Boundedness of commutator generated by fractional integral operator and Orlicz-BMO function

Abstract

For $\alpha \in (0,n)$ and a growth function $\phi :[0,\infty )\to [0,\infty )$, it is proved that the commutator $[b,I_{\alpha }]$ generated by fractional integral operator $I_{\alpha }$ and Orlicz $\mathrm{BMO}$ function $b$ is bounded from Orlicz-Hardy space $H_b^{\phi }({\mathbb{R}}^n)$ to Lebesgue space $L^{\frac{n}{n-\alpha }}({\mathbb{R}}^n)$, where $H_b^{\phi }({\mathbb{R}}^n)$ is a suitable Orlicz-Hardy space. Moreover, the authors also establish that the boundedness of commutator $[b,I_{\alpha }]$ from Orlicz-Hardy space $H^{\phi }({\mathbb{R}}^n)$ to weak Lebesgue space $L^{\frac{n}{n-\alpha },\infty }({\mathbb{R}}^n)$.