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–Ampere operator, differential operators $$d_0$$ and $$d_1$$ and a closed positive current on the Heisenberg group. The quaternionic Monge–Ampere operator is the coefficient of $$ (d_0d_1u)^n$$ . We establish the Chern–Levine–Nirenberg type estimate, the existence of quaternionic Monge–Ampere measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic Monge–Ampere 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 $$ d_0d_1=-d_1d_0 $$ . This is the main reason that we have a good theory for the quaternionic Monge–Ampere operator than $$ (\partial _b{\overline{\partial }}_b)^n$$ .