No abstract

In the present paper, we will characterize the boundedness of the generalized fractional integral operators I ρ and the generalized fractional maximal operators M ρ on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators M ρ and I ρ on generalized Orlicz-Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators M ρ and I ρ on generalized Orlicz-Morrey spaces.

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.

Abstract. We prove the boundedness of the Hardy-Littlewood maximal operator and their commutators with BMO-coefficients in vanishing generalized Orlicz-Morrey spaces V M Φ,ϕ (R n ) including weak versions of these spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators involving the Young function Φ(u) and the function ϕ(x, r) defining the space. No kind of monotonicity condition on ϕ(x, r) in r is imposed.

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.