Weighted Hardy Operators in Complementary Morrey Spaces

Abstract EN

We study the weighted p→q-boundedness of the multidimensional weighted Hardy-type operators Hwα and ℋwα with radial type weight w=w(|x|), in the generalized complementary Morrey spaces ℒ∁{0}p,ψ(ℝn) defined by an almost increasing function ψ=ψ(r).

We prove a theorem which provides conditions, in terms of some integral inequalities imposed on ψ and w, for such a boundedness.

These conditions are sufficient in the general case, but we prove that they are also necessary when the function ψ and the weight w are power functions.

We also prove that the spaces ℒ∁{0}p,ψ(Ω) over bounded domains Ω are embedded between weighted Lebesgue space Lp with the weight ψ and such a space with the weight ψ, perturbed by a logarithmic factor.

Both the embeddings are sharp.