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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 \(\lambda>0\).

35H20 Subelliptic equations
43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
Full Text: DOI
[1] 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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