Some properties of the extremal function for the Fuglede p-modulus

We deal with the Fuglede p-modulus of a system of measures, focusing on three aspects. First, we combine results concerning Badger’s criterion for the extremal function, i.e., the function which realizes the p-modulus, and plans with barycenter in \(L^q\), which give an alternative—in a sense, probabilistic—approach to p-modulus. It seems that the correlation of these results has not yet been established. Second, we deal with families of measures for which the integral of the extremal function is one. On such a family, the p-modulus as well as the optimal plans are concentrated. We consider closures of these families and relate them with generic families of measures for which the extremal function exists. Finally, we compute the p-modulus and extremal function for finite families of measures.