In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator M α on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator M b,α and nonlinear commutator of fractional maximal operator [b, M α ] on Orlicz spaces, when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.

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In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator 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 into . 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 on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.

In this paper, we prove the Spanne-Guliyev type boundedness of the generalized fractional integral operator I ρ from the vanishing generalized local Morrey spaces V LM {x 0 } p,ϕ 1 to V LM {x 0 } q,ϕ 2 , 1 < p < q < ∞, and from the space V LM {x 0 } 1,ϕ 1 to the weak space V W LM {x 0 } q,ϕ 2 , 1 < q < ∞. We also prove the Adams-Guliyev type boundedness of the operator I ρ from the vanishing generalized Morrey spaces V M p,ϕ

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