Let
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be open
- $E$ be a $\mathbb R$-Banach space
- $k\in\mathbb N_0$
- $\gamma\in(0,1]$
The Hölder space $C^{k+\gamma}(\Lambda,E)$ is defined to be $$\left\{f\in C^k(\Lambda,E):\partial^\alpha f\text{ is bounded for all }|\alpha|\le k\text{ and }[\partial^\alpha f]_\gamma<\infty\text{ for all }|\alpha|=k\right\},$$ where $$[f]_\gamma:=\sup_{\substack{x,\:y\:\in\:\Lambda\\x\:\ne\:y}}\frac{\left\|f(x)-f(y)\right\|_E}{|x-y|^\gamma}\;\;\;\text{for }f:\Lambda\to E.$$
Why are only the partial derivatives of order $k$ forced to be $\gamma$-Hölder continuous?
That doesn't make sense to me, unless we can show that, for any $f\in C^{k+\gamma}(\Lambda,E)$, $\partial^\alpha f$ is $\gamma$-Hölder continuous for all $|\alpha|<k$.
It's clear to me that if $g:\Lambda\to E$ is Fréchet differentiable at $x_0\in\Lambda$, then $g$ is locally Lipschitz continuous at $x_0$. Moreover, if $\Lambda$ is convex, $g$ is everywhere Fréchet differentiable and ${\rm D}g$ is bounded by $C\ge0$, then $g$ is Lipschitz continuous with Lipschitz constant $C$.
However, since $\Lambda$ is not assumed to be convex in general, this fact doesn't help.