Let $U$ is a bounded open subset of $\mathbb{R}^n$ where $n >2$ with $C^1$ boundary.
I have to show that $(\int_{\partial U}\vert u \vert ^q)^{\frac{1}{q}} \leq C \|u \|_{W^{1,2}(U)}$ is to hold for all $u$ in $W^{1,2}(U)$, provided that $1 \leq q \leq \frac{2(n-1)}{n-2}$.
I know that $(\int_{\partial U}\vert w \vert) \leq C \|u \|_{W^{1,1}(U)}$ holds for $w$ in $W^{1,1}(U)$. So I insert $w=u^q$ into this inequality. But I cannot proceed my estimates further...
Could anyone help me?