**1. Fundamental solutions **

Exercise (related to the Hopf-Rinow): compute the sub-Riemannan metric associated to vectorfield . Observe that balls are non compact, i.e. metric is not complete.

**2. Existence **

Theorem 1 (Folland)On a Carnot group, all sub-Laplacians have a unique fundamental solution, i.e. a smooth fonction on such that

- , Dirac distribution at the origin,
- .

It is homogeneous of degree under dilations.

\proof of homogeneity. Consider . Then . By hypoellipticity, is smooth and classically harmonic.

By Bony’s maximal principle, since teds to 0 at infinity, . Alternatively, use Liouville’s theorem.

** 2.1. The case of groups of Heisenberg type **

Charles Feffermann, studying several complex variables, suggested the form that the fundamental solution should take in the Heisenberg group. This was implemented by Folland and Kaplan.

Theorem 2 (Folland 1972, Kaplan 1981)Let be of Heiseberg type. The functionis a fundamental solution of . Here, is a suitable constant,

** 2.2. A Lemma **

Lemma 3If is of Heisenberg type,

- ,
- ,
- .

For this, use Baker-Campbell-Hausdorff to compute

Differentiating with respect to at , this gives

This leads rather easily to all 3 formulae.

** 2.3. Proof of the Folland-Kaplan Theorem **

We see that where is a gauge. Let us regularize it,

Then

Given an arbitrary function , differentiate . Then apply it to and observe that this kills a term, yielding

This equation is known as the CR Yamabe equation. This is the conformally invariant form of the sub-Laplacian. It indicates that is critical for the sub-Riemannian Sobolev inequality.

Observe that

Thus

It turns out that . So up to a multiplicative constant, converges to the Dirac distribution as . Indeed, given a test function ,

** 2.4. The CR Yamabe problem **

The problem: let be a compact strictly pseudoconvex CR manifold, find a choice of the contact form , for which the Tanaka-Webster scalar curvature is constant.

This is a sub-Riemannian analogue of a problem posed in 1959 by Yamabe, and which has been solved (Yamabe, Trudinger, Aubin, Schoen).

Theorem 4 (Jerison-Lee 1990)The CR Yamabe problem is solvable when dim and is not locally CR equivalent to the round CR sphere.

The CR version

After a decade, Gamara and Yaccoub, two students of Abbas Bahri, solved the problem when is CR equivalent to the CR round sphere. The 3-dimensional case was later completed by Gamara.

These cases non treated by Jerison and Lee are analogues of the Riemannian cases where the positive mass conjecture in general relativity plays a role. There have been recent progress along similar lines in CR geometry recently. Attend the relevant workshop this fall!

** 2.5. The sub-Riemannian Sobolev embedding theorem **

Observe that

This is an equality case in a Sobolev type inequality. The Euclidean Sobolev inequality reads

The numerology is forced by dilaton invariance.

Theorem 5 (Folland-Stein 1975)In a Carnot group, let . There exists a constant such that, for all smooth compactly supported functions ,

provided .