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.