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Heat Kernels of Lorentz Cones

Published online by Cambridge University Press:  20 November 2018

Hongming Ding
Hongming Ding*
Affiliation:
Department of Mathematics and Computer Science Saint Louis University 221 North Grand Blvd. St. Louis, MO 63103 USA
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Abstract

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We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time $t$ and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

Keywords

35K0543A8535K1580A20Lorentz conesymmetric coneJordan algebraheat kernelheat equationLaplace-Beltrami operatoreigenvalues
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 2 , 01 June 1999 , pp. 169 - 173
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Anker, J.-Ph., Le noyau de la chaleur sur les espaces symétriques U(p, q)/ U(p) × U(q). Lecture Notes inMath. 1359, Springer-Verlag, New York, 1988, 60–82.Google Scholar
[2] Davies, E. B., Heat kernels and spectral theory. Cambridge University Press, 1989.Google Scholar
[3] Ding, H., Gross, K. and Richards, D., Ramanujan's master theorem for symmetric cones. Pacific J. Math. (2) 175 (1996), 447–490.Google Scholar
[4] Faraut, J. and Koranyi, A., Analysis on symmetric cones. Oxford Sci. Publ., 1994.Google Scholar
[5] Helgason, S., Groups and geometric analysis. Academic Press, 1984.Google Scholar
[6] Hong, Y., The heat kernels of some symmetric spaces. In: International Symposium in Memory of Hua Loo Keng, Vol. II (Beijing, 1988), Springer-Verlag, Berlin, 1991, 143–155.Google Scholar
[7] Jordan, P., von Neumann, J. and Wigner, E. P., On algebraic generalizations of the quantum mechanical formalism. Ann.Math. 36 (1934), 29–64.Google Scholar
[8] Lu, Q., The heat kernels of symmetric spaces. In: Differential Geometry: Geometry in Mathematical Physics and Related topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math. 54 (1993), Part 2, Amer. Math. Soc., Providence, RI, 401–409.Google Scholar
[9] Sawyer, P., The heat equation on the spaces of positive definitematrices. Canad. J.Math. (3) 44 (1992), 624–651.Google Scholar
[10] Sawyer, P., On an upper bound for the heat kernel on SU8(2n)= Sp(n). Canad. Bull. Math. (3) 37 (1994), 408–418.Google Scholar
[11] Sawyer, P., Estimates for the heat kernel on SL(n, R)= SO(n). Canad. J. Math. (2) 49 (1997), 359–372.Google Scholar
[12] Veselov, A. P. and Chalykh, O. A., Explicit formulas for spherical functions on symmetric spaces of type AII. Functional Anal. Appl. (1) 26 (1992), 59–60.Google Scholar
[13] Wang, G. and Hong, Y., The heat kernels on the symmetric spaces Sp(n) and GL(n, C)= U(n). J. South China Univ. of Tech. (4) 20 (1992), 324–341.Google Scholar