On the Poincaré inequality for vector fields

Abstract

fieldsAbstract 9 We prove the Poincard inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable "controllable almost exponential maps".

Sobolev spaces on metric-measure spaces

“…51 We say that the function u belongs to the Marcinkiewicz space L p w (X), p > 0 (called also weak L p ), if there is a constant m > 0 such that µ({|u| > t}) ≤ mt −p for all t > 0. 52 x −1 ∈ L 1 w (0, 1), but x −1 ∈ L 1 (0, 1). 53 The lemma is true for any finite measure.…”

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

“…This means that, for instance, one assumes axiomatically the validity of a connectivity theorem, a doubling property for the metric balls, a Poincaré's inequality for the "gradient" defined by the system of vector fields, and proves as a consequence other interesting properties of the metric or of second order PDE's structured on the vector fields. A good deal of papers have been written in this spirit; we just quote some of the Authors and some of the papers on this subject, which are a good starting point for further bibliographic references: Capogna, Danielli, Franchi, Gallot, Garofalo, Gutierrez, Lanconelli, Morbidelli, Nhieu, Serapioni, Serra Cassano, Wheeden; see [1], [9], [14], [15], [22], [23], [24], [33]; see also the already quoted paper [52] and the one by Hajlasz-Koskela [25].…”

“…is X-controllable with hitting time h ′ kj 1/l k j ; by composition, Lemma 4.2 in [33] implies that E X η (x, h ′ ) is X-controllable with hitting time…”

“…As we will see, a key feature of our approach is a suitable mix of the two systems of vector fields {X i } , {S x0 i }, and of the corresponding maps E X η , E S x 0 η . Let us start recalling a definition from [33]. Definition 7.3 Let O be a bounded open set in R p , and Q a neighborhood of the origin.…”

Balls defined by nonsmooth vector fields and the Poincaré inequality

“…In this setting a description of the control balls and Poincar6 inequality was proved by Franchi and Lanconelli [7] in a low regularity situation. In a recent paper Lanconelli and the second author [15] An interesting feature of our result is that the balls are very easy to visualize: they are equivalent to linear images of boxes (see (7)). We also remark that in the present paper we never use the Campbell Hausdorff formula (a powerful tool whose use in analysis of vector fields requires regularity).…”