Homogeneous Carnot groups related to sets of vector fields

Andrea Bonfiglioli

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 1, page 79-107
  • ISSN: 0392-4041

Abstract

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In this paper, we are concerned with the following problem: given a set of smooth vector fields X 1 , … , X m on R N , we ask whether there exists a homogeneous Carnot group G = ( R N , ∘ , δ λ ) such that ∑ 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 non-trivial examples of our construction.

How to cite

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Bonfiglioli, Andrea. "Homogeneous Carnot groups related to sets of vector fields." Bollettino dell'Unione Matematica Italiana 7-B.1 (2004): 79-107. <http://eudml.org/doc/195551>.

@article{Bonfiglioli2004,
abstract = {In this paper, we are concerned with the following problem: given a set of smooth vector fields $X_\{1\}, \ldots , X_\{m\}$ on $\mathbb\{R\}^\{N\}$, we ask whether there exists a homogeneous Carnot group $\mathbb\{G\}=(\mathbb\{R\}^\{N\}, \circ, \delta_\{\lambda\} )$ such that $\sum_\{i\} X_\{i\}^\{2\}$ is a sub-Laplacian on $\mathbb\{G\}$. 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 non-trivial examples of our construction.},
author = {Bonfiglioli, Andrea},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {79-107},
publisher = {Unione Matematica Italiana},
title = {Homogeneous Carnot groups related to sets of vector fields},
url = {http://eudml.org/doc/195551},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Bonfiglioli, Andrea
TI - Homogeneous Carnot groups related to sets of vector fields
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/2//
PB - Unione Matematica Italiana
VL - 7-B
IS - 1
SP - 79
EP - 107
AB - In this paper, we are concerned with the following problem: given a set of smooth vector fields $X_{1}, \ldots , X_{m}$ on $\mathbb{R}^{N}$, we ask whether there exists a homogeneous Carnot group $\mathbb{G}=(\mathbb{R}^{N}, \circ, \delta_{\lambda} )$ such that $\sum_{i} X_{i}^{2}$ is a sub-Laplacian on $\mathbb{G}$. 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 non-trivial examples of our construction.
LA - eng
UR - http://eudml.org/doc/195551
ER -

References

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