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

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.