Weyl calculus for a class of subelliptic operators (Q1597186)

The authors consider a pseudodifferential operator \(P\) with double characteristics, satisfying a sufficient condition for the hypoellipticity with minimal loss of one derivative; namely: on the characteristic manifold, \(p^s_{m-1}+\text{Tr}^+F\) is not a real number \(\leq 0\), where \(p^s_{m-1}\) is the sub-principal symbol of \(P\) and \(\text{Tr}^+F\) is the positive trace of the fundamental matrix associated with the principal symbol of \(P\). The authors are able to construct a parametrix for \(P\) in the frame of the Weyl-Hörmander pseudodifferential operators, without additional assumptions on the geometry of the characteristic set and on transversal hypoellipticity. This represents a generalization of a preceding result of \textit{C. E. Cancelier, J.-Y.Chemin}, and \textit{C. J. Xu} [Ann. Inst. Fourier 43, 1157-1178 (1997; Zbl 0797.35008)], concerning parametrices for step-2 sum-of-squares operators.