Formulas for the representation of soprasoluzioni ultraparaboliche equations on Lie groups.

We investigate several questions in Potential Theory related to a class of hypoelliptic ultraparabolic operators L with underlying homogeneous Lie group structures. Our class is contained in the one of the Hormander operators. We are mainly interested in a characterization of L-superharmonic functions in terms of suitable mean-value operators and in representation formulas. We also characterize the bounded-below L-superharmonic functions in RN+1 and their related L-Riesz measures. Moreover, we prove a more general version of a Riesz representation theorem for Lsuperharmonic functions in terms of L-Green potential of their L-Riesz measures. With this result at hands, we demonstrate the following Poisson-Jensen type representation formula u(z) = integral(partial derivative Omega) u(zeta) d mu(Omega)(z)(zeta) + (G(Omega) * mu), z is an element of Omega Here, u is a L-superharmonic function on a neighborhood of the closure of the arbitrary bounded open set Omega subset of RN+1, mu is the L-Riesz measure for u, mu(Omega)(z) is the L-harmonic measure related to Omega at z, and G(Omega) is the L-Greens function for Omega.