新的齐型BMO, Lipschitz空间上的奇异积分算子

Let(X, d, μ) be a space of homogeneous type, BMO_A(X) and Lip_A(β,X) be the space of BMO type,lipschitz type associated with an approximation to the identity {A_t}_t0 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 BMO(X) to BMO_A(X) and from Lip(β, X) to Lip_A(β, X). As an application, the boundedness of Calderón-Zygmund operators with nonsmooth kernels on BMO(R~n) and Lip(β, R~n) are also obtained.

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

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.