Almost-compact and compact embeddings of variable exponent spaces

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.