Riesz potential and its commutators on Orlicz spaces

Abstract

In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator I-alpha on Orlicz spaces. Cianchi (J. Lond. Math. Soc. 60(1): 247-286, 2011) found necessary and sufficient conditions on general Young functions Phi and Psi ensuring that this operator is of weak or strong type from L-Phi into L-Psi. Our characterizations for the boundedness of the above-mentioned operator are different from the ones in (Cianchi in J. Lond. Math. Soc. 60(1): 247-286, 2011). As an application of these results, we consider the boundedness of the commutators of Riesz potential operator [b,I-alpha] on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.