On boundedness of the Hardy operator in Morrey-type spaces - Burenkov, Victor; Tararykova, Tamara; Jain, Pankaj (Eurasian Mathematical Journal, 2011)
In this paper we study the boundedness of the Hardy operator Hα in local and global Morrey-type spaces LMpθ,w(.), GMpθ,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 Hα from LMp1θ1,w1(.) to LMp2θ2,w2(.) and from GMp1θ1,w1(.) to GMp2θ2,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.