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

Abstract

We consider a family of vector fields

Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius

“…, X q (similarly for other lists of vector fields). Our main assumption is that for every δ ∈ [0, 1) ν , with |δ| sufficiently small, 3 we have:…”

“…Next, we discuss other results concerning CarnotCarathéodory balls in the case when the vector fields are not smooth. In particular, we discuss the recent works [1] and [15]. Third, as motivation for our study of "unit operators" (and maximal functions) in Section 6, we discuss the singular integrals from [28].…”

“…We will see that the main results for multi-parameter Carnot-Carathéodory balls follow from a certain "uniform" version of the theorem of Frobenius on involutive distributions. 1 We will prove this version of the theorem of Frobenius by building on the work of [22] along with work of Tao and Wright [30]. Our primary motivation is to obtain the properties of multi-parameter balls which are relevant for developing a theory of multi-parameter singular integrals, which will be the subject of a future paper.…”

“…3 Throughout the rest of this introduction, δ will always denote a small element of [0, 1) ν . 4 Even in the smooth case, the assumptions in Section 5.1 require less than we outline here.…”

Convex Hull Property and Exclosure Theorems for H-Minimal Hypersurfaces in Carnot Groups

“…to the distribution generated by these vector elds. This result has had many recent generalizations in which the regularity of the vector elds is weakened; see [9], [31], [36] Proof. Observe preliminarily that Theorem 2.19 can be applied to our situation by arguing exactly as in the proof of Corollary 2.18.…”