Heat equations in R×C

Let p :C→R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define Z¯τp = ∂ ∂ ¯z + τ ∂p ∂ ¯z , a closed, densely-defined operator on L2(C). If τp = Z¯τpZ¯∗τp andτ >0, we solve the heat equation ∂u ∂s + τp u = 0, u(0, z) = f (z), on (0,∞)×C. The solution comes via the heat semigroup e−s τp, andwe show that u(s, z) = e−s τp [f ](z) = C Hτp(s, z,w)f (w) dw. We prove that Hτp is C∞ off the diagonal {(s, z,w): s = 0 and z = w} and that Hτp and its derivatives have exponential decay. In particular, we give new estimates for the long time behavior of the heat equation. © 2006 Elsevier Inc. All rights reserved.