A Note on Inclusions of Discrete Morrey Spaces

We give a necessary condition for inclusion relations between discrete Morrey spaces which can be seen as a complement of the results in \cite{GKS,HS2}. We also prove another inclusion property of discrete Morrey spaces which can be viewed as a generalization of the inclusion property of the spaces of $p$-summable sequences. Analogous results for weak type discrete Morrey spaces is also presented. In addition, we show that each of these inclusion relations is proper. Some connections between inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.