Reference for Sobolev-Hölder embedding for unbounded domains 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).
 A: *

*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

V.S. Rychkov. On Restrictions and Extensions of the Besov and
Triebel–Lizorkin spaces with Respect to Lipschitz Domains. Journal
of The London Mathematical Society, 60:237–257, 08 1999.

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

H. Brezis. Functional analysis, Sobolev spaces and partial
differential equations. Universitext. Springer, New York, 2011



*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.
