In this paper, \((X,\rho)\) is a separable complete metric space equipped with a locally finite Radon measure \(\mu\) whose support is \(X\), and \(G(t,x,y)\) is a stochastically complete heat kernel (or transition density) associated to \((X,\rho,\mu)\), satisfying a two-sided estimate. The authors consider particular examples in which \((X,\rho)\) is a fractal set. Under various conditions on the semigroup \(G\), they define Bessel and Riesz potential operators, Besov spaces and (fractional) Sobolev-type spaces associated to such a setting, and they describe some embeddings and isomorphisms between such spaces