Almost-compact and compact embeddings of variable exponent spaces

TL;DR

This paper establishes necessary and sufficient conditions for the almost-compact and compact embeddings of variable exponent Lebesgue and Sobolev spaces, extending existing results and providing a comprehensive framework for understanding these embeddings.

Contribution

It provides a new characterization of almost-compact embeddings for variable exponent spaces, generalizing previous results and including special cases as particular instances.

Findings

Characterization of almost-compact embeddings for variable exponent Lebesgue spaces

Conditions for compact embeddings of variable exponent Sobolev spaces

Extension of known embedding results to more general variable exponent settings

Abstract

Let $\Omega $ be an open subset of $\mathbb{R}^{N}$, and let $p,\, q:\Omega \rightarrow \left[ 1,\infty \right] $ be measurable functions. We give a necessary and sufficient condition for the embedding of the variable exponent space $L^{p(\cdot )}\left( \Omega \right) $ in $L^{q(\cdot )}\left( \Omega \right) $ to be almost compact. This leads to a condition on $\Omega, \, p$ and $q$ sufficient to ensure that the Sobolev space $W^{1,p(\cdot )}\left( \Omega \right) $ based on $L^{p(\cdot )}\left( \Omega \right) $ is compactly embedded in $L^{q(\cdot )}\left( \Omega \right) ;$ compact embedding results of this type already in the literature are included as special cases.