Hardy operators in variable Morrey spaces

Rafeiro, Humberto; Samko, Stefan

http://hdl.handle.net/10400.1/26424

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Authors

Rafeiro, Humberto

Samko, Stefan

Abstract(s)

We study the boundedness of multidimensional Hardy operators over Rn in the framework of variable generalised local and global Morrey spaces with power-type weights, where we admit variable exponents for weights. We find conditions on the domain and target spaces ensuring such boundedness. In case of local spaces, these conditions involved values of variable integrability exponents of the domain and target spaces only at the origin and infinity. Due to the variability of the exponents of weights, the obtained results proved to be different corresponding to two distinct cases, which we called up to borderline and overbordeline case. We also pay special attention to a particular case, when the variable domain and target Morrey spaces are related to each other by Adams-type condition. The proofs are based on certain point-wise estimates for the Hardy operators, which allow, in particular, to get a statement on the boundedness from a local Morrey space to an arbitrary Banach function space with lattice property.