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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group


Authors: Albert Boggess and Andrew Raich
Journal: Proc. Amer. Math. Soc. 137 (2009), 937-944
MSC (2000): Primary 32W30, 33C45, 42C10
Published electronically: October 10, 2008
MathSciNet review: 2457433
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Abstract: Let $ \mathcal{L}_\gamma = -\frac{1}{4} \left( \sum_{j=1}^n(X_j^2+Y_j^2)+i\gamma T \right)$ where $ \gamma \in \mathbb{C}$, and $ X_j$, $ Y_j$ and $ T$ are the left-invariant vector fields of the Heisenberg group structure for $ \mathbb{R}^n \times \mathbb{R}^n\times \mathbb{R}$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the heat equation $ \partial_s\rho = -\mathcal{L}_\gamma\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $ \Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted $ \bar{\partial}$-operator in $ \mathbb{C}^n$ with weight $ \exp(-\tau P(z_1,\dots,z_n))$, where $ P(z_1,\dots,z_n) = \frac 12(\vert\operatorname{Im}z_1\vert^2 + \cdots +\vert\operatorname{Im} z_n\vert^2)$ and $ \tau\in\mathbb{R}$.


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Additional Information

Albert Boggess
Affiliation: Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, Texas 77845-3368
Email: boggess@math.tamu.edu

Andrew Raich
Affiliation: Department of Mathematical Sciences, 1 University of Arkansas, SCEN 327, Fayetteville, Arkansas 72701
Email: araich@uark.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09725-6
PII: S 0002-9939(08)09725-6
Keywords: Heisenberg group, heat equation, fundamental solution, heat kernel, Kohn Laplacian
Received by editor(s): November 27, 2007
Published electronically: October 10, 2008
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.