
Journal of Lie Theory 31 (2021), No. 1, 001014 Copyright Heldermann Verlag 2021 Lie Group Approach to Grushin Operators Jacek Dziubanski Instytut Matematyczny, Uniwersytet Wroclawski, 50384 Wroclaw, Poland jdziuban@math.uni.wroc.pl Adam Sikora Dept. of Mathematics and Statistics, Macquarie University, NSW 2109, Australia adam.sikora@mq.edu.au We consider a finite system {X_{1}, X_{2}, ... , X_{n}} 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 LiYau 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. [ Fulltextpdf (140 KB)] for subscribers only. 