Boundedness of commutators of Riesz potential operators on Musielak–Orlicz Hardy spaces

The fractional integral operators I α I_\alpha play an important role in the theory of Musielak–Orlicz Hardy spaces. This paper studies the boundedness of the commutators [ b , I α ] [b,I_\alpha ] generated by fractional integral operators I α I_\alpha with b b on the Musielak–Orlicz Hardy spaces. We show that, under suitable assumptions on two growth functions φ 1 \varphi _1 and φ 2 \varphi _2 , if b ∈ B M O φ 1 ( R n ) b\in \mathcal {BMO}_{\varphi _1}(\mathbb R^n) , then the commutator [ b , I α ] [b,I_\alpha ] is bounded from the Musielak–Orlicz Hardy spaces H φ 1 ( R n ) H^{\varphi _1}(\mathbb R^n) to the Musielak–Orlicz spaces