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Homogeneous Carnot groups related to sets of vector fields

Andrea Bonfiglioli — 2004

Bollettino dell'Unione Matematica Italiana

In this paper, we are concerned with the following problem: given a set of smooth vector fields $X_{1}, \dots, X_{m}$ on $R^{N}$ , we ask whether there exists a homogeneous Carnot group $G = (R^{N}, \circ, δ_{λ})$ such that $\sum_{i} X_{i}^{2}$ is a sub-Laplacian on $G$ . 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...

Dirichlet problem with L $^{p}$ -boundary data in contractible domains of Carnot groups

Andrea Bonfiglioli; Ermanno Lanconelli — 2006

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let $ℒ$ be a sub-laplacian on a stratified Lie group $G$ . In this paper we study the Dirichlet problem for $ℒ$ with $L^{p}$ -boundary data, on domains $Ω$ which are contractible with respect to the natural dilations of $G$ . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for $ℒ$ . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

A note on lifting of Carnot groups.

Andrea Bonfiglioli; Francesco Uguzzoni — 2005

Revista Matemática Iberoamericana

We prove that every homogeneous Carnot group can be lifted to a free homogeneous Carnot group. Though following the ideas of Rothschild and Stein, we give simple and self-contained arguments, providing a constructive proof, as shown in the examples.

Subharmonic functions in sub-Riemannian settings

Andrea Bonfiglioli; Ermanno Lanconelli — 2013

Journal of the European Mathematical Society

In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution $Γ$ . These characterizations are based on suitable average operators on the level sets of $Γ$ . Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic function...