Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$.

References [Enhancements On Off] (What's this?)

  • [Ber92] B. Berndtsson.
    Weighted estimates for $ \bar\partial$ in domains in $ \mathbb{C}$.
    Duke Math. J., 66(2):239-255, 1992. MR 1162190 (93f:32018)
  • [CT00] Der-Chen Chang and Jingzhi Tie.
    Estimates for powers of the sub-Laplacian on the non-isotropic Heisenberg group.
    J. Geom. Anal., 10:653-678, 2000. MR 1817779 (2001m:58047)
  • [Chr91] M. Christ.
    On the $ \bar\partial$ equation in weighted $ {L}^2$ norms in $ {{\mathbb{C}}}^1$.
    J. Geom. Anal., 1(3):193-230, 1991. MR 1120680 (92j:32066)
  • [FS91] J.E. Fornæss and N. Sibony.
    On $ {L}^p$ estimates for $ \overline\partial$.
    In Several Complex Variables and Complex Geometry, Part 3 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., 52, Part 3, pages 129-163, American Mathematical Society, Providence, R.I., 1991. MR 1128589 (92h:32030)
  • [Gav77] B. Gaveau.
    Principe de moindre action, propogation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents.
    Acta Math., 139:95-153, 1977. MR 0461589 (57:1574)
  • [Has94] F. Haslinger.
    Szegö kernels for certain unbounded domains in $ {{\mathbb{C}}}\sp2$. Travaux de la Conférence Internationale d'Analyse Complexe et du 7e Séminaire Roumano-Finlandais (1993).
    Rev. Roumaine Math. Pures Appl., 39:939-950, 1994. MR 1406110 (97f:32026)
  • [Has95] F. Haslinger.
    Singularities of the Szegö kernel for certain weakly pseudoconvex domains in $ {C}\sp 2$.
    J. Funct. Anal., 129:406-427, 1995. MR 1327185 (96g:32045)
  • [Has98] F. Haslinger.
    Bergman and Hardy spaces on model domains.
    Illinois J. Math., 42:458-469, 1998. MR 1631252 (99f:32004)
  • [Hul76] A. Hulanicki.
    The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group.
    Studia Math., 56:165-173, 1976. MR 0418257 (54:6298)
  • [NS01] A. Nagel and E.M. Stein.
    The $ {\Box}_b$-heat equation on pseudoconvex manifolds of finite type in $ {{\mathbb{C}}}^2$.
    Math. Z., 238:37-88, 2001. MR 1860735 (2002h:32031)
  • [Rai] Andrew Raich.
    Heat equations and the weighted $ \bar{\partial}$-problem with decoupled weights,
  • [Rai06a] Andrew Raich.
    Heat equations in $ {{\mathbb{R}}}\times{{\mathbb{C}}}$.
    J. Funct. Anal., 240(1):1-35, 2006. MR 2259891 (2007h:32059)
  • [Rai06b] Andrew Raich.
    One-parameter families of operators in $ {\mathbb{\mathbb{C}}}$.
    J. Geom. Anal., 16(2):353-374, 2006. MR 2223806 (2007c:32049)
  • [Rai07] Andrew Raich.
    Pointwise estimates for relative fundamental solutions for heat equations in $ {{\mathbb{R}}}\times{{\mathbb{C}}}$.
    Math. Z., 256:193-220, 2007. MR 2282265 (2008b:32030)
  • [Ste93] Elias M. Stein.
    Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.
    Princeton Mathematical Series, 43. Princeton University Press, Princeton, New Jersey, 1993. MR 1232192 (95c:42002)
  • [Tha93] Sundaram Thangavelu.
    Lectures on Hermite and Laguerre Expansions, Mathematical Notes, 42.
    Princeton University Press, Princeton, New Jersey, 1993. MR 1215939 (94i:42001)
  • [Tie06] Jingzhi Tie.
    The twisted Laplacian on $ {{\mathbb{C}}}^n$ and the sub-Laplacian on $ {H}_n$.
    Comm. Partial Differential Equations, 31:1047-1069, 2006. MR 2254603 (2007e:32049)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32W30, 33C45, 42C10

Retrieve articles in all journals with MSC (2000): 32W30, 33C45, 42C10

Additional Information

Albert Boggess
Affiliation: Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, Texas 77845-3368

Andrew Raich
Affiliation: Department of Mathematical Sciences, 1 University of Arkansas, SCEN 327, Fayetteville, Arkansas 72701

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.

American Mathematical Society