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In this paper the authors study the boundedness for a large class of sublinear operators Tα, α ∈ [0, n) generated by Calderón-Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces Mp,ϕ. As an application of the above result, the boundeness of the commutator of sublinear operators T b,α , α ∈ [0, n) on generalized Morrey spaces is also obtained. In the case b ∈ BM O and T b,α is a sublinear operator, we find the sufficient conditions on the pair (ϕ1, ϕ2) which ensures the boundedness of the operators T b,α , α ∈ [0, n) from one generalized Morrey space Mp,ϕ 1 to another Mq,ϕ 2 with 1/p − 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (ϕ1, ϕ2), which do not assume any assumption on monotonicity of ϕ1, ϕ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.Mathematics Subject Classification (2010). Primary 42B20, 42B25, 42B35.

We consider generalized Morrey spaces M p(·),ω ( ) with variable exponent p(x) and a general function ω(x, r) defining the Morrey-type norm. In case of bounded sets ⊂ R n we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(x, r), which do not assume any assumption on monotonicity of ω(x, r) in r.

In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel T on the generalized local Morrey spaces LM

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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.