Weighted Hardy Operators in Complementary Morrey Spaces. (11th November 2012) - Lukkassen, Dag; Persson, Lars-Erik; Samko, Stefan
Abstract : We study the weightedp → q -boundedness of the multidimensional weighted Hardy-type operatorsH w α andℋ w α with radial type weightw = w ( | x | ), in the generalized complementary Morrey spacesℒ ∁ 0 p, ψ ( ℝ n ) defined by an almost increasing functionψ = ψ ( r ) . We prove a theorem which provides conditions, in terms of some integral inequalities imposed onψ andw, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the functionψ and the weightw are power functions. We also prove that the spacesℒ ∁ 0 p, ψ ( Ω ) over bounded domains Ω are embedded between weighted Lebesgue spaceL p with the weightψ and such a space with the weightψ, perturbed by a logarithmic factor. Both the embeddings are sharp.