Heat Equations in $\mathbb{R}\times \mathbb{C}$

Abstract

Let $p:\mathbb{C}\to \mathbb{R}$ be a subharmonic, nonharmonic polynomial and $\tau$ a real parameter. Define ${\bar{Z}}_{\tau p}={\partial }_{\bar{z}}+\tau p_{\bar{z}}$, a closed, densely-defined operator on $L^2(\mathbb{C})$. If ${\square }_{\tau p}={\bar{Z}}_{\tau p}{\bar{Z}}_{\tau p}^{\ast }$ and $\tau >0$, we solve the heat equation $({\partial }_s+{\square }_{\tau p})u=0$, $u(0,z)=f(z)$, on $(0,\infty )\times \mathbb{C}$. The solution comes via the heat semigroup $e^{-s{\square }_{\tau p}}$, and we show that $u(s,z)$ is given as integration of the intial condition against a distributional kernel $H_{\tau p}(s,z,w)$. We prove that $H_{\tau p}$ is $C^{\infty }$ off the diagonal $\{(s,z,w):s=0\text{and }z=w\}$ and that $H_{\tau p}$ and its derivatives have exponential decay.

arXiv commentv3: 29 pages. The main results have been clarified and corrected. An appendix has been added, and many typos have been corrected