A Simplified Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group

Abstract

Let $L=-1/4({\sum }_{j=1}^n(X_j^2+Y_j^2)+i\gamma T)$ where $\gamma$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n\times R^n\times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation ${\partial }_s\rho =-L\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the ${\square }_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operator in $C^n$ with weight $\exp (-\tau P(z_1,...,z_n))$ where $P(z_1,...,z_n)=1/2(x_1^2+>...x_n^2)$, $z_j=x_j+i{y}_j$, and $\tau \in R$.

8 pages