We now turn to the proof of Davie’s estimate. We follow the approach by Friz-Victoir who smartly use interpolations by geodesics in Carnot groups.

**Theorem:*** Let . Assume that are -Lipschitz vector fields in . Let . Let be the solution of the equation
There exists a constant depending only on and such that for every ,
where is the lift of in .*

**Proof:** For , we denote by a path in such that , and , , is a geodesic for the Carnot-Caratheodory distance. We consider then to be the solution of the equation

We can readily observe that from the continuity of Lyons’ lift:

Let us now denote

For fixed , we have then:

To estimate this quantity, we consider the path , , that solves the ordinary differential equation driven by the concatenation of and . We first estimate by observing that and have the same Taylor expansion up to order . Thus by using the lemma of the previous lecture and the triangle inequality, we easily get that:

We then estimate by observing that , . Thus,

This last term is estimated by using basic continuity estimates with respect to the initial condition which gives

We conclude

where

The basic inequality combined with the triangle inequality gives:

.

We are now in position to apply the lemma of the previous lecture (we let the reader check that the assumptions are satisfied). We deduce then

We now keep in mind that

and can be estimated by using basic estimates on differential equations:

From the triangle inequality, we conclude then:

In particular we have for such that ,

This easily gives the required estimate (see Proposition 5.10 in the book by Friz-Victoir)

We can remark that the proof actually also provided the following estimate which is interesting in itself:

**Proposition:** *Let . Assume that are -Lipschitz vector fields in . Let . Let be the solution of the equation
There exists a constant depending only on and such that for every ,
.
*