Pointwise Estimates for Relative Fundamental Solutions of Heat Equations in $\mathbb{R}\times \mathbb{C}$

Let $p:C\to R$ be a subharmonic, nonharmonic polynomial and $\tau \in R$ a parameter. Define $\bar Z_{\tau p} = \partial_{\bar z} + \tau p_{\bar z} = e^{-\tau p} p_{\bar z} e^{\tau p}$,aclosed,denselydefinedoperatoron$L^2©$\mathrm{.}If$\Box_{\tau p} = \bar Z_{\tau p}\bar Z^{\tau p}$and$\tilde\Box{\tau p} = \bar Z^{\tau p}\bar Z{\tau p}$, we solve the heat equations ${\mathrm{\partial }}_{s}u+{\mathrm{\square }}_{\tau p}u=0$, $u(0,z)=f(z)$ and $\partial_s \tilde u + \tilde\Box_{\tau p} \tilde u=0$,$\tilde u(0,z) = \tilde f(z)$. We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are $C^{\mathrm{\infty }}$ off of the diagonal $\{(s,z,w):s=0\text{and}z=w\}$ and find pointwise bounds for the kernels and their derivatives.