What is little Holder space?

Question

In The volume preserving mean curvature flow near spheres, the little Holder space $h^s(U)$ is defined as the closure of $C^\infty(U)$ in the usual Holder norm of $C^s(U)$, where $s >0$. But as I know , the order of Holder space should be $0<s<1$, since it is constant function when $s>1$. But in this paper, there is little Holder space with order more than one. How to understand it ?

Answer 1

As Zachary Selk said, the space $C^{k, \alpha}$ (functions with $k$ derivatives, where the $k$th order derivatives are $\alpha$-Hölder continuous) is sometimes denoted $C^{k+\alpha}$, using a single index equal to the sum of $k$ and $\alpha$. This introduces no ambiguity as long as $k$ is integer and $\alpha\in (0, 1)$.

The little Hölder space is then indeed the closure of smooth functions in $C^{k+\alpha}$ norm.

Aside: the $C^{k+\alpha}$ notation becomes less convenient when $\alpha=1$ (functions with Lipschitz derivatives) since, for example, $C^2$ and $C^{1, 1}$ are different spaces.