Abstract
We study Sobolev embeddings in the Sobolev space $W^{1,p(\cdot)}(\Omega)$ with variable exponent satisfying $1\leqslant p(x) \leqslant n$. Since the exponent is allowed to reach the values $1$ and $n$, we need to introduce new techniques, combining weak- and strong-type estimates, and a new variable exponent target space scale which features a space of exponential type integrability instead of $L^\infty$ at the upper end.
Citation
Petteri Harjulehto. Peter Hästö. "Sobolev inequalities with variable exponent attaining the values $1$ and $n$." Publ. Mat. 52 (2) 347 - 363, 2008.
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