A classical theorem of Montel states that a family of holomorphic functions on a domain Ω⊆C, uniformly bounded on the compact subsets of Ω, is a normal family. The aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to Lu=0, where L belongs to a wide class of real divergence-form PDOs, comprising sub-Laplacians on Carnot groups, subelliptic Laplacians on arbitrary Lie groups, as well as the Laplace-Beltrami operator on Riemannian manifolds. To this end, we extend another remarkable result, due to Koebe: we characterize the solutions to Lu=0 as fixed points of suitable mean-value operators with non-trivial kernels. A suitable substitute for the Cauchy integral formula is also provided. Finally, the local-boundedness assumption is relaxed, by replacing it with L_loc^1-boundedness.

E Battaglia, A Bonfiglioli (2014). Normal families of functions for subelliptic operators and the theorems of Montel and Koebe. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 409(1), 1-12 [10.1016/j.jmaa.2013.06.062].

Normal families of functions for subelliptic operators and the theorems of Montel and Koebe

BATTAGLIA, ERIKA;BONFIGLIOLI, ANDREA
2014

Abstract

A classical theorem of Montel states that a family of holomorphic functions on a domain Ω⊆C, uniformly bounded on the compact subsets of Ω, is a normal family. The aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to Lu=0, where L belongs to a wide class of real divergence-form PDOs, comprising sub-Laplacians on Carnot groups, subelliptic Laplacians on arbitrary Lie groups, as well as the Laplace-Beltrami operator on Riemannian manifolds. To this end, we extend another remarkable result, due to Koebe: we characterize the solutions to Lu=0 as fixed points of suitable mean-value operators with non-trivial kernels. A suitable substitute for the Cauchy integral formula is also provided. Finally, the local-boundedness assumption is relaxed, by replacing it with L_loc^1-boundedness.
2014
E Battaglia, A Bonfiglioli (2014). Normal families of functions for subelliptic operators and the theorems of Montel and Koebe. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 409(1), 1-12 [10.1016/j.jmaa.2013.06.062].
E Battaglia; A Bonfiglioli
File in questo prodotto:
Eventuali allegati, non sono esposti

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/11585/181278
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 6
social impact