Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$ , $k$ be positive integer, and $p \in [1,\infty)$ such that $kp < n$. Let $q\in[1,\dfrac{np}{n-kp}) $ and put $T(u) = u$ for all $u \in W^{k,p} (D)$ . Then $T$ is a bounded linear mapping from $W^{k,p} (D)$ into $L^ q (D)$, and the closure $T(A)$ in $L^ q (D)$ is compact in $L^ q (D)$ for any bounded subset $A$ in $W^{k,p} (D)$ .
Where can I find a proof of this theorem? Or give me some hints.
Thanks a lot.
The case $k=1$ established in Evans is quite enough, since $$ W^{k,p}(D)\overset{I_1}{\hookrightarrow}W^{1,r}(D) \overset{I_2}{\hookrightarrow}L^q(D),\quad r=\frac{np}{n-(k-1)p}\,,\; q\in\bigl[1,\frac{nr}{n-r}\bigr)=\bigl[1,\frac{np}{n-kp}\bigr),$$ with the embedding operators $I_1$ and $I_2\,$, where $I_1$ is just continuous, while $I_2$ is continuous and compact. Hence so is their composition $I=I_1\circ I_2\,$.