Morrey Sequence Spaces: Pitt's Theorem and compact embeddings

TL;DR

This paper introduces Morrey sequence spaces, explores their fundamental properties, and establishes results on embeddings, duality, and a version of Pitt's theorem, advancing the understanding of these less-studied generalizations of classical sequence spaces.

Contribution

It is the first comprehensive study of Morrey sequence spaces, including their basic features, duality, and compactness properties, along with a Pitt-type theorem.

Findings

Morrey sequence spaces are natural generalizations of ll_p spaces.

Established embedding and duality properties of Morrey sequence spaces.

Proved a version of Pitt's compactness theorem for these spaces.

Abstract

Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces $m_{u,p}=m_{u,p}(\mathbb{Z}^d)$, $0<p\leq u<\infty$, which have yet been considered almost nowhere. They are defined as natural generalizations of the classical $\ell_p$ spaces. We consider some basic features, embedding properties, the pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterize the compactness of embeddings of related finite-dimensional spaces.