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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Bilinear decompositions and commutators of singular integral operators


Author: Luong Dang Ky
Journal: Trans. Amer. Math. Soc. 365 (2013), 2931-2958
MSC (2010): Primary 42B20; Secondary 42B30, 42B35, 42B25
Published electronically: November 30, 2012
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Abstract: Let $ b$ be a $ BMO$-function. It is well known that the linear commutator $ [b, T]$ of a Calderón-Zygmund operator $ T$ does not, in general, map continuously $ H^1(\mathbb{R}^n)$ into $ L^1(\mathbb{R}^n)$. However, Pérez showed that if $ H^1(\mathbb{R}^n)$ is replaced by a suitable atomic subspace $ \mathcal H^1_b(\mathbb{R}^n)$, then the commutator is continuous from $ \mathcal H^1_b(\mathbb{R}^n)$ into $ L^1(\mathbb{R}^n)$. In this paper, we find the largest subspace $ H^1_b(\mathbb{R}^n)$ such that all commutators of Calderón-Zygmund operators are continuous from $ H^1_b(\mathbb{R}^n)$ into $ L^1(\mathbb{R}^n)$. Some equivalent characterizations of $ H^1_b(\mathbb{R}^n)$ are also given. We also study the commutators $ [b,T]$ for $ T$ in a class $ \mathcal K$ of sublinear operators containing almost all important operators in harmonic analysis. When $ T$ is linear, we prove that there exists a bilinear operator $ \mathfrak{R}= \mathfrak{R}_T$ mapping continuously $ H^1(\mathbb{R}^n)\times BMO(\mathbb{R}^n)$ into $ L^1(\mathbb{R}^n)$ such that for all $ (f,b)\in H^1(\mathbb{R}^n)\times BMO(\mathbb{R}^n)$ we have

$\displaystyle [b,T](f)= \mathfrak{R}(f,b) + T(\mathfrak{S}(f,b)),$ (1)

where $ \mathfrak{S}$ is a bounded bilinear operator from $ H^1(\mathbb{R}^n)\times BMO(\mathbb{R}^n)$ into $ L^1(\mathbb{R}^n)$ which does not depend on $ T$. In the particular case of $ T$ a Calderón-Zygmund operator satisfying $ T1=T^*1=0$ and $ b$ in $ BMO^{\rm log}(\mathbb{R}^n)$, the generalized $ BMO$ type space that has been introduced by Nakai and Yabuta to characterize multipliers of $ BMO(\mathbb{R}^n)$, we prove that the commutator $ [b,T]$ maps continuously $ H^1_b(\mathbb{R}^n)$ into $ h^1(\mathbb{R}^n)$. Also, if $ b$ is in $ BMO(\mathbb{R}^n)$ and $ T^*1 = T^*b = 0$, then the commutator $ [b, T]$ maps continuously $ H^1_b (\mathbb{R}^n)$ into $ H^1(\mathbb{R}^n)$. When $ T$ is sublinear, we prove that there exists a bounded subbilinear operator $ \mathfrak{R}= \mathfrak{R}_T: H^1(\mathbb{R}^n)\times BMO(\mathbb{R}^n)\to L^1(\mathbb{R}^n)$ such that for all $ (f,b)\in H^1(\mathbb{R}^n)\times BMO(\mathbb{R}^n)$ we have

$\displaystyle \vert T(\mathfrak{S}(f,b))\vert- \mathfrak{R}(f,b)\leq \vert[b,T](f)\vert\leq \mathfrak{R}(f,b) + \vert T(\mathfrak{S}(f,b))\vert.$ (2)

The bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong $ L^1$-estimates.


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Additional Information

Luong Dang Ky
Affiliation: Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong Street, Quy Nhon City, Vietnam
Email: dangky@math.cnrs.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05727-8
PII: S 0002-9947(2012)05727-8
Keywords: Calderón-Zygmund operators, bilinear decompositions, commutators, Hardy spaces, wavelet characterizations, BMO spaces, atoms, bilinear operators
Received by editor(s): June 7, 2011
Published electronically: November 30, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.