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We consider a finite system $\{X_1, X_2, \ldots, X_n\}$ of complete vectorfields acting on smooth manifolds $M$ equipped with a smooth positive measure.We assume that the system satisfies H\"ormander's condition and generates afinite dimensional Lie algebra of type (R). We investigate the sum of squaresof the vector fields operator corresponding to this system which can be viewedas a generalisation of the notion of Grushin operators. In this setting weprove the Poincar\'e inequality and Li-Yau estimates for the corresponding heatkernel as well as the doubling condition for the optimal control metricsdefined by the system. We discuss a surprisingly broad class of examples ofdescribed setting.

Lie group approach to Grushin operators