Weighted Hardy Operators in Complementary Morrey Spaces

Abstract

We study the weightedp→q-boundedness of the multidimensional weighted Hardy-type operatorsHwαandℋwαwith radial type weightw=w(|x|), in the generalized complementary Morrey spacesℒ∁{0}p,ψ(ℝn)defined by an almost increasing functionψ=ψ(r). We prove a theorem which provides conditions, in terms of some integral inequalities imposed onψandw, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the functionψand the weightware power functions. We also prove that the spacesℒ∁{0}p,ψ(Ω)over bounded domains Ω are embedded between weighted Lebesgue spaceLpwith the weightψand such a space with the weightψ, perturbed by a logarithmic factor. Both the embeddings are sharp.