Let $\Sigma(n + \beta, L)$ for $n \in \mathbb{N}_0$, $0 < \beta \le 1$, $L > 0$ be the set of functions $f : \Omega \to \mathbb{R}$ (or $\mathbb{C}$, whatever) whose derivatives up to order $n$ are continuous and where if $r$ is a multi-index with $\sum_j r_j = n$, each of the $r$th derivatives of $f$ are Hölder continuous: $\left\lvert D^r f(x) - D^r f(y)\right\rvert \le L \lVert x - y \rVert^{\beta}$. In my setting, I have $\Omega = [0, 1]^d$.
I think this question shows (based on Bruce K. Driver's lecture notes) that if $\Omega$ is Lipschitz (as is $[0, 1]^d$, of course), then any $f \in \Sigma(n + \beta, L)$ is also contained in $\Sigma(m + \alpha, L')$ for any $m + \alpha < n + \beta$ and some $L'$.
My question is: for a particular $n$, $\beta$, $L$, $m$, and $\alpha$, can we find an $L'$ such that $\Sigma(n + \beta, L) \subseteq \Sigma(m + \alpha, L')$? (Ideally, the smallest such $L'$, but that's not necessary.)
$L'$ needs to be: $$ L' \ge \sup_{f \in \Sigma(n+\beta, L)} \max_{|r|=m} \sup_{x \ne y \in \Omega} \frac{\lvert D^r f(x) - D^r f(y) \rvert}{\lVert x - y \rVert^\alpha} .$$
So even if $n = m$, for $\alpha < \beta$ it's not obvious to me whether there's a single value of $L'$ that works.