arXiv 1105.04862011-05math.CAmath.FA

奇异积分算子的双线性分解与分解子

Bilinear decompositions and commutators of singular integral operators

Luong Dang Ky

备注：: Accepted to Trans. Amer. Math. Soc

摘要：

设$b$是一个$BMO$函数。众所周知，Calderón-Zygmund算子$T$的线性交换子$[b, T]$一般不会连续地从$H^1(\mathbb R^n)$映射到$L^1(\mathbb R^n)$。然而，佩雷斯表明，如果$H^1(\mathbb R^n)$被替换为一个合适的原子子空间$\mathcal H^1_b(\mathbb R^n)$，那么从$\mathcal H^1_b(\mathbb R^n)$到$L^1(\mathbb R^n)$的换位子是连续的。在本文中，我们找到了最大的子空间$H^1_b(\mathbb R^n)$，使得Calderón-Zygmund算子的所有扩张子从$H^1_b(\mathbb R^n)$到$L^1(\mathbb R^n)$是连续的.给出了$H^1_b(\mathbb R^n)$的一些等价刻画.我们还研究了次线性算子类$\mathcal K$中的$[b,T]$对$T$的分解子$[b,T]$，这类算子包含了调和分析中几乎所有重要的算子。当$T$是线性的时，我们证明了存在一个双线性算子$\mathfrak R= \mathfrak R_T$连续地将$H^1(\mathbb R^n)\times BMO(\mathbb R^n)$映射到$L^1(\mathbb R^n)$，使得对于所有的$(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)$，我们有\label{abstract 1} [b，T]（f）= \mathfrak R（f，b）+ T（\mathfrak S（f，b）），其中$\mathfrak S$是一个从$H^1(\mathbb R^n)\times BMO(\mathbb R^n)$到$L^1(\mathbb R^n)$的不依赖于$T$的有界双线性算子.在$T$的特殊情况下，一个Calderón-Zygmund算子满足$T1=T^*1=0$和$b$在$BMO^{\rm log}(\mathbb R^n)$ -广义$\BMO$型空间，已引入Nakai和Yabuta的特征乘子的$\BMO(\bR^n)$ -我们证明了交换子$[b,T]$连续映射$H^1_b(\mathbb R^n)$到$h^1(\mathbb R^n)$。此外，如果$b$位于$BMO(\mathbb R^n)$和$T^*1 = T^*b = 0$中，则转换器$[b, T]$将连续地将$H^1_b (\mathbb R^n)$映射到$H^1(\mathbb R^n)$。当$T$是次线性算子时，我们证明了存在有界次双线性算子$\mathfrak R= \mathfrak R_T: H^1(\mathbb R^n)\times BMO(\mathbb R^n)\to L^1(\mathbb R^n)$，使得对所有$(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)$，我们有\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））|.双线性分解（\ref{abstract 1}）和次双线性分解（\ref{abstract 2}）使我们能够给出所有已知的弱和强$L^1$ -估计的一般概述。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文