It is the cache of It is a snapshot of the page. The current page could have changed in the meantime.
Tip: To quickly find your search term on this page, press Ctrl+F or ⌘-F (Mac) and use the find bar.

Cubo (Temuco) - The Semigroup and the Inverse of the Laplacian on the Heisenberg Group

SciELO - Scientific Electronic Library Online

vol.12 número3L -Random and Fuzzy Normed Spaces and Classical TheorySelf-Dual and Anti-Self-Dual Solutions of Discrete Yang-Mills Equations on a Double Complex índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.3 Temuco  2010 

CUBO A Mathematical Journal Vol.12, N°03, (83–97). October 2010


The Semigroup and the Inverse of the Laplacian on the Heisenberg Group1



Department of Mathematics, Indian Institute of Science, Bangalore–560012, India email:

Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada email:


By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operators L t ,t ∈ R \ {0}, and using parametrized Fourier-Wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by L t, and the inverse of L t . Using these formulas and estimates, we obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Laplacian.

Key words and phrases: Heisenberg group, Laplacian, parametrized partial differential operators, Hermite functions, Fourier-Wigner transforms, heat equation, one parameter semigroup, inverse of Laplacian, Sobolev spaces.


Mediante descomposición del Laplaceano sobre el grupo de Heisenberg en una familia de operadores diferenciales parciales parametrizados L t, t ∈ R \{0}, y usando transformada de Fourier-Wigner parametrizada, damos fórmulas y estimativas para la continuidad fuerte del semigrupo generado por L t, y la inversa de L t. Usando esas fórmulas y estimativas obtenemos estimativas de Sobolev para el semigrupo a un parámetro y la inversa del Laplaceano.

Math. Subj. Class.: 47F05, 47G30, 35J70.


1This research has been supported by the Natural Sciences and Engineering Research Council of Canada.


[1] DASGUPTA, A AND WONG, M.W.,Weyl transforms and the inverse of the sub-Laplacian on the Heisenberg group, in Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Institute Communications, 52, American Mathematical Society, 2007, 27–36.

[2] DASGUPTA, A. AND WONG, M.W., Weyl transforms and the heat equation for the sub- Laplacian on the Heisenberg group, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications, 189, Birkhäuser, 2009, 33–42.

[3] DAVIES, E.B., One-Parameter Semigroups, Academic Press, 1980.        [ Links ]

[4] DAVIES, E.B., Linear Operators and their Spectra, Cambridge University Press, 2007.        [ Links ]

[5] FRIEDMAN, A., Partial Differential Equations, Holt, Reinhart and Winston, 1969.        [ Links ]

[6] FURUTANI, K., The heat kernel and the spectrum of a class of manifolds, Comm. Partial Differential Equations, 21 (1996), 423–438.

[7] GAVEAU, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139 (1977), 95–153.

[8] HILLE, E. AND PHILLIPS, R.S., Functional Analysis and Semi-groups, Third Printing of Revised Version of 1957, American Mathematical Society, 1974.        [ Links ]

[9] IANCU, G.M. AND WONG, M.W., Global solutions of semilinear heat equations in Hilbert spaces, Abstr. Appl. Anal., 1 (1996), 263–276.

[10] KIM, J. AND WONG, M.W., Positive definite temperature functions on the Heisenberg group, Appl. Anal., 85 (2006), 987–1000.

[11] SHUBIN, M.A., Pseudodifferential Operators and Spectral Theory, Springer-Verlag, 1987.        [ Links ]

[12] STEIN, E.M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.        [ Links ]

[13] WONG, M.W., Weyl Transforms, Springer-Verlag, 1998.        [ Links ]

[14] WONG, M.W., Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global Anal. Geom., 28 (2005), 271–283.

[15] WONG, M.W., The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J., 34 (2005), 393–404.

[16] YOSIDA, K., Functional Analysis, Reprint of Sixth Edition, Springer-Verlag, 1995        [ Links ]