Heat Equations in mathbb{R}timesmathbb{C}

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 $\Box_{\tau p} = \bar{Z}_{\tau p}\bar{Z}_{\tau p}^*$ and $\tau>0$, we solve the heat equation $ (\partial_s + \Box_{\tau p}) u =0$, $u(0,z) = f(z)$, on $(0,\infty)\times\mathbb{C}$. The solution comes via the heat semigroup $e^{-s\Box_{\tau p}}$, 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.