Egmont Porten: Hypoellipticity and Extension of CR functions

In the first part, we consider subelliptic estimates for systems of complex vector fields. In the real case, a classical theorem of Hörmander yields subelliptic estimates if the bracket condition is satisfied (meaning that iterated brackets generate the tangent bundle). A naïve generalization to the complex case is known to fail. However one gets sufficient conditions by suitably restricting the vector fields allowed in the brackets. Moreover these restrictions disappear in the real case.

In the second part, we explain applications to tangential Cauchy-Riemann operators on CR manifolds. In this context, the above mentioned restricted bracket condition can be interpreted as higher-order Levi pseudoconcavity condition. In consequence one derives regularity results on weakly pseudoconcave CR manifolds. We apply these to the long-standing open problem to give a geometric characterization of those CR manifolds for which all CR functions are restrictions of holomorphic functions. We obtain an analytic characterization in terms of hypoellipticity, which permits to derive the extension property for a large family of weakly pseudoconcave homogeneous CR manifolds. For the subclass of essentially pseudoconcave CR manifolds introduced by Hill and Nacinovich, we solve the geometric characterization problem.