Frechet differentiability of molecular distribution functions I. $L^\infty$ analysis

For a grand canonical ensemble of classical point-like particles at equilibrium in continuous space we investigate the functional relationship between a stable and regular pair potential describing the interaction of the particles and the corresponding molecular distribution functions. For certain admissible perturbations of the pair potential and sufficiently small activity we rigorously establish Frechet differentiability with respect to the supremum norm in the image space -- both for bounded domains and in the thermodynamical limit.