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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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Submitted on : Wednesday, March 9, 2011 - 12:36:57 PM
Last modification on : Friday, April 12, 2019 - 4:46:03 PM
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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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