Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality
-
,
Abstract.
We consider a family of vector fields
(
) defined in some bounded domain 
and assume that the Xi satisfy Hörmander's rank
condition of some step r in 
, and 
. We extend to this nonsmooth context some results
which are well known for smooth Hörmander's vector fields, namely: some
basic properties of the distance induced by the vector fields, the doubling
condition, Chow's connectivity theorem, and, under the stronger assumption
, Poincaré's inequality. By
known results, these facts also imply a Sobolev embedding. All these tools
allow us to draw some consequences about second order differential operators
modeled on these nonsmooth Hörmander's vector fields:
where 
is a uniformly elliptic matrix of
functions.
© 2013 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- A note on the L-theory of infinite product categories
- 𝔬K0-quasi-abelian varieties with complex multiplication
- Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality
- Continuity property for the commutators of multilinear Calderón–Zygmund operators
- Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces
- Ritt's theory on the unit disk
- A short note on the existence of nontrivial solutions to the vectorial equation in H01(Ω)N
- Characterization of finitely generated infinitely iterated wreath products
Articles in the same Issue
- Masthead
- A note on the L-theory of infinite product categories
- 𝔬K0-quasi-abelian varieties with complex multiplication
- Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality
- Continuity property for the commutators of multilinear Calderón–Zygmund operators
- Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces
- Ritt's theory on the unit disk
- A short note on the existence of nontrivial solutions to the vectorial equation in H01(Ω)N
- Characterization of finitely generated infinitely iterated wreath products
