Generalised Morrey sequence spaces

Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces $m_{\phi ,p}=m_{\phi ,p}({\mathbb{Z}}^d)$. They are natural generalisations of the classical Morrey sequence spaces $m_{u,p}$, $0<p\le u<\mathrm{\infty }$, which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.