In this lecture we prove that most of the results that were proven for Laplace-Beltrami operators may actually be generalized to any locally subelliptic operator.

Let be a locally subelliptic diffusion operator defined on . For every smooth functions , we recall that the carre du champ operator is the symmetric first-order differential form defined by:

A straightforward computation shows that if

then,

As a consequence, for every smooth function ,

**Definition**: An absolutely continuous curve is said to be subunit for the operator if for every smooth function we have . We then define the subunit length of as .

Given , we indicate with

In these lectures we always assume that

If is an elliptic operator or if is a sum of squares operator that satisfies Hormander’s condition, then this assumption is satisfied.

Under such assumption it is easy to verify that

defines a true distance on . This is the intrinsic distance associated to the subelliptic operator . A beautiful result by Fefferman and Phong relates the subellipticity of to the size of the balls for this metric:

**Theorem**: Let . There exist constants , and such that for ,

where denotes here the ball for the metric and the ball for the Euclidean metric on .

A corollary of this result is that the topology induced by coincides with the Euclidean topology of . The distance can also be computed using the following definition:

**Proposition**: For every ,

Proof: Let . We denote

Let be a sub-unit curve such that

We have , therefore, if ,

As a consequence

We now prove the converse inequality which is trickier. We already know that if is elliptic then . If is only subelliptic, we consider the sequence of operators where is the usual Laplacian. We denote by the distance associated to . It is easy to see that increases with and that . We can find a curve , such that and for every ,

where is the carre du champ operator of . Since , we see that the sequence is uniformly equicontinuous. As a consequence of the Arzela-Ascoli theorem, we deduce that there exists a subsequence which we continue to denote that converges uniformly to a curve , such that and for every ,

By definition of , we deduce . As a consequence, we proved that . Since it is clear that

we finally conclude that , hence .

A straightforward corollary of the previous proposition is the following useful result:

**Corollary**: If satisfies , then is constant.

The Hopf-Rinow theorem is still true with an identical proof in the case of subelliptic operators.

**Theorem**: The metric space is complete (i.e. Cauchy sequences are convergent) if and only the compact sets are the closed and bounded sets.

Similarly, we also have the following key result:

**Proposition**: There exists an increasing sequence , , such that on , and , as if and only if the metric space is complete.