Reference request: interpolation of Hölder spaces

Question

On the Wikipedia page on interpolation space, it is written that the space $C^\theta([0, 1])$ is the (real) interpolation of $C^0([0, 1])$ and $C^1([0, 1])$, where $C^\theta([0, 1])$ denotes the space of $\theta$-H\"older continuous functions on $[0, 1]$.

However, I was unable to find the detailed proof of this fact (all books I checked treat $L^p$-spaces and their derived spaces). Do you have a reference for this?

To be more precise, the result I would be interested in is the following:

Given $\Omega$ a (non-empty) open subset of $\mathbb{R}^n$, is it true that $C^{k, \alpha}(\Omega)$ is the (real) interpolation space of $C^0(\Omega)$ and $C^l(\Omega)$ for any $l > k+\alpha$ and (possibly) $\alpha \in (0, 1)$?

Answer 1

I'd recommend the works of Alessandra Lunardi for many results about the Hölder spaces. Check for instance her book "Analytic Semigroups and Optimal Regularity in Parabolic Problems" [1, Ch. I.1.2.4], or of course the other book, "Interpolation theory" [2, e.g. Example 1.3.7].

There is also some stuff in Triebel's book [3, Ch. 4.5.2].

[1] Lunardi, Alessandra, Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics. Basel: Birkhäuser (ISBN 978-3-0348-0556-8/pbk; 978-3-0348-0557-5/ebook). xvii, 424 p. (2013). ZBL1261.35001.

[2] Lunardi, Alessandra, Interpolation theory, ZBL06861769.

[3] Triebel, Hans, Interpolation theory, function spaces, differential operators., Leipzig: Barth. 532 p. (1995). ZBL0830.46028.