Luca Rizzi

We prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. As an application, we derive (p-)Hardy-type and isoperimetric-type inequalities for a compact domain with sufficiently regular boundary. Moreover, we prove an universal (i.e. curvature independent) lower bound for the first Dirichlet eigenvalue of the intrinsic sub-Laplacian, All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations.

(joint work with D. Prandi and M. Seri)