The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space

Abstract

In this paper, we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic Monge–Ampere operator. By introducing a new quaternionic capacity, we prove a sufficient condition which implies the weak convergence of quaternionic Monge–Ampere measures \((\triangle u_j)^n\rightarrow (\triangle u)^n\). We also obtain an equivalent condition of “convergence in \(C_{n-1}\)-capacity” by using methods from Xing (Proc Am Math Soc 124(2):457–467, 1996). As an application, the range of the quaternionic Monge–Ampere operator is discussed.