Suppose $0<\alpha <\beta$. Then, the Hölder space $C^\beta$ is compactly imbedded to $C^\alpha$. See the Wikipedia article Hölder condition.
However, I could not find precise reference from some books on functional analysis.
Can anybody indicate a precise reference for this theorem?
If possible, I would like to know a reference on the similar result on parabolic Hölder space.
It's Lemma 6.33 in Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger. A proof is also presented by Ngô Quốc Anh on his blog. It's quite simple. Take a bounded family of functions in $C^\beta(\Omega)$. Extend them continuously to $\overline{\Omega}$ (uniform continuity allows that). Apply the Ascoli-Arzelà theorem to extract a uniformly convergent subsequence. Since $$\|u_m-u_n\|_{C^\alpha} \le \|u_m-u_n\|_{C^\beta}^{\alpha/\beta} (\sup |u_m-u_n|)^{1-\alpha/\beta} \tag1$$ it follows that $(u_m)$ is Cauchy in $C^\alpha$.
A parabolic Hölder space is a Hölder space with respect to a particular metric on $\mathbb R^n\times\mathbb R$, like $\rho((x,t),(x',t'))=|x-x'|+|t-t'|^{1/2}$. The above argument applies irrespective of the metric. All that matters is that the domain has a compact closure, so that the Ascoli-Arzelà theorem can be applied.
