Can "abstract potential theory" be used to establish maximum principles for more general PDOs?

Question

As I understand it, this paper uses ideas from "abstract potential theory" (harmonic spaces, sheaves etc.) to find a maximum principle applying to unbounded domains. But its assumptions on the form of the PDO seem rather restrictive—I am interested in a maximum principle that would apply for more general elliptic and parabolic operators. Is it worth my while learning about this machinery of harmonic sheaves and such? Does it have application beyond this particular class of operators? Any reference would be much appreciated.