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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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