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
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.