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Geometric and Potential Theoretic Results on Lie Groups

Published online by Cambridge University Press:  20 November 2018

N. Th. Varopoulos
N. Th. Varopoulos*
Affiliation:
Institut Universitaire de France, Université Paris VI, Département de Mathématiques, 4, place Jussieu, 75005 Paris, France
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Abstract

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The main new results in this paper are contained in the geometric Theorems 1 and 2 of Section 0.1 below and they are related to previous results of M. Gromov and of myself (cf. [11], [29]). These results are used to prove some general potential theoretic estimates on Lie groups (cf. Section 0.3) that are related to my previous work in the area (cf. [28], [34]) and to some deep recent work of G. Alexopoulos (cf. [3], [4]).

Keywords

22E3043A8060J6060J65
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 2 , 01 April 2000 , pp. 412 - 437
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Alexopoulos, G., Fonctions harmoniques bornées sur les groupes résolubles. C. R. Acad. Sci. Paris Sér. I Math. 305(1987), 777–779.Google Scholar
[2] Alexopoulos, G., An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Canad. J. Math. (4) 44(1992), 691–727.Google Scholar
[3] Alexopoulos, G., Sous-laplaciens et densités centrées sur les groupes de Lie à croissance polynomiale du volume. C. R. Acad. Sci. Paris Sér. I Math. 326(1998), 539–542.Google Scholar
[4] Alexopoulos, G., Sublaplacians on groups of polynomial growth.Mem. Amer. Math. Soc., to appear.Google Scholar
[5] Anker, J.-Ph., Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. (2) 65(1992), 257–297.Google Scholar
[6] Chevaley, C., Théorie de groupes de Lie. Tomes II, III. Hermann, 1955.Google Scholar
[7] Davies, E. B., Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. Math. 109(1986), 319–333.Google Scholar
[8] Folland, G. and Stein, E., Hardy spaces on homogeneous groups. Princeton University Press, 1982.Google Scholar
[9] Gangolli, R., Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces. Acta Math. 121(1968), 151–192.Google Scholar
[10] Gangolli, R. and Varadarajan, V. S., Harmonic analysis of spherical funtions on real reductive groups. Ergeb. Math. Grenzgeb. 101, Springer-Verlag, 1988.Google Scholar
[11] Gromov, M., Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (eds. Niblo, G. A. and Roller, M. A.), LondonMath. Soc. Lecture Note Ser. 181, Cambridge University Press, 1993.Google Scholar
[12] Hall, M., Theory of groups. Chelsea, New York, 1976.Google Scholar
[13] Helgason, S., Groups and geometric analysis. Pure Appl. Math. 113, Academic Press, 1984.Google Scholar
[14] Hochschild, G., The structure of Lie Groups. Holden-Day, 1965.Google Scholar
[15] Hörmander, L., Hypoelliptic second order equations. Acta Math. 119(1967), 147–171.Google Scholar
[16] Jacobson, N., Lie algebras. Interscience, 1962.Google Scholar
[17] Kaimanovich, V. A. and Vershik, A. M., Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11(1983), 457–490.Google Scholar
[18] Lepage, E. and Peigné, M., A local limit theorem on the semi-direct product of R + and Rn. Ann. Inst. H. Poincaré Probab. Statist. (2) 33(1997), 223–252.Google Scholar
[19] Magnus, W., Karrass, A. and Solitar, D., Combinational group theory. Interscience, 1966.Google Scholar
[20] Montgomery, D. and Zippin, L., Topological transformation groups. Interscience Tracts in Pure and Applied Mathematics 1, 1955.Google Scholar
[21] Mustapha, S., Gaussian estimates for heat kernels on Lie groups. Math. Proc. Cam. Phil. Soc., to appear.Google Scholar
[22] Pontrjagin, L., Topological groups. Princeton University Press, 1939.Google Scholar
[23] Reiter, H., Classical harmonic analysis and locally compact groups. Oxford Math.Monographs, 1968.Google Scholar
[24] Varadarajan, V. S., Lie groups, Lie algebras and their representations. Prentice Hall, 1984.Google Scholar
[25] Varopoulos, N. Th., Information theory and harmonic functions. Bull. Sci. Math. 110(1986), 347–389.Google Scholar
[26] Varopoulos, N. Th., Convolution powers on locally compact groups. Bull. Sci. Math 111(1987), 333–342.Google Scholar
[27] Varopoulos, N. Th., Analysis on Lie groups. Rev. Mat. Iberoamericana (3) 12(1996), 791–917.Google Scholar
[28] Varopoulos, N. Th., The local theorem for symmetric diffusion on Lie groups, an overview. CMS Conf. Proc. 21(1997), 143–152.Google Scholar
[29] Varopoulos, N. Th., Distance distortion on Lie groups. Institute Mittag-Leffler Report 31, 1995/96, and Random walks discrete potential theory Proceedings Costona (1997), Symposia Mathematica 39(1999) (eds. Picardello, M. A and Woess, W.), Cambridge University Press.Google Scholar
[30] Varopoulos, N. Th., A geometric classification of Lie groups. Rev. Mat. Iberoamericana, to appear 2000.Google Scholar
[31] Varopoulos, N. Th., Analysis on Lie groups, II. To appear.Google Scholar
[32] Varopoulos, N. Th., Saloff-Coste, L. and Coulhon, T., Analysis and geometry on groups. Cambridge Tracts in Mathematics 100, Cambridge University Press, 1992.Google Scholar
[33] Varopoulos, N. Th., Wiener-Hopf theory and nonunimodular groups. J. Funct. Anal. (2) 120(1994), 467–483.Google Scholar
[34] Varopoulos, N. Th., Diffusion on Lie groups I, II, III. Canad. J. Math. 46(1994), 438–448. 1073–1992; 48(1996), 641–672.Google Scholar
[35] Varopoulos, N. Th. and S. Mustapha, forthcoming book. Cambridge University Press.Google Scholar