On the positive parts of second order symmetric pseudodifferential operators
The authors consider a spectral decomposition of self-adjoint pseudodifferential operators \(P(x,D)\) on \(\mathbb R^n\) of Hörmander class. Considering a degenerate elliptic operator of second order, they assume three condtions. First, the principal symbol vanishes exactly of order two on a symplectic submanifold \(\Sigma\subset T^* \mathbb R^n\). Secondly, the eigenvalues of the fundamental matrix are distinctive at a point \(\rho\in\Sigma\). Thirdly, considering a kind of indicial polynomial determined by the subprincipal symbol of \(P(x,D)\), all the length \(|\alpha|\) of zero points \(\alpha\) should be either odd or even, at this point \(\rho\in\Sigma\). Under these assumptions, they prove that one can microlocally construct the positive part operator \(\Pi^+\) and negative part operator \(\Pi^-\) of \(P(x,D)\), and proves a sharp Garding inequality using \(\Pi^{\pm}\). This problem was previously studied by D.~Fujiwara for first order pseudodifferential operators, and the present article is an attempt of its generalization.
scientific article; zbMATH DE number 5633112