2013 | OriginalPaper | Buchkapitel
The Heat Kernel and Green Function of the Sub-Laplacian on the Heisenberg Group
We give a construction of the heat kernel and Green function of a hypoelliptic operator on the one-dimensional Heisenberg group
$$\mathbb{H}$$
, the sub-Laplacian
$$\mathcal{L}$$
. The explicit formulas are developed using Fourier–Wigner transforms, pseudo-differential operators of the Weyl type, i.e., Weyl transforms, and spectral analysis. These formulas are obtained by first finding the formulas for the heat kernels and Green functions of a family of twisted Laplacians
$${L}_{\tau}$$
for all non-zero real numbers
$${\tau}$$
. In the case when
$${\tau=1, {L}_{1}}$$
is just the usual twisted Laplacian.
