Let X = {X_0,..., X_m} be a family of smooth vector fields on an open set A ⊆ R^N. Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets A, we find necessary and sufficient conditions on X for the existence of a Lie group (A, ∗) such that the operator L = X_1^2 + ... + X_m^2 + X_0 is left-invariant with respect to the operation ∗. Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance.

Left-Invariance for Smooth Vector Fields and Applications / Biagi, Stefano; Bonfiglioli, Andrea; Polidoro, Sergio. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 34:9(2024), pp. 1-21. [10.1007/s12220-024-01733-3]

Left-Invariance for Smooth Vector Fields and Applications

Polidoro, Sergio
2024

Abstract

Let X = {X_0,..., X_m} be a family of smooth vector fields on an open set A ⊆ R^N. Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets A, we find necessary and sufficient conditions on X for the existence of a Lie group (A, ∗) such that the operator L = X_1^2 + ... + X_m^2 + X_0 is left-invariant with respect to the operation ∗. Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance.
2024
10-lug-2024
34
9
1
21
Left-Invariance for Smooth Vector Fields and Applications / Biagi, Stefano; Bonfiglioli, Andrea; Polidoro, Sergio. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 34:9(2024), pp. 1-21. [10.1007/s12220-024-01733-3]
Biagi, Stefano; Bonfiglioli, Andrea; Polidoro, Sergio
File in questo prodotto:
File Dimensione Formato  
s12220-024-01733-3.pdf

Open access

Tipologia: Versione pubblicata dall'editore
Dimensione 370.14 kB
Formato Adobe PDF
370.14 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

Licenza Creative Commons
I metadati presenti in IRIS UNIMORE sono rilasciati con licenza Creative Commons CC0 1.0 Universal, mentre i file delle pubblicazioni sono rilasciati con licenza Attribuzione 4.0 Internazionale (CC BY 4.0), salvo diversa indicazione.
In caso di violazione di copyright, contattare Supporto Iris

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1346446
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
social impact