A parametrix at noninvolutively crossing characteristic points (Q2367463)

We construct a parametrix for hyperbolic equations near a very plain double characteristic, namely, at a neighborhood of singular points of two non-involutively crossing hypersurfaces. We reduce the problem to constructing a parametrix for the following simple equation of second order: \[ P=\partial_ z \bigl( \partial_ z-i \Lambda (z, x, D) + a(z,x,D) \bigr) + c(z,x,D) + d(z,x,D) , \] where the principal part \(p_ 2\) is \(-(\zeta-\Lambda) \zeta\) and the following conditions hold for the symbol of this operator. 1) \(a,b,c, \Lambda\) are classical pseudo- differential operators in \((x,\xi)\) with the smooth parameter \(z\) belonging to \(S^ 0_{1,0}\), \(S^ 0_{1,0}\), \(S^ 1_{1,0}\), and \(S^{-1}_{1,0}\), respectively. 2) \(\Lambda\) is real and positively homogeneous of order 1 in \(\xi\). 3) \(\partial_ z\Lambda \neq 0\) at the points where \(\Lambda=0\). 4) \(\partial_ z [c/ \partial_ z \Lambda] \equiv 0\) near \(\Lambda = 0\).