I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (higher order) Hölder spaces that would not come from the extension theorem for Sobolev spaces.
If $(X, d)$ is a metric space and $A \subseteq X$, McShane's Extension Theorem states that every $L$-Lipschitz function $f \colon A \to \mathbb{R}$ can be extended to $L$-Lipschitz function $f' \colon X \to \mathbb{R}$. By changing metric $d$ to $d^{\alpha}$, where $\alpha \in (0,1)$, we can make this statement about $\alpha$-Hölder functions. We see that there are no requirements on the regularity of $A$ (as in, we do not require anything about the boundary of $A$) to perform this extension.
Now, suppose $X = \mathbb{R}^n$ and we would like to extend $f$ which is of class $C^{k, \alpha}(A)$ to an element $C^{k, \alpha}(X)$. I'm aware that if $A$ is open then we can think of elements of $C^{k, 1}(A)$ as elements of the Sobolev space $W^{k+1, \infty}(A)$. Hence, we can have an extension for sufficiently regular $A$'s. However, I'm more interested in whether this regularity condition can be relaxed much like in the case $k = 0$. Is there an extension theorem for Hölder functions with weaker conditions on the regularity of $A$ than the ones from the Sobolev extension theorem?