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-Ampère 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-Ampère operator to locally bounded quaternionic plurisubharmonic functions and prove the corresponding convergence theorem.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2014
- DOI:
- 10.48550/arXiv.1401.5291
- arXiv:
- arXiv:1401.5291
- Bibcode:
- 2014arXiv1401.5291W
- Keywords:
-
- Mathematics - Complex Variables;
- Mathematics - Analysis of PDEs
- E-Print:
- 32 pages