Singular Integral Operators on New BMO and Lipschitz Spaces of Homogeneous Type

DOI：10.3770/j.issn:2095-2651.2016.01.012

 作者 单位 李朋 新疆大学数学与系统科学学院, 新疆 乌鲁木齐 830046 周疆 新疆大学数学与系统科学学院, 新疆 乌鲁木齐 830046

设$(X,d,\mu)$是一个齐型空间, ${\rm BMO}_A(X)$和${\rm Lip}_A(\beta,X)$是分别被Duong, Yan和Tang引进的与恒等逼近算子$\{A_t\}_{t>0}$有关的BMO,Lipschitz型空间. 假设$T$是$L^2(X)$上的线性有界算子, 作者找到了使得$T$从${\rm BMO}(X)$到${\rm BMO}_A(X)$和从${\rm Lip}(\beta, X)$到${\rm Lip}_A(\beta,X)$的充分条件. 作为应用, Calder\'on-Zygmund 算子在${\rm BMO}(X)$和${\rm Lip}(\beta, X)$上的有界性也被得到.

Let $(X,d,\mu)$ be a space of homogeneous type, ${\rm BMO}_A(X)$ and ${\rm Lip}_A(\beta,X)$ be the space of BMO type, lipschitz type associated with an approximation to the identity $\{A_t\}_{t>0}$ and introduced by Duong, Yan and Tang, respectively. Assuming that $T$ is a bounded linear operator on $L^2(X)$, we find the sufficient condition on the kernel of $T$ so that $T$ is bounded from ${\rm BMO}(X)$ to ${\rm BMO}_A(X)$ and from ${\rm Lip}(\beta, X)$ to ${\rm Lip}_A(\beta,X)$. As an application, the boundedness of Calder\'on-Zygmund operators with nonsmooth kernels on ${\rm BMO}(\mathbb{R}^n)$ and ${\rm Lip}(\beta, \mathbb{R}^n)$ are also obtained.