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

Abstract

Let $p:C\to R$ be a subharmonic, nonharmonic polynomial and $\tau\in R$ a parameter. Define $\bar Z_{\tau p} = \partial_{\bar z} + \tau p_{\bar z} = e^{-\tau p} p_{\bar z} e^{\tau p}$, a closed, densely defined operator on $L^2(C)$. 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 $\partial_s u + \Box_{\tau p} u=0$, $u(0,z)=f(z)$ 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 $C^\infty$ off of the diagonal $\{(s,z,w) : s=0 \text{and} z=w\}$ and find pointwise bounds for the kernels and their derivatives.