Heat equations in R×C

Let be a subharmonic, nonharmonic polynomial and τ∈R a parameter. Define , a closed, densely-defined operator on L2(C). If and τ>0, we solve the heat equation , u(0,z)=f(z), on (0,∞)×C. The solution comes via the heat semigroup e−s□τp, and we show that . We prove that Hτp is C∞ off the diagonal and that Hτp and its derivatives have exponential decay. In particular, we give new estimates for the long time behavior of the heat equation.