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Almost compact and compact embeddings of variable exponent spaces

Volume 268 / 2023

D. E. Edmunds, A. Gogatishvili, A. Nekvinda 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.

Authors

  • D. E. EdmundsDepartment of Mathematics
    Pevensey 2 Building
    University of Sussex
    Brighton BN1 9QH, UK
    e-mail
  • A. GogatishviliInstitute of Mathematics CAS
    Žitná 25
    115 67 Praha 1, Czech Republic
    e-mail
  • A. NekvindaDepartment of Mathematics
    Faculty of Civil Engineereng
    Czech Technical University
    Thákurova 7
    16629 Praha 6, Czech Republic
    e-mail

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