Minimal Gevrey regularity for Hörmander operators (Q6623136)

The authors consider the sum of squares type operator \N\[ \NP(x,D) = \sum_{j=1} ^N (X_j (x,D))^2 + i X_0 (x,D) + g(x), \N\] \Nwhere \(g\) belongs to a Gevrey class \(G^s (\Omega)\), \(X_j (x,D)\), \( j =0, \dots, N\), are vector fields in \(\Omega\) with real \(G^s\) coefficients. In particular, \(X_0 (x,D)\) is a real vector field.\N\NIf the vector fields satisfy the Hörmander hypothesis, then the operator \(P(x,D)\) is hypoelliptic.\N\NIn this paper, the authors give the Gevrey regularity of the solutions to \(Pu = f,\) \(f\) real analytic. The minimal Gevrey regularity for this type of equations is derived, in the sense that it is obtained for every operator in the class. It is given in terms of the maximum length of the commutators generating the Lie algebra on the set \(U\) arising in the Hörmander condition.\N\NThe technique of the proof is based on inductive arguments and a careful examination of a priori estimates for \(P\).\N\NThe crucial role of the vector field \(X_0\) is highlighted by relevant examples.