The quaternionic Monge-Amp\`{e}re operator and plurisubharmonic functions on the Heisenberg group

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-Amp\`{e}re operator, differential operators $d_0$ and $d_1$ and a closed positive current on the Heisenberg group. The quaternionic Monge-Amp\`{e}re operator is the coefficient of $(d_0d_1u)^n$. We establish the Chern-Levine-Nirenberg type estimate, the existence of quaternionic Monge-Amp\`{e}re measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic Monge-Amp\`{e}re operator. Unlike the tangential Cauchy-Riemann operator $\overline{\partial}_b$ on the Heisenberg group which behaves badly as $\partial_b\overline{\partial}_b\neq -\overline{\partial}_b\partial_b$, the quaternionic counterpart $d_0$ and $d_1$ satisfy $d_0d_1=-d_1d_0$. This is the main reason that we have a better theory for the quaternionic Monge-Amp\`{e}re operator than $(\partial_b\overline{\partial}_b)^n$.