Górka, PrzemysławKarak, NijjwalPons, Daniel J.2024-05-272024-05-272023Revista Matematica Complutense. Open Access. 20231139-1138https://repositorio.unab.cl/handle/ria/57143Indexación: Scopus.We would like to thank the reviewers for the valuable comments and suggestions. P. G. has been supported by NCN grant 2018/02/X/ST1/02133. N. K. thanks DST-SERB (Project SRG/2021/000118) and BITS Pilani (BITS/GAU/RIG/2020/H0749) for financial support.We provide necessary conditions on Euclidean domains for inclusions W1,p(·)(Ω) ↪ Lq(·)(Ω) of variable exponent Sobolev spaces. The conditions on the exponent p(·) are log-Hölder and log-log-Hölder continuity, while those on the domain Ω are the measure and the log measure density conditions. Restrictions on the exponents q(·) and p(·) appearing in Górka et al. (J. Geom. Anal. 310: 7304-7319, 2021) are relaxed, improving the results obtained in that work. © 2023, The Author(s).enMeasure Density ConditionSobolev EmbeddingSobolev spacesArtículoCC BY 4.0 DEED Attribution 4.0 International10.1007/s13163-023-00470-5