Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be defined by, for example, $f(x) = \frac{1}{2}x^2$. Then $f$ belongs to the Holder space $C^{1,1}(\mathbb{R})$.
Since $C^{1,1}(\mathbb{R})\subset C^{1,\alpha}(\mathbb{R})$ for $\alpha < 1$, then $f \in C^{1,\alpha}(\mathbb{R})$.
But this implies that $\frac{|f'(x) – f'(y)|}{|x-y|^{\alpha}} = |x-y|^{1-\alpha} \leq C$, for all $x, y$.
What is the obvious thing I am missing here? It seems the inclusion $C^{1,1}(\mathbb{R})\subset C^{1,\alpha}(\mathbb{R})$ is false, but I am looking at it right now on page 16 of the book `Introduction to the Calculus of Variations' by Bernard Dacorogna: http://www.amazon.com/Introduction-To-The-Calculus-Variations/dp/1860945082.
The precise statement in there is
Let $\Omega \subset \mathbb{R}^n$ be open. If $0 \leq \alpha \leq \beta \leq 1$ and $k \geq 0$ is an integer, then
\begin{equation}
C^{k,1}(\bar{\Omega}) \subset C^{k,\beta}(\bar{\Omega}) \subset C^{k,\alpha}(\bar{\Omega})\subset C^{k}(\bar{\Omega}).
\end{equation}
Best Answer
The inclusion is false on unbounded domains, unless one modifies the definition of the Hölder space, e.g., by replacing $$\frac{|u(x)-u(y)|}{|x-y|^\alpha}$$ with $$\frac{|u(x)-u(y)|}{|x-y|^\alpha+|x+y| }$$ (which would not be an unreasonable thing to do. Estimates of the kind $\lesssim |x-y|^\alpha$ are usually meant to be local; the consideration of large distances is mostly a nuisance.)
When reading Dacorogna's book, assume throughout that $\Omega$ is bounded. This came up before.