Generalized version of the Lions-type lemma

TL;DR

This paper revisits the history of Lions-type lemmas addressing compactness issues in unbounded domains and introduces a generalized vanishing lemma applicable to Lebesgue-measurable functions, extending results to Orlicz-Sobolev spaces.

Contribution

It provides a generalized Lions-type lemma that focuses on integral behavior, broadening applicability to Orlicz-Sobolev spaces and unbounded domains.

Findings

Proves a vanishing Lions-type lemma for Lebesgue-measurable functions.

Extends classical results to Orlicz-Sobolev spaces.

Highlights the importance of integral behavior over space properties.

Abstract

In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz-Sobolev spaces. What matters here is the behavior of the integral, not the space.