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.
We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces 𝐿𝑝(·), λ(·)(Ω) over a bounded open set Ω ⊂ ℝ𝑛 and a Sobolev type 𝐿𝑝(·), λ(·) → 𝐿𝑞(·), λ(·)-theorem for potential operators 𝐼
α(·), also of variable order. In the case of constant α, the limiting case is also studied when the potential operator 𝐼
α
acts into BMO space.
We consider local "complementary" generalized Morrey spacesis a variable exponent, and no monotonicity type conditio is imposed onto the function ω(r) defining the "complementary" Morrey-type norm. In the case where ω is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case 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 type,ω {x 0 } (Ω) -theorem for the potential operators I α(·) , also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(r) , which do not assume any assumption on monotonicity of ω(r) .
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