A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group
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- by Albert Boggess and Andrew Raich PDF
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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 $P(z_1,\dots ,z_n) = \frac 12(|\operatorname {Im}z_1|^2 + \cdots +|\operatorname {Im} z_n|^2)$ and $\tau \in \mathbb {R}$.References
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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
- MR Author ID: 634382
- ORCID: 0000-0002-3331-9697
- Email: araich@uark.edu
- Received by editor(s): November 27, 2007
- Published electronically: October 10, 2008
- Communicated by: Mei-Chi Shaw
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 937-944
- MSC (2000): Primary 32W30, 33C45, 42C10
- DOI: https://doi.org/10.1090/S0002-9939-08-09725-6
- MathSciNet review: 2457433