Gevrey Hypoellipticity for Grushin Operators

Gevrey hypoellipticity for sums of squares of vector fields in $ \R^2 $ with quasi-homogeneous polynomial vanishing

“…In this case, Char(P)∩Ω is a symplectic submanifold of the co-tangent bundle T * Ω, and the operator P is then known to be analytic hypoelliptic (ahe). This result is due to Tartakoff, [12], [13], and Trèves, [14].…”

Hypoellipticity in the infinitely degenerate regime

“…x +(x∂ t ) 2 is analytic hypoelliptic [15]. A parallel result for Gevrey hypoellipticity is that when a(x) = x q and b(x) = x p with q ≥ p ≥ 1, L 3 is hypoelliptic in the Gevrey class of order s if and only if b • a −1/s is bounded as x → 0 [7]; see also [5] and [23].…”

Examples pertaining to Gevrey hypoellipticity

“…The following variant of Theorem 2 can be proved by the same technique, and was also obtained by Bernardi, Bove and Tartakoff [1] and Matsuzawa [10]. Consider…”

“…After this paper was circulated we received preprints of Bernardi, Bove and Tartakoff [1] and of Matsuzawa [10] containing Theorems 2 and 3, with different methods of proof. The latter paper contains more general results as well.…”