Axiomatic Potential Theory,Asymptotic Values and Elliptic Differential Operators

In 14] Fernández, Heinonen and Llorente extend the Hornblower's results, about boundary behaviour of subharmonic functions in the unit disc of the complex plane, to subharmonic functions in the unit ball or the upper half space in higher dimensions. In this paper we establish that those results are also valid in the much more general setting of linear axiomatic potential theory. The interest of our general formulation relies on the applications to differential operators. We apply our result to Laplace–Beltrami operator and some uniformly elliptic second order operators in divergence form.