I have recently started working with Sobolev Spaces and I wanted to ask the following:
Let $1 \leq p \leq \infty, \Omega \subset \mathbb{R}^d$. Does $ W^{n,p}(\Omega) \subset W^{1,p}(\Omega)$ hold?
Here $W^{n,p}(\Omega)$ is the space of $L^p$ "functions" such that their weak derivatives up to order n are also in $L^p$.
I do not know whether this seems trivial or I am missing some technical details here.
Further I have seen that $W^{1,\infty}(\Omega)$ can be described as the set of (locally) Lipschitz functions for bounded $\Omega$. That would also imply that all functions in $W^{n,\infty}$ would have derivatives up to order $n-1$ (including themselves), which are Lipschitz, is that correct?
Thanks for the answers in advance!