Let X =(X1,..., Xm) be a set of Hormander vector fields in Rn, where any Xj is homogeneous of degree 1 with respect to a family of nonisotropic dilations in Rn. If N is the dimension of Lie(X), we can either lift X to a system of generators of a higher dimensional Carnot group on RN (if N > n), or we can equip Rn with a Carnot group structure with Lie algebra equal to Lie(X) (if N = n). We shall deduce these facts via a local-to-global procedure (available in the homogeneous setting), starting from general results on the lifting of finite-dimensional Lie algebras of vector fields. The use of the Baker-Campbell-Hausdorff Theorem is crucial. Due to homogeneity, the lifting procedure is simpler than Rothschild-Stein's lifting technique. We finally provide applications to the study of the fundamental solution G for the Hormander sum of squares including global pointwise estimates of G and of its Xderivatives in terms of the Carnot-Caratheodory distance induced by X.

Hormander vector fields equipped with dilations: Lifting, lie-group construction, applications

Bonfiglioli A.
2020

Abstract

Let X =(X1,..., Xm) be a set of Hormander vector fields in Rn, where any Xj is homogeneous of degree 1 with respect to a family of nonisotropic dilations in Rn. If N is the dimension of Lie(X), we can either lift X to a system of generators of a higher dimensional Carnot group on RN (if N > n), or we can equip Rn with a Carnot group structure with Lie algebra equal to Lie(X) (if N = n). We shall deduce these facts via a local-to-global procedure (available in the homogeneous setting), starting from general results on the lifting of finite-dimensional Lie algebras of vector fields. The use of the Baker-Campbell-Hausdorff Theorem is crucial. Due to homogeneity, the lifting procedure is simpler than Rothschild-Stein's lifting technique. We finally provide applications to the study of the fundamental solution G for the Hormander sum of squares including global pointwise estimates of G and of its Xderivatives in terms of the Carnot-Caratheodory distance induced by X.
2020
Bonfiglioli A.
File in questo prodotto:
File Dimensione Formato  
Bonfiglioli_Matematiche_2019.pdf

accesso aperto

Tipo: Versione (PDF) editoriale
Licenza: Licenza per Accesso Aperto. Creative Commons Attribuzione (CCBY)
Dimensione 217.82 kB
Formato Adobe PDF
217.82 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/816792
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 1
social impact