Fractional integrals and their commutators on martingale Orlicz spaces

Abstract

On the dyadic martingales, Chao and Ombe (1985) proved the Lp-Lq boundedness of the fractional integrals Iα and their commutators [b,Iα], where b is in BMO or Lipschitz spaces. We extend these results to LΦ-LΨ boundedness of generalized fractional integrals Iγ and their commutators [b,Iγ], where LΦ and LΨ are Orlicz spaces and b is in generalized martingale Campanato spaces on more general martingales. We also prove the LΦ-FLΨϕ boundedness of [b,Iγ], where FLΨϕ is the Triebel-Lizorkin-Orlicz space. Moreover, we characterize b such that the commutator is bounded or compact.