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 H n . 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 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-Ampère operator to locally bounded quaternionic plurisubharmonic functions and prove the corresponding convergence theorem.

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

“…Then Alesker and Verbitsky studied the quaternionic Monge-Ampère operator on general quaternionic manifolds ( [4,16] and references therein). In Wan and Wang [18] we established an explicit expression of the quaternionic MongeAmpère operator: it can be defined as ( u) n , where the Baston operator = d 0 d 1 is the first operator of 0-Cauchy-Fueter complex:…”

“…By introducing the quaternionic version of differential forms, we defined in Wan and Wang [18] the notions of closed positive forms and closed positive currents in the quaternionic case and our definition of closedness matches positivity well. We proved that u is a closed positive 2-current for any plurisubharmonic function u, and showed that when functions u 1 , .…”

Potential theory for quaternionic plurisubharmonic functions

“…∧ △u n = µΩ 2n for a well defined positive Radon measure µ. See [17] for the detailed information about the closed positive currents in H n .…”

“…Motivated by this, Wang and the author introduced in [17] ∂, ∂ and ∂∂ in several complex variables. The quaternionic Monge-Ampère operator can be defined as (△u) n and has a simple explicit expression, which is much more convenient than the definition by using Moore determinant.…”

“…The theory of quaternionic closed positive currents established recently in [17] allows us to treat the quaternionic Monge-Ampère operator as an operator of divergence form, and so we can integrate by parts. Since this can avoid the inconvenience in using Moore determinant, we established several useful quaternionic versions of results in the complex pluripotential theory in [1].…”