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Journal of Lie Theory 31 (2021), No. 1, 001--014
Copyright Heldermann Verlag 2021

Lie Group Approach to Grushin Operators

Jacek Dziubanski
Instytut Matematyczny, Uniwersytet Wroclawski, 50-384 Wroclaw, Poland

Adam Sikora
Dept. of Mathematics and Statistics, Macquarie University, NSW 2109, Australia

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.

Keywords: Lie groups, degenerate elliptic operators, Grushin operators, heat kernels, Riesz transform.

MSC: 22E30, 43A15; 22E25, 35A30, 35J70, 43A65.

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