Characterization of functions via commutators of bilinear fractional integrals on Morrey spaces
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For $b\in L^1_{\rm loc}(\mathbb{R}^n)$, let $\mathcal{I}_\alpha$ be the bilinear fractional integral operator, and $[b,\,\mathcal{I}_\alpha]_i$ be the commutator of $\mathcal{I}_\alpha$ with pointwise multiplication $b$ ($i=1,\,2$). This paper shows that if the commutator $[b, \mathcal{I}_\alpha]_i$ for $i=1$ or $2$ is bounded from the product Morrey spaces $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to the Morrey space $L^{q,\lambda}(\mathbb{R}^n)$ for some suitable indexes $\lambda,\,\lambda_1$, $\lambda_2$ and $p_1,\,p_2,\,q$, then $b\in BMO(\mathbb{R}^n)$, as well as that the compactness of $[b,\,\mathcal{I}_\alpha]_i$ for $i=1$ or $2$ from $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to $L^{q,\lambda}(\mathbb{R}^n)$ implies that $b\in CMO(\mathbb{R}^n)$ (the closure in $BMO(\mathbb{R}^n)$ of the space of $C^\infty(\mathbb{R}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO(\mathbb{R}^n)$ functions or $CMO(\mathbb{R}^n)$ functions in essential ways.
Abstract
For $b\in L^1_{\rm loc}(\mathbb{R}^n)$, let $\mathcal{I}_\alpha$ be the bilinear fractional integral operator, and $[b,\,\mathcal{I}_\alpha]_i$ be the commutator of $\mathcal{I}_\alpha$ with pointwise multiplication $b$ ($i=1,\,2$). This paper shows that if the commutator $[b, \mathcal{I}_\alpha]_i$ for $i=1$ or $2$ is bounded from the product Morrey spaces $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to the Morrey space $L^{q,\lambda}(\mathbb{R}^n)$ for some suitable indexes $\lambda,\,\lambda_1$, $\lambda_2$ and $p_1,\,p_2,\,q$, then $b\in BMO(\mathbb{R}^n)$, as well as that the compactness of $[b,\,\mathcal{I}_\alpha]_i$ for $i=1$ or $2$ from $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to $L^{q,\lambda}(\mathbb{R}^n)$ implies that $b\in CMO(\mathbb{R}^n)$ (the closure in $BMO(\mathbb{R}^n)$ of the space of $C^\infty(\mathbb{R}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO(\mathbb{R}^n)$ functions or $CMO(\mathbb{R}^n)$ functions in essential ways.
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