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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.

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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.

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  • $\begingroup$ Thanks. This is exactly what i was looking for. $\endgroup$ – Adrián González-Pérez Feb 16 at 12:10
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The result is proven in a more general setting, which includes noncompact $X$, in Theorem 4.38 of my book Lipschitz Algebras, second edition.

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  • $\begingroup$ I suspected as much, but I don't have a copy of your book, so I thought I would go with the reference that I happened to know for sure :) $\endgroup$ – Yemon Choi Feb 16 at 20:19
  • $\begingroup$ Thanks. Thant was a nice read! $\endgroup$ – Adrián González-Pérez Mar 4 at 14:42

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