An Elementary Symmetry-Based Derivation of the Heat Kernel on Heisenberg Group

Abstract

Using symmetry arguments, we propose a simple derivation of a fundamental solution of the operator $\partial_t \Delta_H$ in which $\Delta_H$ is Kohn-Laplace operator on the Heisenberg group $H^{2n+1}$. Our derivation extends that of Craddock and Lennox [J. Differential Equations 232(2007), 652-674]. Indeed, these authors solved the same problem by employing a symmetry approach in the case $n=1$ . We demonstrate that the case $n=1$ is quite peculiar from a symmetry standpoint and the extension of symmetry arguments to the case $n>1$ requires some intermediate results.