In this Lecture, we prove one of the fundamental estimates of rough paths theory. This estimate is due to Davie. It provides a basic estimate for the solution of the differential equation

in terms of the -variation of the lift of in the free Carnot group of step .

We first introduce the somehow minimal regularity requirement on the vector fields ‘s to study rough differential equations.

**Definition.** *A vector field on is called -Lipschitz if it is times continuously differentiable and there exists a constant such that the supremum norm of its th derivatives and the Holder norm of its th derivative are bounded by . The smallest that satisfies the above condition is the -Lipschitz norm of and will be denoted .*

The fundamental estimate by Davie is the following;

**Definition:** *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 .
*

We start with two preliminary lemmas, the first one being interesting in itself.

**Lemma:*** Let . Assume that are -Lipschitz vector fields in . Let . Let be the solution of the equation
There exists a constant depending only on such that,
where is the identity map.
*

**Proof:** For notational simplicity, we denote . An iterative use of the change of variable formula leads to

Since are -Lipschitz, we deduce that

Since,

we deduce that

The result follows then easily by plugging this estimate into the integral

The second lemma is an analogue of a result already used in previous lectures (Young-Loeve estimate, estimates on iterated integrals).

**Lemma:** *Let . Let us assume that:*

- There exists a control such that

- There exists a control and such that for ,

*Then, for all ,
*

**Proof:**

For , consider then the control

Define now

If is such that , we can find a such that , . Indeed, the continuity of forces the existence of a such that . We obtain therefore

which implies by maximization,

We have and an iteration easily gives

We deduce

and the result follows by letting