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.