When $\Omega\subset%
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\mathbb{R}
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^{n}$, $n\geq2$ is a bounded domain, we have by the Sobolev embedding theorems
that $W_{0}^{k,p}\left( \Omega\right) \subset L^{q}\left( \Omega\right)
,~1\leq q\leq\frac{np}{n-kp},~kp<n$ and that $W_{0}^{k,\frac{n}{k}}\left(
\Omega\right) \subset L^{q}\left( \Omega\right) ,~1\leq q<\infty$. However,
there are counterexamples to the embedding $W_{0}^{k,\frac{n}{k}}\left(
\Omega\right) \subset L^{\infty}\left( \Omega\right) $. In this case, it
was proposed independently by Yudovich, Pohozaev and
Trudinger that $W_{0}^{1,n}\left( \Omega\right) \subset
L_{\varphi_{n}}\left( \Omega\right) $ where $L_{\varphi_{n}}\left(
\Omega\right) $ is the Orlicz space associated with the Young function
$\varphi_{n}(t)=\exp\left( \beta_{n} \left\vert t\right\vert ^{n/(n-1)}%
\right) -1$ for some positive $\beta_{n}$. Moreover, Moser explored more in
this direction and further found the largest positive real number $\beta_{n}$.
In fact, in his 1971 paper [A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71), 1077–1092.], J. Moser proved the following result: There exist sharp constant $\beta_{n}%
=n\omega_{n-1}^{\frac{1}{n-1}}$, where $\omega_{n-1}$ is the
area of the surface of the unit $n-$ball, such that
$$
\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\exp\left( \beta
\left\vert u\right\vert ^{\frac{n}{n-1}}\right) dx\leq c_{0}%
$$
for any $\beta\leq\beta_{n}$, any $u\in W_{0}^{1,n}\left(
\Omega\right) $ with $\int_{\Omega}\left\vert \nabla u\right\vert
^{n}dx\leq1$. This constant $\beta_{n}$ is sharp in the sense
that if $\beta>\beta_{n}$, then the above inequality can no longer
hold with some $c_{0}$ independent of $u$.
The same result for $W_{0}^{k,p}\left( \Omega\right)$ was proved by D. Adams in his paper [A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. (2) 128 (1988), no. 2, 385–398.]
