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

We study the continuity properties of the generalized fractional integral operatorIρon the generalized local Morrey spacesLMp,φ{x0}and generalized Morrey spacesMp,φ. We find conditions on the triple(φ1,φ2,ρ)which ensure the Spanne-type boundedness ofIρfrom one generalized local Morrey spaceLMp,φ1{x0}to anotherLMq,φ2{x0},1<p<q<∞, and fromLM1,φ1{x0}to the weak spaceWLMq,φ2{x0},1<q<∞. We also find conditions on the pair(φ,ρ)which ensure the Adams-type boundedness ofIρfromMp,φ1/ptoMq,φ1/qfor1<p<q<∞and fromM1,φtoWMq,φ1/qfor1<q<∞. In all cases the conditions for the boundedness ofIρare given in terms of Zygmund-type integral inequalities on(φ1,φ2,ρ)and(φ,ρ), which do not assume any assumption on monotonicity ofφ1(x,r),φ2(x,r), andφ(x,r)inr.

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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, the necessary and sufficient conditions are found for the boundedness of the rough Bfractional integral operators from the Lorentz spaces L p,s,γ to L q,r,γ , 1 < p < q < ∞, 1 r s ∞, and from L 1,r,γ to L q,∞,γ ≡ W L q,γ , 1 < q < ∞, 1 r ∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.

We establish two inequalities of Stein-Weiss type for the Riesz potential operator I α,γ ( B− Riesz potential operator) generated by the Laplace-Bessel differential operator Δ B in the weighted Lebesgue spaces L p,|x| β ,γ . We obtain necessary and sufficient conditions on the parameters for the boundedness of I α,γ from the spaces L p,|x| β ,γ to L q,|x| −λ ,γ , and from the spaces L 1,|x| β ,γ to the weak spaces W L q,|x| −λ ,γ . In the limiting case p = Q/α we prove that the modified B− Riesz potential operator I α,γ is bounded from the spaces L p,|x| β ,γ to the weighted B − BMO spaces BMO |x| −λ ,γ .As applications, we get the boundedness of I α,γ from the weighted B -Besov spaces B s pθ ,|x| β ,γ to the spaces B s qθ ,|x| −λ ,γ . Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue L p,|x| β ,γ and weighted B -Besov spaces B s pθ ,|x| β ,γ by using the fundamental solution of the B -elliptic equation Δ α/2 B . Mathematics subject classification (2010): 42B20, 42B25, 42B35.