On the compactness of embeddings for a class of weighted Orlicz-Sobolev sequence spaces

TL;DR

Addresses compact embeddings for weighted Orlicz-Sobolev sequence spaces defined on counting measures, extending Sobolev embedding theory to discrete settings. Builds a framework around the nonlinear class $c_{k,Φ,w}$ and two Banach spaces $l_{k,Φ,w}$ and $s_{k,Φ,w}$, proving $\mathrm{span}\,c_{k,Φ,w}=l_{k,Φ,w}$ and that $s_{k,Φ,w}$ is a separable Banach space and the largest subspace of $c_{k,Φ,w}$, and derives continuous embeddings via domination of Orlicz functions; under a local $Δ_2$-condition at the origin, shows $s_{k,Φ,w}=c_{k,Φ,w}=l_{k,Φ,w}$ and that $s_{k',Φ,w} \hookrightarrow s_{k,Φ,w}$ is compact when $k'>k$. The paper then combines these results to generate a spectrum of compact-embedding chains and illustrates with power-type and exponential Orlicz functions, discussing connections to Lebesgue-based theory and open questions.

Abstract

In this article we introduce a new scale of weighted Orlicz-Sobolev sequence spaces generated by a class of suitable Orlicz functions and prove various continuity and compactness criteria for them. In a nutshell, continuity is a consequence of pointwise comparison between Orlicz functions while compactness follows from the combination of the existence of a Schauder basis in the spaces under consideration with a condition on the generating Orlicz functions regarding their local behavior in a small neighborhood of the origin. We illustrate our results by means of several concrete examples and also mention some open questions along the way.