Lie Group Approach to Grushin Operators
Journal of Lie theory, Tome 31 (2021) no. 1, pp. 1-14
We consider a finite system {X1, X2, ... , Xn} of complete vector fields acting on a smooth manifold M equipped with a smooth positive measure. We assume that the system satisfies Hörmander's condition and generates a finite dimensional Lie algebra of type (R). We investigate the sum of squares of the vector fields operator corresponding to this system which can be viewed as a generalisation of the notion of Grushin operators. In this setting we prove the Poincaré inequality and Li-Yau estimates for the corresponding heat kernel as well as the doubling condition for the optimal control metrics defined by the system. We discuss a surprisingly broad class of examples of the described setting.
Classification :
22E30, 43A15, 22E25, 35A30, 35J70, 43A65
Mots-clés : Lie groups, degenerate elliptic operators, Grushin operators, heat kernels, Riesz transform
Mots-clés : Lie groups, degenerate elliptic operators, Grushin operators, heat kernels, Riesz transform
@article{JLT_2021_31_1_JLT_2021_31_1_a0,
author = {J. Dziubanski and A. Sikora},
title = {Lie {Group} {Approach} to {Grushin} {Operators}},
journal = {Journal of Lie theory},
pages = {1--14},
year = {2021},
volume = {31},
number = {1},
url = {http://geodesic.mathdoc.fr/item/JLT_2021_31_1_JLT_2021_31_1_a0/}
}
J. Dziubanski; A. Sikora. Lie Group Approach to Grushin Operators. Journal of Lie theory, Tome 31 (2021) no. 1, pp. 1-14. http://geodesic.mathdoc.fr/item/JLT_2021_31_1_JLT_2021_31_1_a0/