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

In this paper,we obtain that b∈ BMO(R~n) if and only if the commutatorb,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β(R~n) if and only if the commutatorb,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.

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

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 $p,q,\lambda,\mu$.