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.

The authors study the boundedness for a large class of sublinear operatorTgenerated by Calderón-Zygmund operator on generalized Morrey spacesMp,φ. As an application of this result, the boundedness of the commutator of sublinear operatorsTaon generalized Morrey spaces is obtained. In the casea∈BMO(ℝn),1<p<∞andTais a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operatorTafrom one generalized Morrey spaceMp,φ1to anotherMp,φ2. In all cases, the conditions for the boundedness ofTaare given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity ofφ1,φ2inr. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

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In this paper we study the boundedness for a large class of sublinear operators T generated by Calderón-Zygmund operators on generalized weighted Morrey spaces Mp,φ(w) with the weight function w(x) belonging to Muckenhoupt's class Ap. We find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the operator T from one generalized weighted Morrey space Mp,φ 1 (w) to another Mp,φ 2 (w) for p > 1 and from M1,φ 1 (w) to the weak space W M1,φ 2 (w). In all 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 pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

In this paper, we study the boundedness of the Hardy operators on Musielak–Orlicz spaces.