We introduce non-homogeneous equilibrium states which include the classical equilibrium states for subshifts of finite type, Riesz products and G-measures. We prove a rather precise estimate for ∥Pnf∥∞ where Pn are the averaging operators. Applications are given to estimate ∥Lngf∥∞ (Lg being a transfer operator on a subshift of finite type) and then to prove a central limit theorem for f(Tnx) (T being the shift), to study the almost everywhere convergence of some lacunary series with respect the equilibrium state and then to compute the Hausdorff dimension of the equilibrium state etc.