Fractional integrals and their commutators on martingale Orlicz spaces

On the dyadic martingales, Chao and Ombe (1985) proved the L-p-L-q boundedness of the fractional integrals I-alpha and their commutators [b, I-alpha], where bis in BMO or Lipschitz spaces. We extend these results to L-Phi-L-Psi boundedness of generalized fractional integrals I-gamma and their commutators [b, I-gamma], where L-Phi and L-Psi are Orlicz spaces and bis in generalized martingale Campanato spaces on more general martingales. We also prove the L-Phi-F-L Psi(phi) boundedness of [b, I-gamma], where F-L Psi(phi) is the Triebel-Lizorkin-Orlicz space. Moreover, we characterize bsuch that the commutator is bounded or compact. (c) 2020 Elsevier Inc. All rights reserved.