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Families of diffeomorphic sub-Laplacian and free Carnot groups. (English) Zbl 1065.35102
Let $$\{X_1,\dots, X_m\}$$ be a set of vector fields which satisfying Hörmander’s condition for a free homogeneous Carnot group $${\mathbb G}=({\mathbb R}^N,\circ, \delta_\lambda)$$. The canonical sub-Laplacian on $${\mathbb G}$$ is the second order differential operator $\Delta_{\mathbb G}=\sum_{j=1}^m X_j^2.$ Let $$A$$ be a $$m\times m$$ symmetric matrix such that $\Lambda^{-1}| {\mathbf x}| ^2\leq \langle A\xi,\xi\rangle\leq \Lambda| {\mathbf x}| ^2,$ where $$\Lambda\geq 1$$ is a fixed constant. In this paper, the authors prove that there exists a Lie group automorphism $$T_A$$ of $${\mathbb G}$$ such that $\Big(\sum_{j=1}^m \big(A^{1/2}\big)_{j,k}X_k\Big) (u\circ T_A)=(X_j u)\circ T_A,\quad j=1,\dots, m$ for every smooth function $$u$$. As a consequence, the authors show that ${\mathcal L}_A(u\circ T_A)=(\Delta_{\mathbb G}u)\circ T_A.$ Here ${\mathcal L}=\sum_{j=1}^m Y_j^2 ,$ and $$\{Y_1,\dots,Y_m\}$$ is any basis for the $${\text{span}}\{X_1,\dots, X_m\}$$. Moreover, $$T_A$$ has polynomial component funcitons and it commutes with the dilations of $${\mathbb G}$$: $\delta_\lambda({\mathbf x})= \delta_\lambda(x^{(1)},\dots,x^{(r)})= \big(\lambda x^{(1)},\lambda^2 x^{(2)}, \ldots, \lambda^r x^{(r)}\big)$ for all $$\lambda>0$$.

##### MSC:
 35H20 Subelliptic equations 43A80 Analysis on other specific Lie groups 22E30 Analysis on real and complex Lie groups
##### Keywords:
automorphism; homogeneous dimension
Full Text:
##### References:
  Jerison, Math Sa nchez - Calle A : Estimates for the heat kernel for a sum of squares of vector fields Indiana Univ Math : Applications of the Malliavin calculus III Sci Univ, Res Lett 6 pp 645– (1999)
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