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Analysis on Lie groups. (English) Zbl 0634.22008

This paper deals mainly with inequalities of Sobolev type, estimates for the kernel of the operator of heat conduction as \(t\to 0\) and as \(t\to \infty\), and their interrelations. Let G be a connected real Lie group and \(H=\{X_ 1,...,X_ k\}\) be left-invariant vector fields on G that, together with their brackets, generate the Lie algebra of G. Define for \(f\in C_ 0^{\infty}(G):\) \(D(f)=\int_{G}| \nabla f|^ 2dx\), \(| \nabla f|^ 2=\sum | X_ jf|^ 2\), where dx is right Haar measure. D induces a semi-group \(T_ t=\exp (-t\Delta)\), \(\Delta =-\sum^{k}_{1}X_ j^ 2\), with symmetric kernel \(p_ t(x,y)\), which the author re-scales when necessary to \(r_ t(x,y)\) through the “modular function” that relates left and right Haar measure.
There is a fundamental alternative for the Lie groups in question: If \(B_ t\) denotes the ball of radius t in the Carathéodory metric defined through the vector fields H, then the function \(\gamma (t)=left\) Haar measure of \(B_ t\) satisfies either \(\gamma (t)\approx e^ t\) (t\(\to \infty)\) or there exists \(a\geq 0\) such that \(\gamma (t)\approx t^ a\) (t\(\to \infty)\) [Y. Guivarc’h, Bull. Soc. Math. Fr. 101(1973), 333-379 (1974; Zbl 0294.43003)]. An inequality of Sobolev type is, with Lebesgue norms, \((Sob_ n):\| f\|_{n/n-1}\leq Const.\| \nabla f\|_ 1\), \(\forall f\in C_ 0^{\infty}(G)\). The author relates such inequalities to estimates \(r_ t(e,e)=O(t^{-a/2})\) (t\(\to \infty)\), \(e=identity\) of G, as well as to lower bounds for all \(t\geq 0:\gamma (t)\geq Const.\cdot t^ a\), \(\exists a\geq 0\). Necessary and sufficient conditions that \((Sob_ n)\) should hold are obtained, conditions that depend upon the growth of \(\gamma\) (t)\(\to \infty.\)
The proofs of these results involve a wealth of ideas and techniques and constitute an extraordinary tour de force.
Reviewer: E.J.Akutowicz

MSC:

22E30 Analysis on real and complex Lie groups
58J65 Diffusion processes and stochastic analysis on manifolds
35K05 Heat equation
58J35 Heat and other parabolic equation methods for PDEs on manifolds

Citations:

Zbl 0294.43003
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References:

[1] Carathéodory, C., Math. Ann., 67, 355-386 (1909)
[2] Bony, J. M., Ann. Inst. Fourier (Grenoble), 277 (1969)
[3] Deny, J., Potential theory, (C.I.M.E. 1er cycle, Stresa (1969)), dal 2 al 10 Luglio
[4] Fukushima, M., Dirichlet Forms and Markov Processes (1980), North-Holland/Kodansha: North-Holland/Kodansha Amsterdam · Zbl 0422.31007
[5] McKean, H. P., Stochastic Integrals (1969), Academic Press: Academic Press Orlando, FL · Zbl 0191.46603
[6] Yosida, K., Functional Analysis (1978), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0152.32102
[7] Dynkin, E. B., Markov Processes (1965), Springer-Verlag: Springer-Verlag Berlin · Zbl 0132.37901
[8] Sanches-Calle, A., Invent. Math., 78, 143-160 (1984) · Zbl 0582.58004
[9] Rothschild, L. P.; Stein, E. M., Acta Math., 137, 247-320 (1977) · Zbl 0346.35030
[10] Hörmander, L.; Melin, A., Ark. Mat., 16, 83-88 (1978) · Zbl 0383.35013
[11] Nagel, A.; Stein, E. M.; Wainger, S., Acta Math., 55, 103-147 (1985) · Zbl 0578.32044
[12] Varopoulos, N. Th, J. Funct. Anal., 63, 215-239 (1985) · Zbl 0573.60059
[13] Varopoulos, N. Th, J. Funct. Anal., 63, 240-260 (1985) · Zbl 0608.47047
[14] Varopoulos, N. Th, C.R. Acad. Sci. Paris Sér. I Math., 300, 617-620 (1985) · Zbl 0589.47046
[15] Varopoulos, N. Th, C. R. Acad. Sci. Paris Sér. I Math., 301, 865 (1985) · Zbl 0586.47039
[16] Davies, E. B., One Parameter Semigroups (1980), Academic Press: Academic Press Orlando, FL · Zbl 0457.47030
[17] Gromov, M., Inst. Hautes Études Sci. Publ. Math., 53 (1981) · Zbl 0474.20018
[18] Guivarc’h, Y., Bull. Soc. Math. France, 101, 333-379 (1973) · Zbl 0294.43003
[19] Jenkins, J. W., J. Funct. Anal., 12, 113-127 (1973) · Zbl 0247.43001
[21] Varopoulos, N. Th, J. Funct. Anal., 66, 406-431 (1986) · Zbl 0595.22008
[22] Varopoulos, N. Th, Bull. Sci. Math. (2), 109, 113-119 (1985) · Zbl 0583.60008
[23] Benedek, A.; Panzone, R., Duke Math. J., 28, 301 (1961) · Zbl 0107.08902
[24] Hörmander, L., Acta Math., 119, 147-171 (1967) · Zbl 0156.10701
[25] Varopoulos, N. Th, (Lecture notes (1986), Université Paris VI), Notes taken by L. Saloff-Coste and T. Coulhon
[28] Davies, E. B., Explicit constants for Gaussian upper bounds on Heat kernels, Amer. J. Math., 109, 319-333 (1976) · Zbl 0659.35009
[29] Hochschild, G., The Structure of Lie Groups (1965), Holden-Day: Holden-Day San Francisco · Zbl 0131.02702
[30] Varopoulos, N. Th, C.R. Acad. Sci. Paris Sér. I. Math., 299, 651-654 (1984) · Zbl 0566.31006
[31] Federer, H., Geometric Measure Theory (1969), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0176.00801
[33] Stein, E. M., Singular Integrals and Differentiability of Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[34] Chevalley, C., Théorie des Groupes de Lie. III. Théorèmes généraux sur les algébres de Lie (1955), Hermann: Hermann Paris · Zbl 1469.17032
[35] Auslander, L.; Green, L. W., Amer. J. Math., 88, 43-60 (1966) · Zbl 0149.19903
[36] Guivarc’h, Y., Loi des grands nombres et rayon spectral d’une marche aléatoire sur ungroupe de Lie, Astérisque, 74, 47-98 (1979) · Zbl 0448.60007
[38] Hunt, R. A., Enseign. Math., 12, 249-275 (1966) · Zbl 0181.40301
[39] Varopoulos, N. Th, C.R. Acad. Sci. Paris Sér. I Math. (1987)
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