Lie group approach to Grushin operators

2019/06/02 by Dziubański, Jacek, Sikora, Adam
#22E25 #35A30 #35H20 #35J70 (primary) #43A65 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1906.00371

Abstract

We consider a finite system \X₁, X₂, …, X_n\ of complete vector fields acting on smooth manifolds 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 described setting.

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