Abstract
We give a construction of the heat kernel and Green function of a hypoelliptic operator on the one-dimensional Heisenberg group \(\mathbb{H}\), the sub-Laplacian \(\mathcal{L}\). The explicit formulas are developed using Fourier–Wigner transforms, pseudo-differential operators of the Weyl type, i.e., Weyl transforms, and spectral analysis. These formulas are obtained by first finding the formulas for the heat kernels and Green functions of a family of twisted Laplacians \({L}_{\tau}\) for all non-zero real numbers \({\tau}\). In the case when \({\tau=1, {L}_{1}}\) is just the usual twisted Laplacian.
Mathematics Subject Classification (2010). 47F05, 47G30.
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References
R. Beals, Geometric and PDE on the Heisenberg group: a case study, in The Geometric Study of Differential Equations, Contemp. Math. 285, American Mathematical Society, 2001, 21–27.
A. Dasgupta and M.W. Wong, Essential self-adjointness and global hypoellipticity of the twisted Laplacian, Rend. Sem. Mat. Univ. Pol. Torino 66 (2008), 75–85.
G.B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc 79, (1973), 373–376.
M. de Gosson, Spectral properties of a class of generalized Landau operators, Comm. Partial Differential Equations 33 (2008), 2096–2104.
B. Gaveau, Principe de moindre action, propagation de la chaleur et estim´ees sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95–153.
M. de Gosson, Phase-space Weyl calculus and global hypoellipticity of a class of degenerate elliptic partial differential operators, in New Developments in Pseudo- Differential Operators, Operator Theory: Advances and Applications 189, Birkhäuser, 2009, 1–14.
M. de Gosson, Symplectic Methods in Harmonic Analysis and Mathematical Physics, Birkhäuser, 2011.
T. Gramchev, S. Pilipovi´c and L. Rodino, Classes of degenerate elliptic operators in Gelfand–Shilov spaces, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkhäuser, 2009, 15–31.
T. Gramchev, S. Pilipovi´c and Luigi Rodino, Global Regularity and Stability in SSpaces for Classes of Degenerate Shubin Operators, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications 205, Birkhäuser, 2010, 81–90.
A. Grossmann, G. Loupias and E. M. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1968), 343–368.
L. Hörmander, Hypoelliptic second-order differential equations, Acta Math. 119 (1967), 147–171.
L. Hörmander, The Analysis of Linear Partial Differential Operators II, Springer- Verlag, 1983.
A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic hypoellipticity of certain sub-elliptic operators operators on the Heisenberg group, Studia Math. 56 (1976), 165–173.
H. Koch and F. Ricci, Spectral projections for the twisted Laplacian, Studia Math. 180 (2007), 103–110.
A. Klinger, New derivation of the Heisenberg kernel, Comm. Partial Differential Equations 22 (1997), 2051–2060.
E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.
S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Birkhäuser, 1998.
M.W. Wong, Weyl Transforms, Springer-Verlag, 1998.
M.W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global Anal. Geom. 28 (2005), 271–283.
M.W.Wong,Weyl transforms and convolution operators on the Heisenberg group, in Pseudo-Differential Operators and Related Topics, Operator Theory: Advances and Applications 164, Birkhäuser, 2006, 115–120.
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Duan, X. (2013). The Heat Kernel and Green Function of the Sub-Laplacian on the Heisenberg Group. In: Molahajloo, S., Pilipović, S., Toft, J., Wong, M. (eds) Pseudo-Differential Operators, Generalized Functions and Asymptotics. Operator Theory: Advances and Applications, vol 231. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0585-8_3
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DOI: https://doi.org/10.1007/978-3-0348-0585-8_3
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