Quasicontinuity and maximality of quaternionic plurisubharmonic functions

Abstract

In this paper, we establish the quaternionic versions of several results in the complex pluripotential theory by using the quaternionic closed positive currents, which were introduced in [24]. We show that quasicontinuity, one of the most important properties of complex plurisubharmonic functions, holds also for quaternionic plurisubharmonic functions in Hn. Moreover, we prove an equivalent characterization of the maximality, which is central to the pluripotential theory. A locally bounded plurisubharmonic function u is maximal if and only if it satisfies the homogeneous quaternionic Monge–Ampère equation (Δu)n=0.