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

Abstract

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