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 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.
