Subharmonic functions in sub-Riemannian settings

Abstract

In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution $\Gamma$. These characterizations are based on suitable average operators on the level sets of $\Gamma$. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic function in the weak sense of distributions, and we show how to approximate subharmonic functions by smooth ones. The classes of operators involved are wide enough to comprise, as very special cases, the sub-Laplacians on Carnot groups. The results presented here generalize and carry forward former results of the authors in [6, 8].