Sobolev inequalities with variable exponent attaining the values $1$ and $n$

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.