Homogeneous Carnot groups related to sets of vector fields

Bonfiglioli, Andrea

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 1, pp. 79-107.

Abstract (VO)

In this paper, we are concerned with the following problem: given a set of smooth vector fields $X_{1}, \ldots , X_{m}$ on $\mathbb{R}^{N}$, we ask whether there exists a homogeneous Carnot group $\mathbb{G}=(\mathbb{R}^{N}, \circ, \delta_{\lambda} )$ such that $\sum_{i} X_{i}^{2}$ is a sub-Laplacian on $\mathbb{G}$. We find necessary and sufficient conditions on the given vector fields in order to give a positive answer to the question. Moreover, we explicitly construct the group law i as above, providing direct proofs. Our main tool is a suitable version of the Campbell-Hausdorff formula. Finally, we exhibit several non-trivial examples of our construction.

Riassunto

In questo articolo ci occupiamo del seguente problema: data una famiglia di campi vettoriali regolari $X_{1}, \ldots , X_{m}$ su $\mathbb{R}^{N}$, ci chiediamo se esiste un gruppo omogeneo di Carnot $\mathbb{G}=(\mathbb{R}^{N}, \circ, \delta_{\lambda} )$ tale che $\sum_{i} X_{i}^{2}$ sia un sub-Laplaciano su $\mathbb{G}$. A tale proposito troviamo condizioni necessarie e sufficienti sugli assegnati campi vettoriali affinchè la risposta alla suddetta domanda sia positiva. Inoltre esibiamo una costruzione esplicita della legge di gruppo i che verifica i requisiti di cui sopra, fornendo dimostrazioni dirette. La prova è essenzialmente basata su una opportuna versione della formula di Campbell-Hausdorff. Per finire, mostriamo svariati esempi non banali del nostro metodo costruttivo.

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