On quaternionic Monge-Ampere operator, closed positive currents and Lelong-Jensen type formula on quaternionic space

Abstract

In this paper, we introduce the first-order differential operators $d_0$ and $d_1$ acting on the quaternionic version of differential forms on the flat quaternionic space $\mathbb{H}^n$. The behavior of $d_0,d_1$ and $\triangle=d_0d_1$ is very similar to $\partial,\overline{\partial}$ and $\partial \overline{\partial}$ in several complex variables. The quaternionic Monge-Amp\`{e}re operator can be defined as $(\triangle u)^n$ and has a simple explicit expression. We define the notion of closed positive currents in the quaternionic case, and extend several results in complex pluripotential theory to the quaternionic case: define the Lelong number for closed positive currents, obtain the quaternionic version of Lelong-Jensen type formula, and generalize Bedford-Taylor theory, i.e., extend the definition of the quaternionic Monge-Amp\`{e}re operator to locally bounded quaternionic plurisubharmonic functions and prove the corresponding convergence theorem.