Question: $\Omega$ Lipschitz bounded domain in $\mathbb{R}^n$, show that for $ q\geq 1$, $$ \|v\|_q \leq C(n)q^{1-1/n} |\Omega|^{1/q}\|v\|_{W^{1,n}} $$ for $ v\in W_0^{1,n} $.
I think it's correct, since when $n=1$, the exponent on $q$ vanish, which is consistent with the fact that $W^{1,1}\subseteq L^\infty$ in 1d.
My attempt: I tried to use potential estimate like $$ v(x)\leq C(n)\int_\Omega \frac{|Dv(y)|}{|y-x|^{n-1}}dy $$ as in the proof of Morrey inequality. I tried to use Minkovski inequality to bound the $L^q$ norm, but it failed, since the potential term $|y-x|^{-n+1}$ can't have too big exponent to be integrable with respect to $x$. I don't know what to do next.
I also tried to bound the $L^q$ norm by some $W^{1,p}$norm, $p<n$ and then use Holder. But in the first step I only get exponent 1 of $q$, not $1-1/n$ (I use Sobolev inequality in Evans PDE, where the constant is $\gamma=q(n-1)/n \cdot C(n)$).
My work on the 2nd attempt: Let $\gamma=q(n-1)/n$, and apply the inequality (in comment) to $|u|^\gamma$, \begin{align*} &\left(\int_\Omega |u|^{\gamma\frac{n}{n-1}}\,dx\right)^{(n-1)/n} \\ \leq &C(n)\int_\Omega |D(|u|^\gamma)|\,dx = C(n)\gamma \int_\Omega |u|^{\gamma-1}|Du|\,dx \\ \leq &C(n)\gamma\left(\int_\Omega |u|^{(\gamma-1)\frac{p}{p-1}}\right)^{(p-1)/p}\left(\int_\Omega |Du|^p \right)^{1/p} \end{align*} choose $p=\frac{nq}{n+q}$ and we get $$ \left(\int_\Omega |u|^q\,dx\right)^{(n-1)/n} \leq C(n)\gamma \left(\int_\Omega |u|^q\,dx\right)^{(nq-n-q)/nq} \|Du\|_p, $$ therefore $$ \|u\|_q \leq C(n)q\frac{n-1}{n} \|Du\|_p \leq C(n)q\frac{n-1}{n}|\Omega|^{1/q}\|Du\|_n, $$ where the last step use Holder. I still only get exponent 1 on $q$.
Would anyone like to help me or give me some hints? Thanks very much!
