A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group

Abstract

Let L = −1/4 The purpose of this note is to present a simplified calculation of the Fourier transform of fundmental solution of theb-heat equation on the Heisenberg group. The Fourier transform of the fundamental solution has been computed by a number of authors (Gav77, Hul76, CT00, Tie06). We use the approach of (CT00, Tie06) and compute the heat kernel using Hermite functions but differ from the earlier approaches by working on a different, though biholomorphically equivalent, version of the Heisenberg group. The simplification in the computation occurs because the differential operators on this equivalent Heisenberg group take on a simpler form. Moreover, in the proof of Theorem 1.2, we reduce the n-dimensional heat equation to a 1-dimensional heat equation, and this technique would also be useful when analyzing the heat equation on the nonisotropic Heisenberg group (e.g., see (CT00)). We actually use the same version of the Heisenberg group as Hulanicki (Hul76), but he computes the fundamental solution of the heat equation associated to the sub-Laplacian and not the Kohn Laplacian acting on (0,q)-forms.