On the quaternionic Monge–Ampère operator, closed positive currents and Lelong–Jensen type formula on the quaternionic space

Abstract

In this paper, we introduce the first-order differential operators d0 and d1 acting on the quaternionic version of differential forms on the flat quaternionic space Hn. The behavior of d0,d1 and △=d0d1 is very similar to ∂,∂‾ and ∂∂‾ in several complex variables. The quaternionic Monge–Ampère operator can be defined as (△u)n and has a simple explicit expression. We define the notion of a closed positive current in the quaternionic case, and extend several results in complex pluripotential theory to the quaternionic case: define the Lelong number of a closed positive current, obtain the quaternionic version of Lelong–Jensen type formula, and generalize Bedford–Taylor theory, i.e., extend the definition of the quaternionic Monge–Ampère operator to locally bounded quaternionic plurisubharmonic functions and prove the corresponding convergence theorem.