Riesz potential and its commutators on Orlicz spaces

Abstract

Abstract In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator I α $I_{\alpha}$ on Orlicz spaces. Cianchi (J. Lond. Math. Soc. 60(1):247-286, 2011) found necessary and sufficient conditions on general Young functions Φ and Ψ ensuring that this operator is of weak or strong type from L Φ $L^{\Phi}$ into L Ψ $L^{\Psi}$ . 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 α ] $[b,I_{\alpha }]$ on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.

Keywords

• Orlicz space
• Riesz potential
• commutator
• BMO
• Lipschitz space