双线性分数次积分算子交换子在Morrey空间上有界的充分必要条件

Necessary and Sufficient Conditions for Boundedness of Commutators of Bilinear Fractional Integral Operators on Morrey Spaces

中文关键词:  分数次积分算子  Morrey空间  交换子  BMO空间  Lipschitz空间

英文关键词:fractional integral operator  Morrey spaces  commutators  ${\rm BMO}$  Lipschitz

中文摘要:

      在本文中,我们得到了双线性分数次交换子由乘积Morrey空间$L^{p_1,\lambda_1}(\mathbb{R}^n)\times L^{p_2,\lambda_2}(\mathbb{R}^n)$到$L^{q,\lambda}(\mathbb{R}^n)$有界,当且仅当$b$属于BMO空间,其中$p,q,\lambda,\mu$满足一定的条件.同时,我们也证明了双线性分数次交换子由乘积Morrey空间 $L^{p_1,\lambda_1}(\mathbb{R}^n)\times L^{p_2,\lambda_2}(\mathbb{R}^n)$到$L^{q,\lambda}(\mathbb{R}^n)$, 当且仅当$b$属于Lipschitz空间,其中$p,q,\lambda,\mu$满足另一合适的条件.

英文摘要:

      In this paper, we obtain that $b\in {\rm BMO}(\mathbb{R}^n)$ if and only if the commutator $[b,I_{\alpha}]$ is bounded from the Morrey spaces $L^{p_1,\lambda_1}(\mathbb{R}^n)\times L^{p_2,\lambda_2}(\mathbb{R}^n)$ to $L^{q,\lambda}(\mathbb{R}^n)$, for some appropriate indices $p,q,\lambda,\mu$. Also we show that $b\in {\rm Lip}_{\beta}(\mathbb{R}^{n})$ if and only if the commutator $[b,I_{\alpha}]$ is bounded from the Morrey spaces $L^{p_1,\lambda_1}(\mathbb{R}^n)\times L^{p_2,\lambda_2}(\mathbb{R}^n)$ to $L^{q,\lambda}(\mathbb{R}^n)$, for some appropriate indices $p,q,\lambda,\mu$.