In the first part of our paper we consider the fundamental solution \(p_ s(z,t)\) of the heat equation \(\Delta _ K-\partial /\partial s\), where \(\Delta _ K\) is the Laplace-Kohn operator on the Heisenberg group \({\mathbb{H}}\) with coordinates (z,t)\(\in {\mathbb{C}}\times {\mathbb{R}}\). We present a complete picture of the asymptotic behaviour of \(p_ 1(z,t)\) as \(| z| +| t|\) tends to infinity. This generalizes results of B. Gaveau [Acta Math. 139, 59-153 (1977; Zbl 0366.22010)], who describes the asymptotic behaviour of \(p_ s(z,t)\) for fixed (z,t) and \(s\to 0.\)
In the second part of the paper we describe the asymptotic behaviour of the fundamental solution K(z,t) of the equation \(\Delta _ K-2\) on \({\mathbb{H}}\). Using these results we are able to determine the Martin compactification of \({\mathbb{H}}\) with respect to \(\Delta _ K-2\), which is done in the last section. We show that the Martin boundary is homeomorphic to the Martin compactification of \({\mathbb{R}}^ 2 \)with respect to \(\Delta\)-2, where \(\Delta\) is the usual Laplacian on \({\mathbb{R}}^ 2.\) Especially we find, that every positive \((\Delta _ K- 2)\)-harmonic function on \({\mathbb{H}}\) is indepenent of t. Hence it is a (\(\Delta\)-2)-harmonic function of z.