Variable exponent Sobolev spaces and regularity of domains-II

Abstract

AbstractWe provide necessary conditions on Euclidean domains for inclusions $$W^{1,p(\cdot )}(\Omega ) \hookrightarrow L^{q(\cdot )}(\Omega ) $$ W 1 , p ( · ) ( Ω ) ↪ L q ( · ) ( Ω ) of variable exponent Sobolev spaces. The conditions on the exponent $$ p(\cdot ) $$ p ( · ) are log-Hölder and log-log-Hölder continuity, while those on the domain $$ \Omega $$ Ω are the measure and the log measure density conditions. Restrictions on the exponents $$ q(\cdot ) $$ q ( · ) and $$ p(\cdot )$$ p ( · ) appearing in Górka et al. (J. Geom. Anal. 310: 7304-7319, 2021) are relaxed, improving the results obtained in that work.