On inclusion properties of discrete Morrey spaces

We discuss a necessary condition for inclusion relations of weak type discrete Morrey spaces which can be seen as an extension of the results in [H. Gunawan, E. Kikianty and C. Schwanke, Discrete Morrey spaces and their inclusion properties, Math. Nachr. 291 2018, 8–9, 1283–1296] and [D. D. Haroske and L. Skrzypczak, Morrey sequence spaces: Pitt’s theorem and compact embeddings, Constr. Approx. 51 2020, 3, 505–535]. We also prove a proper inclusion from weak type discrete Morrey spaces into discrete Morrey spaces. In addition, we give a necessary condition for this inclusion. Some connections between the inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.

Pitt's theorem for operators between general Lorentz sequence spaces

We characterize the pairs of general Lorentz sequence spaces $\ell^{u,v}(\nu),$ $\ell^{p,q}(\mu), 0 < u, v, p, q < \infty$ such that all continuous linear maps from the first space into the second one are compact.