# Bilinear decompositions and commutators of singular integral operators

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 operators $\mathfrak R= \mathfrak R_T$ 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 $$\label{abstract 1} [b,T](f)= \mathfrak R(f,b) + T(\mathfrak S(f,b)),$$ 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 $\BMO(\bR^n)$ --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 $$\label{abstract 2} |T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|.$$ The bilinear decomposition (\ref{abstract 1}) and the subbilinear decomposition (\ref{abstract 2}) allow us to give a general overview of all known weak and strong $L^1$-estimates.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-00589824
Contributor : Luong Dang Ky <>
Submitted on : Friday, January 13, 2012 - 10:19:35 PM
Last modification on : Thursday, May 3, 2018 - 3:32:06 PM
Long-term archiving on: : Saturday, April 14, 2012 - 2:33:34 AM

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• HAL Id : hal-00589824, version 5
• ARXIV : 1105.0486

### Citation

Luong Dang Ky. Bilinear decompositions and commutators of singular integral operators. Transactions of the American Mathematical Society, American Mathematical Society, 2013, 28 p. ⟨hal-00589824v5⟩

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