In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator I α $I_{lpha}$ 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 ] $[b,I_{lpha }]$ on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.
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机译：在本文中，我们将给出关于Orlicz空间上Riesz势算子Iα$ I _ { alpha} $的强和弱有界性的充要条件。 Cianchi（J.Lond.Math.Soc.60（1）：247-286，2011）发现了一般Young函数Φ和necessary的充要条件，从而确保该算子是LΦ$ L ^ { Phi} $变成LΨ$ L ^ { Psi} $。我们对上述算子的有界性的表征与（Cianchi in J.Lond。Math。Soc.60（1）：247-286，2011）中的表征不同。作为这些结果的应用，当b分别属于BMO空间和Lipschitz空间时，我们考虑了Oriesz空间上Riesz势算子[b，Iα] $ [b，I _ { alpha}] $的交换子的有界性。
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