I would like to know whether the following is true and references:
If $\Omega\subset\mathbb R^n$ is open (not necessarily bounded) and $k = n/p + r+\alpha$, and $\alpha \in (0,1), r\in \mathbb N, k\geq 0,p\geq 1$, then there is a continuous embedding of the Sobolev space
$$ W^{k,p}(\Omega)\to C^{r,\alpha}(\Omega)$$
into the space of Hölder functions.
I have found in several places the case the case $k=1$, and in other places, the case when $\Omega$ is bounded. In this generality it is stated in Wikipedia for $\Omega=\mathbb R^n$. I am looking for a more reliable source.
It is possible that $\Omega$ open is not enough, in that case I would be interested to the case when $\Omega = \mathbb R^n$ or to know what are the most general hypothesis (e.g. $\Omega$ star domain).
Best Answer
First, it suffices to prove your result for $\mathbb{R}^n$ if the open set $\Omega$ satifies the extension property, i.e. the existence of bounded linear operator $E\,:\,W^{k,p}(\Omega)\longrightarrow W^{k,p}(\mathbb{R}^n)$ such that $Eu_{|_{\Omega}}=u$. It is true for sufficiently general $\Omega$, in particular in the case of Lipschitz domains not necessarily bounded, see
Noticing that Triebel-Lizorkin and Besov spaces contains the whole scales of Sobolev spaces. Various other simpler extension operators exist, but Rychkov's is the only one that covers, in my knowlegde, unbounded Lipschitz domains.
To prove the result on $\mathbb{R}^n$, in its full generality you just need the result
$$W^{1,p}(\mathbb{R}^n)\hookrightarrow C^{0,1-\frac{n}{p}}(\mathbb{R}^n).$$
This is exactly the Theorem 9.12 of
To recover the general case, apply it iteratively (hence prove it by induction) using $\nabla u$ instead $u$, and use the fact that Sobolev spaces decrease as the regularity index increase and sum with the previous estimate.