# A PoincarĂ© inequality for Lipschitz intrinsic vector fields in the Heisenberg group

### (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 PoincarĂ© 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 PoincarĂ© 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 PoincarĂ©

inequality for the nonlinear vector fields.