Lie group approach to Grushin operators

Jacek Dziubański, Adam Sikora

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)1-14
Number of pages14
JournalJournal of Lie Theory
Volume31
Issue number1
Publication statusPublished - 2021

Keywords

  • Lie groups
  • degenerate elliptic operators
  • Grushin operators
  • heat kernels
  • Riesz transform

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