Yes, indeed every Lipschitz space is a dual space, a fact which has been rediscovered (in varying levels of generality) several times. The earliest proof is due to Arens and Eells.

Holder spaces are special cases of Lipschitz spaces because a function is $\alpha$-Holder continuous for a metric $\rho$ if and only if it is Lipschitz for the metric $\rho^\alpha$.

${\rm Lip}_0(X)$ is the space of Lipschitz functions on $X$ vanishing at a base point. This is the most general class of Lipschitz spaces; other Lipschitz spaces are special cases of these. The predual of ${\rm Lip}_0(X)$ is simply characterized as the universal Banach space containing an isometric copy of $X$ such that the base point of $X$ corresponds to $0$.

There's a lot of information about the predual in my book Lipschitz Algebras (and there will be even more in a soon forthcoming second edition). A recent result is that for a large class of metric spaces the predual of ${\rm Lip}_0(X)$ is unique.

Yes, indeed every Lipschitz space is a dual space, a fact which has been rediscovered (in varying levels of generality) several times. The earliest proof is due to Arens and Eells.

Holder spaces are special cases of Lipschitz spaces because a function is $\alpha$-Holder continuous for a metric $\rho$ if and only if it is Lipschitz for the metric $\rho^\alpha$.

${\rm Lip}_0(X)$ is the space of Lipschitz functions on $X$ vanishing at a base point. This is the most general class of Lipschitz spaces; other Lipschitz spaces are special cases of these. The predual of ${\rm Lip}_0(X)$ is simply characterized as the universal Banach space containing an isometric copy of $X$ such that the base point of $X$ corresponds to $0$.

There's a lot of information about the predual in my book Lipschitz Algebras (and there will be even more in a soon forthcoming second edition). A recent result is that for a large class of metric spaces the predual of ${\rm Lip}_0(X)$ is unique.