Weighted Endpoint Estimates for Commutators of Calder\'on-Zygmund Operators
Weighted Endpoint Estimates for Commutators of Calder\'on-Zygmund Operators
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DOI：
10.1090/proc/13130
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发表时间：
2015-10
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影响因子：
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通讯作者：
Yiyu Liang;Luong Dang Ky;Dachun Yang
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文献类型：
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作者：
Yiyu Liang;Luong Dang Ky;Dachun Yang
Let $\delta\in(0,1]$ and $T$ be a $\delta$-Calder\'on-Zygmund operator. Let $w$ be in the Muckenhoupt class $A_{1+\delta/n}({\mathbb R}^n)$ satisfying $\int_{{\mathbb R}^n}\frac {w(x)}{1+|x|^n}\,dx<\infty$. When $b\in{\rm BMO}(\mathbb R^n)$, it is well known that the commutator $[b, T]$ is not bounded from $H^1(\mathbb R^n)$ to $L^1(\mathbb R^n)$ if $b$ is not a constant function. In this article, the authors find out a proper subspace ${\mathop\mathcal{BMO}_w({\mathbb R}^n)}$ of $\mathop\mathrm{BMO}(\mathbb R^n)$ such that, if $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$, then $[b,T]$ is bounded from the weighted Hardy space $H_w^1(\mathbb R^n)$ to the weighted Lebesgue space $L_w^1(\mathbb R^n)$. Conversely, if $b\in{\rm BMO}({\mathbb R}^n)$ and the commutators of the classical Riesz transforms $\{[b,R_j]\}_{j=1}^n$ are bounded from $H^1_w({\mathbb R}^n)$ into $L^1_w({\mathbb R}^n)$, then $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$.
Let $\delta\in(0,1]$ and $T$ be a $\delta$-Calder\'on-Zygmund operator. Let $w$ be in the Muckenhoupt class $A_{1+\delta/n}({\mathbb R}^n)$ satisfying $\int_{{\mathbb R}^n}\frac {w(x)}{1+|x|^n}\,dx<\infty$. When $b\in{\rm BMO}(\mathbb R^n)$, it is well known that the commutator $[b, T]$ is not bounded from $H^1(\mathbb R^n)$ to $L^1(\mathbb R^n)$ if $b$ is not a constant function. In this article, the authors find out a proper subspace ${\mathop\mathcal{BMO}_w({\mathbb R}^n)}$ of $\mathop\mathrm{BMO}(\mathbb R^n)$ such that, if $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$, then $[b,T]$ is bounded from the weighted Hardy space $H_w^1(\mathbb R^n)$ to the weighted Lebesgue space $L_w^1(\mathbb R^n)$. Conversely, if $b\in{\rm BMO}({\mathbb R}^n)$ and the commutators of the classical Riesz transforms $\{[b,R_j]\}_{j=1}^n$ are bounded from $H^1_w({\mathbb R}^n)$ into $L^1_w({\mathbb R}^n)$, then $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$.