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Analysis on nilpotent groups. (English) Zbl 0595.22008
G denotes a simply-connected real nilpotent Lie group, L the Lie algebra of G, $$L_ 1=L$$, $$L_{k+1}=[L_ k,L]$$, $$d=\sum k \dim (L_ k/L_{k+1})=$$ the Dirichlet dimension of G, $$\delta =\sum (m+1) \dim (K_{m+1}/K_ m)$$, where $$K_ m$$ (m$$\geq 0)$$ denotes the subspace of L generated by all commutators of length $$\leq m$$. Take $$X_ 1,...,X_ k\in L$$ and assume that $$X_ j mod[L,L]$$ $$(j=1,...,k)$$ span L mod[L,L]. The main results are easily stated as follows:
(1) If $$d\geq 3$$, there exists a constant $$C>0$$ such that $\| f\|_{2d/(d-2)}\leq C\sum^{k}_{j=1}\| X_ jf\|_ 2,\quad \forall f\in C_ 0^{\infty}(G).$ (2) In general (when $$d\geq 1)$$, there exists $$C>0$$ such that $\| f\|_{d/(d-1)}\leq C\sum^{k}_{j=1}\| X_ jf\|_ 1.$ The various $$L^ p$$ norms are with respect to Haar measure on G. The proofs are another matter. The arguments are put together from several earlier papers of the author [cf. the author’s notes in C. R. Acad. Sci., Paris, Sér. I since 1984: 299, 651-654 (1984; Zbl 0566.31006); 301, 143-144, 559-560, 865-868 (1985; Zbl 0582.43002, Zbl 0582.43003, Zbl 0586.47039); also J. Funct. Anal. 63, 215-239, 240-260 (1985; Zbl 0573.60059)] and others and also bring in new ideas. The usual ingredients of Lie theory are combined in an ingenious step-wise way with delicate estimates involving the semigroup exp(-t$$\sum X^ 2_ j)$$ of heat conduction. The optimal nature of the norm indices is discussed and the Dirichlet dimension d is related to the growth of volume on G, as measured through the $$X_ j's$$ in a natural way.
Reviewer: E.J.Akutowicz

##### MSC:
 22E25 Nilpotent and solvable Lie groups 22E30 Analysis on real and complex Lie groups 58J35 Heat and other parabolic equation methods for PDEs on manifolds 43A80 Analysis on other specific Lie groups
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