Compact embeddings and Pitt's property for weighted sequence spaces of Sobolev type

TL;DR

This paper develops a framework of weighted Sobolev-type sequence spaces $h_{\ ext{C},w}^{k,s}$ with norm $\left\Vert \mathsf{p} \right\Vert_{k,s,w} = \left( \sum_{m\in\mathbb{Z}} w_m \left(1+|m|^{s}\right)^{k} |p_m|^{s} \right)^{1/s}$ and proves several compact embedding theorems among these spaces and related weighted $\ell^s$ spaces. A concrete Schauder basis $\{e_m\}$ is constructed to facilitate norm-convergent expansions and the analysis. The main results include (i) compact embeddings when the Sobolev order decreases ($k<k'$), (ii) compact embeddings between spaces with different summability and weight relations (involving $w_m$ and a secondary weight $\hat w_m$), and (iii) a Pitt-type theorem showing that every bounded operator between two scales $h_{ ext{C},w}^{k,s}$ and $h_{ ext{C},w}^{k,t}$ with $s>t\ge 1$ is compact. These results generalize embedding properties used in the spectral analysis of master equations with non-constant coefficients and provide a versatile tool for applications in mathematics and mathematical physics, including non-equilibrium statistical mechanics. The work highlights the role of weights and Sobolev-type indices in securing compactness and operator-norm control across a scale of weighted sequence spaces. All mathematical notation is presented with explicit $...$ delimiters to ensure precision in search and indexing.

Abstract

In this article we introduce a new class of weighted sequence spaces of Sobolev type and prove several compact embedding theorems for them. It is our contention that the chosen class is general enough so as to allow applications in various areas of mathematics and mathematical physics. In particular, our results constitute a generalization of those compact embeddings recently obtained in relation to the spectral analysis of a class of master equations with non-constant coefficients arising in non-equilibrium statistical mechanics. As a byproduct of our considerations, we also prove a theorem of Pitt's type asserting that under some conditions every linear bounded transformation from one weighted sequence space of the class into another is compact.