##
**Analysis on Lie groups.**
*(English)*
Zbl 0881.22009

Let \(G\) be a real connected Lie group, let \(X_j\), \(j=0,\ldots,n\), be left invariant vector fields on \(G\), which are generators of the Lie algebra of \(G\) (i.e. together with all their successive brackets they span the Lie algebra of \(G\)). Let also \(dg\) (\(d^r g\)) be a left (right) Haar measure on \(G\). The principal object of the paper under review is the subelliptic left invariant Laplacian
\[
\Delta=-\sum _{j=1^n} X_j^2 +X_0
\]
together with the associated heat diffusion semigroup \(T_t=e^{-t\Delta}\) and the corresponding heat diffusion kernel \(\varphi_t(g)\) that is defined by
\[
T_t f(x)= \int_G f(y) \varphi_t(y^{-1}x) dy, \quad t>0,\;x \in G,\;f \in C_0^\infty(G).
\]
If \(X_0=0\) then \(\Delta=\Delta_0\) is called driftless. The heat semigroup has an important characteristic: the spectral gap of \(\Delta_0\) is defined by
\[
\lambda= \inf \{ \int_G \left |\nabla f \right|^2 d^r g: \int_G f ^2 d^r g\}.
\]
Particularly, \(|T_t |_{2\rightarrow 2} = e^{-\lambda t}\). The author gives an algebraic classification of Lie algebras (resp. Lie groups) into two classes: the \(B\)-algebras (\(B\)-groups) and the \(NB\)-algebras (\(NB\)-groups). This classification is “algebraically very natural but fairly long to explain”. Roughly, it depends on whether the dynamical system \(Ad(P)\) is “hyperbolic” or not, where \(P\) is the minimal parabolic subgroup of \(G\). The main results of the paper are the following.

Theorem A. Let \(G\) be a Lie group as above and let \(\Delta_0\) be a driftless Laplacian, let \(\varphi_t \in C^\infty (G)\), \(\lambda>0\), be the corresponding heat diffusion kernel and spectral gap, respectively. Then: (A\(_1\)) If we assume that \(G\) is a \(B\)-group then there exist \(C\), \(c_1\), \(c_2>0\) such that \[ C^{-1} \exp (-\lambda t-c_2 t^{1/3}) \leq \varphi_t(e) \leq C \exp (-\lambda t-c_1 t^{1/3}), \quad t\geq 1. \] (A\(_2\)) If we assume that \(G\) is an \(NB\)-group then there exist \(C>0\), \(\nu\geq0\) such that \[ C^{-1} t^{-\nu} e^{-\lambda t} \leq \varphi_t(e) \leq C t^{-\nu} e^{-\lambda t}, \quad t \geq 1. \] Theorem B. Let \(G\) be a \(B\)-group as above and let \(\mu\in {\mathbf P}(G)\) be a Gaussian probability measure on \(G\). Then there exists \(c>0\) such that for all \(\varphi \in C^\infty_0\) we have \[ \langle \varphi, \mu^{*n} \rangle = O(\parallel \mu^n \parallel_{2\rightarrow 2} e^{-cn^{1/3}}). \] The paper is rather lengthy, besides its essential research value it also has expositional and instructional merits and contains a lot of supplementary information. It is carefully structured, the material is separated in hierarchically ordered logical units and provides reading advices. This, together with the personal language, helps to keep the attention of the reader alive.

Theorem A. Let \(G\) be a Lie group as above and let \(\Delta_0\) be a driftless Laplacian, let \(\varphi_t \in C^\infty (G)\), \(\lambda>0\), be the corresponding heat diffusion kernel and spectral gap, respectively. Then: (A\(_1\)) If we assume that \(G\) is a \(B\)-group then there exist \(C\), \(c_1\), \(c_2>0\) such that \[ C^{-1} \exp (-\lambda t-c_2 t^{1/3}) \leq \varphi_t(e) \leq C \exp (-\lambda t-c_1 t^{1/3}), \quad t\geq 1. \] (A\(_2\)) If we assume that \(G\) is an \(NB\)-group then there exist \(C>0\), \(\nu\geq0\) such that \[ C^{-1} t^{-\nu} e^{-\lambda t} \leq \varphi_t(e) \leq C t^{-\nu} e^{-\lambda t}, \quad t \geq 1. \] Theorem B. Let \(G\) be a \(B\)-group as above and let \(\mu\in {\mathbf P}(G)\) be a Gaussian probability measure on \(G\). Then there exists \(c>0\) such that for all \(\varphi \in C^\infty_0\) we have \[ \langle \varphi, \mu^{*n} \rangle = O(\parallel \mu^n \parallel_{2\rightarrow 2} e^{-cn^{1/3}}). \] The paper is rather lengthy, besides its essential research value it also has expositional and instructional merits and contains a lot of supplementary information. It is carefully structured, the material is separated in hierarchically ordered logical units and provides reading advices. This, together with the personal language, helps to keep the attention of the reader alive.

Reviewer: V.V.Kisil (Gent)

### MSC:

22E30 | Analysis on real and complex Lie groups |

22E15 | General properties and structure of real Lie groups |