Does anybody know the definition of $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$, where $0<\alpha<1$? I hope someone can give me the definition of the following:

$C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$ for some $0<\alpha<1$ and some domain $\Omega$.

In this context they also talk about $\Omega$ as a bounded and $C^{2+\alpha}$ domain. What does this mean? . (bounded is clear ;) )
I found this in a paper about non-local diffusion.
(It is used in many theorems, but there is no definition...)
Does this rooms have an specific name? Sorry if this something I should know, but if you don't know the name if this it is really hard to find with Google ore something else ;).
Thanks for your help.
 A: $C^{2,1}(\Omega)$ usually refer to the set of function that are $C^2$ on $\Omega$ with respect to the first variable, and $C^1$ with respect to the second one.
In case you have "fractional" numbers, it usually refers to the  Holder condition as mentioned in the other answer.
We say that a certain domain (open and convex) $\Omega$ is $C^2$ if some specific conditions are met[0]; basically it tells you that locally, the boundary of $\Omega$ can be represented by the graph of a certain $C^2$ function
[0] 
Formal definition

We say that $\Omega$ is a $C^1$ domain if for every $x \in \partial
 \Omega$, there exists a coordinate system $(y_1, \dots, y_n) \equiv
 (\bar y', y_n)$ with origin in $x$, a sphere $B(x)$ and a function
  $\varphi$, defined in a neighborhood $\mathcal N \subset \mathbb
 R^{n-1}$ of $\bar y' = 0$, such that 
$$\varphi \in C^1(\mathcal N), \varphi(0) = 0$$ $$\partial \Omega \cap
 B(x) = \{(\bar y', y_n): y_n = \varphi(\bar y'), \bar  y' \in \mathcal
 N\}$$ $$\Omega \cap B(x) = \{(\bar y', y_n): y_n > \varphi(\bar y'),
 \bar  y' \in \mathcal N\}$$

The second condition reflects the fact that $\partial \Omega$ is locally the graph of a $C^1$ function, and the third that $\Omega$ is locally on the same side with respect to $\partial \Omega$.
If you want a $C^2$ domain or a Lipschitz domain replace the above $C^1$ with the corresponding space of functions ;-) 
P.S. Definition taken from "Partial differential equation" by Salsa
A: My guess would be a Hölder condition.
