Let \(P\) be a classical pseudodifferential operator of order \(m\) defined on the open set \(X\subset \mathbb{R}^{n},\) with complete symbol \[ P(x,\xi )\sim p_{m}(x,\xi )+p_{m-1}(x,\xi )\dots,(x,\xi )\in T^{\ast }X\backslash 0. \] We say that \(P\) satisfies the Hörmander inequality, if for any \(K\)compact on \(X\) there exists \(C_{K}>0\) such that \[ (Pu,u)\geq -C_{K}\left\| u\right\| _{\frac{m-2}{2}}^{2},\forall u\in C_{0}^{\infty }(K). \tag{HI} \] A necessary condition for (HI) to hold is the following: \[ p_{m}(x,\xi )\geq 0,\forall (x,\xi )\in T^{\ast }X\backslash 0, \tag{NC1} \]
\[ p_{m}(x,\xi )=0\Rightarrow p_{m-1}^{s}(x,\xi )+Tr^{+}F_{(x,\xi )}\geq 0, \tag{NC2} \] where \(p_{m-1}^{s}(x,\xi )\) is the subprincipal symbol of \(P\) and \(Tr^{+}F_{(x,\xi )}\) is the positive trace of the fundamental matrix \(F_{(x,\xi )}\) of \(P.\)
Under the condition (NC1), the authors treat some classes of classical pseudodifferential operators such that (NC2) is also sufficient for Hörmander inequality (HI) to hold.