Kolmogorov compactness criterion in variable exponent Lebesgue spaces

Abstract

The well-known Kolmogorov compactness criterion is extended to the case of variable exponent Lebesgue spaces $L^{p(\cdot )}(\Omega )$, where $\Omega$ is a bounded open set in ${\mathbb{R}}^n$ and $p(\cdot )$ satisfies some "standard" conditions. Our final result should be called Kolmogorov-Tulajkov Sudakov compactness criterion, since it includes the case $p_{-}=1$ and requires only the "uniform" condition.

8 pages