Abstract
We first generalize a decomposition of functions on Carnot groups as linear combinations of the Dirac delta and some of its derivatives, where the weights are the moments of the function. We then use the decomposition to describe the large time behavior of solutions of the hypoelliptic heat equation on Carnot groups. The solution is decomposed as a weighted sum of the hypoelliptic fundamental kernel and its derivatives the coefficients being the moments of the initial datum.
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Agrachev A., Boscain U., Gauthier J.-P., Rossi F.: The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256, 2621–2655 (2009)
Alexopoulos G., Lohoué N.: On the large time behavior of heat kernels on Lie groups. Duke Math. J. 120(2), 311–351 (2003)
Barut, A.O., Raczka, R.: Theory of Group Representations and Applications. World Scientific, Singapore (1986)
Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer, Berlin (2007)
Bonfiglioli A.: Taylor formula for homogenous groups and applications. Math. Z. 262(2), 255–279 (2009)
Cygan, J.: Heat kernels for class 2 nilpotent groups. Stud. Math. 64(3):227–238 (1979)
Duoandikoetxea J., Zuazua E.: Moments, masses de Dirac et développements de fonctions. C.R. Acad. Sci. Paris 315, 693–698 (1992)
Eldredge N.: Precise estimates for the subelliptic heat kernel on H-type groups. J. Math. Pures. Appl. 92, 52–85 (2009)
Folland G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(2), 161–207 (1975)
Gaveau B.: Principe de moindre action, propagation de la chaleur etéestimes sous elliptiques sur certains groupes nilpotents. Acta Math. 139(1–2), 95–153 (1977)
Hewitt E., Ross K.A.: Abstract Harmonic Analysis I. Springer, New York/Heidelberg (1963)
Hörmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
Hulanicki A.: The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group. Stud. Math. 56(2), 165–173 (1976)
Ostellari P.: Global behavior of the heat kernel associated with certain sub-Laplacians on semisimple Lie groups. J. Funct. Anal. 199, 521–534 (2003)
Varopoulos, N., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge University Press, Cambridge (1992)
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The author is grateful to D. Barilari, C. Mora-Corral and E. Zuazua for useful discussions. He also acknowledges the useful suggestions by the anonymous reviewer.
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Rossi, F. Large time behavior for the heat equation on Carnot groups. Nonlinear Differ. Equ. Appl. 20, 1393–1407 (2013). https://doi.org/10.1007/s00030-012-0215-9
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DOI: https://doi.org/10.1007/s00030-012-0215-9