx <\infty.$ The two bilinear operators can be defined in terms of paraproducts. As a consequence, we find an endpoint estimate involving the space $\H^{\log}(\bR^n)$ for the $\div$-$\curl$ lemma.
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Article dans une revue
Journal de Mathématiques Pures et Appliquées, Elsevier, 2012, 97, pp.230-241
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https://hal.archives-ouvertes.fr/hal-00575012
Contributeur : Sandrine Grellier <>
Soumis le : mercredi 9 mars 2011 - 12:36:57
Dernière modification le : jeudi 3 mai 2018 - 15:32:06
Document(s) archivé(s) le : vendredi 10 juin 2011 - 02:31:40

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• HAL Id : hal-00575012, version 1
• ARXIV : 1103.1822

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Aline Bonami, Sandrine Grellier, Luong Dang Ky. Paraproducts and Products of functions in $BMO(\mathbb R^n)$ and $H^1(\mathbb R^n)$ through wavelets. Journal de Mathématiques Pures et Appliquées, Elsevier, 2012, 97, pp.230-241. 〈hal-00575012〉

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