Reference request: $\alpha$-Hölder spaces as double duals

Question

If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that $$ \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}. $$ Similarly, we can define the Little Hölder space $\lambda_\alpha(X)$ as the subset of functions of $\Lambda_\alpha(X)$ satisfying that $$ \lim_{x \to y} \frac{|f(x) - f(y)|}{|x - y|^\alpha} = 0. $$ I recall a result stating that, at least in classical contexts like $X = \mathbb{T}^n$, $\Lambda_\alpha(X)$ is isomorphic to the double dual of $\lambda_\alpha(X)$.

Question: Is there any reference for that duality in the context of more general metric spaces? An initial google search didn't yield anything.

Answer 1

First of all, I am not sure what you mean by $L^\infty(X)$ for a general complete metric space $X$. Don't you want $C_b(X)$?

Secondly: when $X$ is compact and $0<\alpha<1$, the result you want is Theorem 3.5 in

W. G. Bade, P. C. Curtis, Jr, and H. G. Dales, Amenabilty and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. 55 (1987) 359–377.

Answer 2

The result is proven in a more general setting, which includes noncompact $X$, in Theorem 4.38 of my book Lipschitz Algebras, second edition.