Definition of the space of Hölder continuous functions

Question

I have a question concerning the definition of the space of Hölder continuous functions $C^{k,\alpha}(\Omega)$, where $\Omega\subset\mathbb{R}^n$ is an open set and $k\in\mathbb{N}$ and $\alpha\in (0,1]$.

Often the following definition is used: $f\in C^{k,\alpha}(\Omega)$ if and only if $f\in C^{k}(\Omega)$ such that all its derivatives up to order $k$ are bounded functions and the $k-th$ order derivatives are Hölder continuous.

I am wondering why only $D^{\beta}f:U\rightarrow \mathbb{R}$ with $\vert \beta \vert=k$ has to be Hölder continuous? Does it maybe follow that $f$ and its derivatives up to order $k-1$ have to be Hölder continuous as well? Otherwise the space of Hölder continuous functions would contain functions which are not Hölder continuous, which sounds very awkward to me.

Best wishes

Answer 1

It is possible to further weaken the assumptions given on $\Omega$ as being "only" a bounded domain with $C^1$ boundary (even Lipschitz should still work).

It suffices to prove that if $u \in C^{1,\alpha}_b(\Omega)$, i.e. $u$, and $\partial_i u$ are bounded and $\partial_i u$ are $\alpha$-Hölder, then $u$ is $\alpha$-Hölder as well.

Since $\Omega$ is a bounded $C^1$-domain, there exist finitely many bounded diffeomorphisms $\Phi_i \colon \mathbb R^n \to \mathbb R^n$ such that $$\Phi_i(\Omega \cap U_i) \subset B_i^+, \quad \Phi(\partial\Omega \cap U_i) = B_i^+ \cap (\mathbb R^{n-1} \times \{0\})$$ Where $(U_i)_{i = 1,\dots,N}$ is an open covering of $\partial\Omega$ and $$ B_i^+ := \{ x \in \mathbb R^n \colon |x| < r_i, x_n \geq 0\}. $$ Correct me if something seems odd here, as this should be standard for $C^1$-submanifolds.

Now, take $x,y \in \Omega$ and let $\delta > 0$ be the Lebesgue number of the covering $(U_i)$. Now we distinguish the following cases

1. If $|x - y| < \delta$ and $x,y \in \overline{B_\delta(x)} \subset \Omega$: In this case, the mean value theorem is applicable and yields $$ |u(x) - u(y)| \leq \|\nabla u\|_\infty |x - y|. $$

2. If $|x - y| < \delta$ and $x,y \in \Omega \cap U_i$: Consider the convex hull $$ W := \{(1 - t) \Phi_i(x) + t \Phi(y) \colon t \in [0,1]\} \subset B_i^+. $$ Then \begin{align*} |u(x) - u(y)| &= |u(\Phi_i^{-1}\circ \Phi_i(x)) - u(\Phi_i^{-1}\Phi_i(y))| \\ &= \Big|\int_0^1 \frac{d}{dt} \Big((u\circ\Phi_i^{-1})\big((1 - t)\Phi_i(x) + t \Phi_i(y)\big) \Big) d t \Big| \\ &\leq \| \nabla(u \circ \Phi_i^{-1})\|_\infty \cdot |\Phi_i(x) - \Phi_i(y)| \\ &\leq \| \nabla u\|_{\infty} \cdot \| D\Phi_i^{-1}\|_\infty \cdot \|D\Phi_i\|_{\infty} |x - y|, \end{align*} where we applied the mean value theorem to $\Phi_i$.

3. If $|x - y| \geq \delta$, then $$ |u(x) - u(y)| \leq 2 \|u\|_\infty \cdot 1 \leq 2 \|u\|_{\infty} \frac{|x - y|}{\delta}. $$

Hence, $$ \sup_{x, y \in \Omega, x \neq y} \frac{|u(x) - u(y)|}{|x - y|} \leq C (\|u\|_\infty + \|\nabla u\|_\infty) $$ for some constant $C > 0$ and thus $u$ is Lipschitz, i.e. in particular $\alpha$-Hölder.