Representation in $C(K)$ by Lipschitz functions

The isometric universality of the spaces $C(K)$ for K a non scattered Hausdorff compact does not take into account the ``quality'' of the representation. Indeed, the existence of an isometric copy of a separable Banach space X into $C(K)$ made of regular enough functions, say Lipschitz with respect to a lower semicontinuous metric defined on K, imposes severe restrictions to both X and K. In this paper, we present a systematic treatment of the representation of Banach spaces into $C(K)$ by Lipschitz functions improving previous results of the author.