An application of the almost-positivity of a class of fourth-order pseudodifferential operators (Q1361052)

The author proves the following result. Let \(P=P^*\) be a classical pseudodifferential operator of order 2 with Weyl symbol \(p_2+ p^s_1+\cdots\)\ . Assume \(p_2\geq 0\) is transversally elliptic with respect to the characteristic manifold \(\Sigma\), and the symplectic form has constant rank on \(\Sigma\). Moreover, \(p^s_1+\text{Tr}^+ F_p>0\), the positive trace being defined as standard; then \[ |Pu|+|u|\geq C(|Xu|_{1/2}+|u|_1) \] for any pseudodifferential operator \(X\) of order 1 with principal symbol vanishing at \(\Sigma\). The proof is deduced from \textit{C. Parenti}, and \textit{A. Parmeggiani} [J. Anal. Math. 69, 55-65 (1996; Zbl 0864.35131)]. Applications are given to some operators of sum-of-squares type.