A Poincaré inequality for Lipschitz intrinsic vector fields in the Heisenberg group

Giovanna Citti

(Dipartimento di Matematica, Universita' di Bologna)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 13, 2011, time: 14:30

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. This result is a joint work with M.Manfredini, A.Pinamonti, F.Serra Cassano. We prove a Poincaré inequality for Lipschitz
intrinsic vector fields in any Heisenberg group of dimension \(n>1\).
If a subriemannian metric is defined in this group, a regular surface
implicitly defines a graph \(\phi\), which is regular with respect
to non linear vector fields, defined in terms of \(\phi\) itself.
Geometric equations can be written in terms of these nonlinear vector fields.
Hence it is necessary to establish a Poincaré formula for vector
fields with minimal assumptions on the coefficients.
This inequality has been already
established in case of coefficients Lipchitz continuous
in the standard Euclidean sense, but the intrinsic Lipschitz
condition is weaker. Hence we will use a different technique
based on approximation of the given vector fields
with polynomial ones. These approximating vector fields
satisfy a representation formula, from which we get the Poincaré
inequality for the nonlinear vector fields.