Abstract
We consider a second order operator with analytic coefficients whose principal symbol vanishes exactly to order two on a symplectic real analytic manifold. We assume that the first (non degenerate) eigenvalue vanishes on a symplectic submanifold of the characteristic manifold. In the C ∞ framework this situation would mean a loss of 3/2 derivatives (see Helffer, 1977). We prove that this operator is analytic hypoelliptic. The main tool is the FBI transform. A case in which C ∞ hypoellipticity fails is also discussed.
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