On boundedness of the Hardy operator in Morrey-type spaces

Abstract

In this paper we study the boundedness of the Hardy operator H_α in local and global Morrey-type spaces LM_pθ,w(.), GM_pθ,w(.) respectively, characterized by numerical parameters p, θ and a functional parameter w. We reduce this problem to the problem of a continuous embedding of one local Morrey-type space to another one. This allows obtaining, for all admissible values of the numerical parameters α, p₁, p₂, θ₁, θ₂, sufficient conditions on the functional parameters w₁ and w₂ ensuring the boundedness of H_α from LM_p1θ1,w1(.) to LM_p2θ2,w2(.) and from GM_p1θ1,w1(.) to GM_p2θ2,w2(.). Moreover, for a certain range of the numerical parameters and under certain a priori assumptions on w₁ and w₂ these sufficient conditions coincide with the necessary ones.