Almost compact and compact embeddings of variable exponent spaces
Volume 268 / 2023
Studia Mathematica 268 (2023), 187-211
MSC: Primary 46E30; Secondary 26D15.
DOI: 10.4064/sm211206-24-2
Published online: 28 July 2022
Abstract
Let $\Omega $ be an open subset of $\mathbb R^{N}$, and let $p$, $q:\Omega \rightarrow [ 1,\infty ] $ be measurable functions. We give a necessary and sufficient condition for the embedding of the variable exponent space $L^{p(\cdot )}( \Omega ) $ in $L^{q(\cdot )}( \Omega ) $ 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 )}( \Omega ) $ based on $L^{p(\cdot )}( \Omega ) $ is compactly embedded in $L^{q(\cdot )}( \Omega )$; compact embedding results of this type already in the literature are included as special cases.
