Pointwise estimates for relative fundamental solutions of heat equations in ${\mathbb{R} \times \mathbb{C}}$

Abstract

Let ${p : \mathbb{C} \to \mathbb{R}}$ be a subharmonic, nonharmonic polynomial and ${\tau \in \mathbb{R}}$ a parameter. Define $\bar{Z}_{\tau p} = \frac{\partial}{\partial\bar{z}} + \tau\frac{\partial p}{\partial \bar{z}} = e^{-\tau p}\frac{\partial}{\partial\bar{z}} e^{\tau p}$ , a closed, densely defined operator on ${L^2(\mathbb{C})}$ . If ${\square}_{\tau p}=\bar{Z}_{\tau p}\bar{Z}_{\tau p}^{*}$ and ${\square}_{\tau p}=\bar{Z}_{\tau p}^{*}\bar{Z}_{\tau p}$ , we solve the heat equations ${\partial_s u + \tilde{\square}_{\tau p} u=0}$ , u(0,z) = f(z) and ${\partial_s \tilde{u} + \tilde{\square}_{\tau p} \tilde{u}=0}$ , ${\tilde{u}(0,z) = \tilde{f}(z)}$ . 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 ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.