×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.