Generalised Morrey sequence spaces
Dorothee D. Haroske, Leszek Skrzypczak
TL;DR
The paper develops a comprehensive theory for generalised Morrey sequence spaces $m_{\varphi,p}$ on $\mathbb Z^d$, extending the classical Morrey sequence spaces and linking them to generalized Morrey function spaces ${\mathcal M}_{\varphi,p}$. It establishes the fundamental structure, including nontriviality, inclusions, separability, and connections to $\ell_p$ and $\ell_\infty$, and provides a sharp, necessary-and-sufficient criterion for the continuity of embeddings $m_{\varphi_1,p_1}\hookrightarrow m_{\varphi_2,p_2}$ with a fixed exponent relation, while proving embeddings are never compact. The work also develops finite-dimensional analogues $m^{2^{jd}}_{\varphi,p}$ and derives explicit norms for the identity maps between them, enabling entropy/approximation analysis and aiding the study of Morrey-type embeddings on bounded domains. Finally, it characterizes strict singularity of infinite-dimensional embeddings, revealing when such maps fail to be isomorphisms on any infinite-dimensional subspace and connecting discrete results to classical Morrey-space theory via the asymptotics of $\varphi$.
Abstract
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 $m_{\varphi,p}=m_{\varphi,p}(\mathbb{Z}^d)$. They are natural generalisations of the classical Morrey sequence spaces $m_{u,p}$, $0<p\le u<\infty$, which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.