The heat kernel on H-type groups

Yang, Qiaohua; Zhu, Fuliu

doi:10.1090/S0002-9939-07-09257-X

The heat kernel on H-type groups
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by Qiaohua Yang and Fuliu Zhu PDF

Proc. Amer. Math. Soc. 136 (2008), 1457-1464 Request permission

Abstract:

In this paper we present an explicit calculation of the heat kernel for the sub-Laplacian on an H-type group $G$ by using irreducible unitary representations of $G$ and the heat kernel for the associated Hermite operator.

References

F. Astengo, M. Cowling, B. Di Blasio, and M. Sundari, Hardy’s uncertainty principle on certain Lie groups, J. London Math. Soc. (2) 62 (2000), no. 2, 461–472. MR 1783638, DOI 10.1112/S0024610700001186
Richard Beals, Bernard Gaveau, and Peter C. Greiner, Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians. I, Bull. Sci. Math. 121 (1997), no. 1, 1–36. MR 1431098
A. Bonfiglioli and F. Uguzzoni, Nonlinear Liouville theorems for some critical problems on H-type groups, J. Funct. Anal. 207 (2004), no. 1, 161–215. MR 2027639, DOI 10.1016/S0022-1236(03)00138-1
Der Chen Chang and Jing Zhi Tie, A note on Hermite and subelliptic operators, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 4, 803–818. MR 2156956, DOI 10.1007/s10114-004-0336-0
Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. MR 1070979
Jacek Cygan, Heat kernels for class $2$ nilpotent groups, Studia Math. 64 (1979), no. 3, 227–238. MR 544727, DOI 10.4064/sm-64-3-227-238
Bernard Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), no. 1-2, 95–153. MR 461589, DOI 10.1007/BF02392235
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
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 (1976), no. 2, 165–173. MR 418257, DOI 10.4064/sm-56-2-165-173
Aroldo Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), no. 1, 147–153. MR 554324, DOI 10.1090/S0002-9947-1980-0554324-X
Wolfgang Staubach, Wiener path integrals and the fundamental solution for the Heisenberg Laplacian, J. Anal. Math. 91 (2003), 389–400. MR 2037416, DOI 10.1007/BF02788796
Sundaram Thangavelu, An introduction to the uncertainty principle, Progress in Mathematics, vol. 217, Birkhäuser Boston, Inc., Boston, MA, 2004. Hardy’s theorem on Lie groups; With a foreword by Gerald B. Folland. MR 2008480, DOI 10.1007/978-0-8176-8164-7
Fuliu Zhu, The heat kernel and the Riesz transforms on the quaternionic Heisenberg groups, Pacific J. Math. 209 (2003), no. 1, 175–199. MR 1973940, DOI 10.2140/pjm.2003.209.175