Balls defined by nonsmooth vector fields and the Poincaré inequality

Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality

“…Nonsmooth vector fields of step two. The two papers by MontanariMorbidelli [39], [40] consider vector fields with Lipschitz continuous coefficients, satisfying Hörmander's condition of step two, plus some other structural condition. The goal of these papers is to prove Poincaré's and Sobolev' type inequalities for these vector fields.…”

A note on the Poincaré inequality for Lipschitz vector fields of step two

“…. , X m } does not have the regularity required in [10]; then also in this setting, the result of [10] cannot be directly applied.…”

“…their commutators of length at most two span the tangent space at every point. Nevertheless, due to the minimal regularity of the coefficients (the function u is only a euclidean Lipschitz continuous function), the result of [10] cannot be applied in this setting since the authors required that the commutators be Lipschitz continuous.…”

“…In many situations the regularity assumptions in [10] on the commutators are not satisfied, and it does not seem to be enough to get the Poincaré inequality.…”

“…In [8] the authors do not need the smoothness of the vector fields, but they require that the control balls are representable by a controllable almost exponential maps introduced in [11]. In the recent paper [10] the Poincaré inequality is proved by developing the method of [8] for non-smooth and non-diagonal vector fields of step two, assuming the Lipschitz condition on the vector fields and on their commutators.…”