The Poincaré inequality for vector fields satisfying Hörmander's condition (Q1820325)

From author's summary: The purpose of this paper is to prove a Poincaré-type inequality of the form \[ \int_{B(r)}| f- f_{B(r)}|^ 2\leq Cr^ 2\int_{B(r)}\sum^{m}_{i=1}| X_ if|^ 2,\text{ for all } f\in C^{\infty}(\overline{B(r)}), \] where \(X_ 1,...,X_ m\) denote vector fields on \(R^ d\) satisfying Hörmander's condition, B(r) denotes a ball of radius r with respect to a natural metric associated with \(X_ 1,...,X_ m\) and \(f_{B(r)}\) denotes the average value of f on the ball B(r). This inequality is the same as finding a lower bound \(1/Cr^ 2\) of the least nonzero eigenvalue in the Neumann problem for \(L=\sum^{m}_{i=1}X^*_ iX_ i\) of B(r).