Bilinear decompositions and commutators of singular integral operators

Details Bilinear decompositions and commutators of singular integral operators Bilinear decompositions and commutators of singular integral operators

2011-05-03

arxiv.org/abs/1105.0486v5

Mathematics

Classical Analysis and ODEs

Accepted to Trans. Amer. Math. Soc

Scientific paper

Let $b$ be a $BMO$-function. It is well-known that the linear commutator $[b, T]$ of a Calder\'on-Zygmund operator $T$ does not, in general, map continuously $H^1(\mathbb R^n)$ into $L^1(\mathbb R^n)$. However, P\'erez 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 Calder\'on-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 Calder\'on-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.

Affiliated with

Also associated with

No associations

Rating

Bilinear decompositions and commutators of singular integral operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Bilinear decompositions and commutators of singular integral operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Bilinear decompositions and commutators of singular integral operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-692422