Heat Equations in $\mathbb{R}\times\mathbb{C}$

Details Heat Equations in $\mathbb{R}\times\mathbb{C}$ Heat Equations in $\mathbb{R}\times\mathbb{C}$

2005-08-29

arxiv.org/abs/math/0508571v3

J. Funct. Anal., 240:1-35, 2006

Mathematics

Complex Variables

v3: 29 pages. The main results have been clarified and corrected. An appendix has been added, and many typos have been correct

Scientific paper

Let $p:\mathbb{C}\to\mathbb{R}$ be a subharmonic, nonharmonic polynomial and $\tau$ a real parameter. Define $\bar{Z}_{\tau p} = \partial_{\bar z} + \tau p_{\bar z}$, a closed, densely-defined operator on $L^2(\mathbb{C})$. If $\Box_{\tau p} = \bar{Z}_{\tau p}\bar{Z}_{\tau p}^*$ and $\tau>0$, we solve the heat equation $ (\partial_s + \Box_{\tau p}) u =0$, $u(0,z) = f(z)$, on $(0,\infty)\times\mathbb{C}$. The solution comes via the heat semigroup $e^{-s\Box_{\tau p}}$, and we show that $u(s,z)$ is given as integration of the intial condition against a distributional kernel $H_{\tau p}(s,z,w)$. We prove that $H_{\tau p}$ is $C^\infty$ off the diagonal $\{(s,z,w):s=0 \text{and }z=w\}$ and that $H_{\tau p}$ and its derivatives have exponential decay.

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