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

The fractional integral operators I α I_α 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_α ] generated by fractional integral operators I α I_α with b b on the Musielak–Orlicz Hardy spaces. We show that, under suitable assumptions on two growth functions φ 1 φ _1 and φ 2 φ _2 , if b ∈ B M O φ 1 ( R n ) b∈ BMOφ _1( R^n) , then the commutator [ b , I α ] [b,I_α ] is bounded from the Musielak–Orlicz Hardy spaces H φ 1 ( R n ) Hφ _1( R^n) to the Musielak–Orlicz spaces