The quaternionic Monge-Ampère operator and plurisubharmonic functions on the Heisenberg group

Abstract

Many fundamental results of pluripotential theory on the quaternionic space ${\mathbb{H}}^n$ are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge-Ampère operator, differential operators $d_0$ and $d_1$ and a closed positive current on the Heisenberg group. The quaternionic Monge-Ampère operator is the coefficient of $(d_0{d}_{1}u{)}^n$. We establish the Chern-Levine-Nirenberg type estimate, the existence of quaternionic Monge-Ampère measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic Monge-Ampère operator. Unlike the tangential Cauchy-Riemann operator ${\overline{\partial }}_b$ on the Heisenberg group which behaves badly as ${\partial }_b{\overline{\partial }}_b\ne -{\overline{\partial }}_b{\partial }_b$, the quaternionic counterpart $d_0$ and $d_1$ satisfy $d_0{d}_1=-d_1{d}_0$. This is the main reason that we have a better theory for the quaternionic Monge-Ampère operator than $({\partial }_b{\overline{\partial }}_b{)}^n$.