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

Abstract:

Let $\mathcal {L}_\gamma = -\frac {1}{4} \left ( \sum _{j=1}^n(X_j^2+Y_j^2)+i\gamma T \right )$ where $\gamma \in \mathbb {C}$, and $X_j$, $Y_j$ and $T$ are the left-invariant vector fields of the Heisenberg group structure for $\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the heat equation $\partial _s\rho = -\mathcal {L}_\gamma \rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\Box _b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted $\bar {\partial }$-operator in $\mathbb {C}^n$ with weight $\exp (-\tau P(z_1,\dots ,z_n))$, where $P(z_1,\dots ,z_n) = \frac 12(|\operatorname {Im}z_1|^2 + \cdots +|\operatorname {Im} z_n|^2)$ and $\tau \in \mathbb {R}$.