Semiclassical hypoelliptic estimates with a loss of many derivatives

We study the pseudospectral properties of general pseudodifferential operators around a doubly characteristic point and provide necessary and sufficient conditions for semiclassical hypoelliptic a priori estimates with a loss of many derivatives. In the case of the first author, the research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM