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

Abstract:

      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 $p,q,\lambda,\mu$.