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 $τ\in R$ a parameter. Define $\bar Z_{τp} = \partial_{\bar z} + τp_{\bar z} = e^{-τp} p_{\bar z} e^{τp}$, a closed, densely defined operator on $L^2(C)$. If $\Box_{τp} = \bar Z_{τp}\bar Z^*_{τp}$ and $\tilde\Box_{τp} = \bar Z^*_{τp}\bar Z_{τp}$, we solve the heat equations $\partial_s u + \Box_{τp} u=0$, $u(0,z)=f(z)$ and $\partial_s \tilde u + \tilde\Box_{τp} \tilde u=0$, $\tilde u(0,z) = \tilde f(z)$. 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.