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.
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 function
is a fundamental solution of . Here, is a suitable constant,
2.2. A Lemma
Lemma 3 If 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,
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.
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
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 ,