Bonfiglioli, Andrea:
Homogeneous Carnot groups related to sets of vector fields
Bollettino dell'Unione Matematica Italiana Serie 8 7-B (2004), fasc. n.1, p. 79-107, Unione Matematica Italiana (English)
pdf (350 Kb), djvu (371 Kb). | MR2044262 | Zbl 1178.35140
Sunto
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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Con il contributo del Ministero per i Beni e le Attività Culturali