Heat Kernel for the Kohn Laplacian on the Heisenberg Group

Abstract

We shall deal next with the nonsymmetric form of the Heisenberg group. The Heisenberg group considered in this section will be the set ℍ _n = ℝ ⁿ ×ℝ ⁿ ×ℝ with the following group law:

$$(x,y,t) {_\ast} ({x}^{{\prime}},{y}^{{\prime}},{t}^{{\prime}}) = (x + {x}^{{\prime}},y + {y}^{{\prime}},t + {t}^{{\prime}} + x \cdot {y}^{{\prime}}),$$

where (x, y, t), (x ^′, y ^′, t ^′) ∈ ℝ ⁿ ×ℝ ⁿ ×ℝ and

$$x \cdot {y}^{{\prime}} ={ \sum \nolimits }_{k=1}^{n}{x}_{ k}{y}_{k}^{{\prime}}.$$