On the Levi condition for Goursat problem (Q1099324)

The author treats the following differential operator \[ (1)\quad P(D_ t,D_ x,D_ y)=\sum^{m}_{j=\ell}C_ j(D_ x,D_ y)D_ t^{m- j}=P_ m+\sum^{m}_{k=1}P_{m-k}, \] where \(C_ j(\zeta,\eta)\) is a polynomial with constant coefficients of order \(\leq j\), and \(P_ j\) is a homogeneous part of degree j of P. Let \[ P_ m(\tau,\zeta,\eta)=\prod^{n'}_{j=1}(\zeta -\lambda_ j(\eta))^{\nu_ j}\prod^{n''}_{i=1\quad}(\tau -\tau_ i(\zeta,\eta))^{\rho_ i}. \] The author proves that the Goursat problem for (1) is \({\mathfrak E}\)-wellposed if and only if \(P_{m-k}\) has the following form: \[ P_{m-k}=\sum_{k_ 1+k_ 2=k}q_{k_ 1k_ 2}(\tau,\zeta,\eta)\prod^{n'}_{j=1}(\zeta -\quad \lambda_ j)^{\nu_ j-k_ 1}\prod^{n''}_{i=1}(\tau -\tau_ i)^{\rho_ i-k_ 2}. \] In the case of variable coefficients the author gives a sufficient condition for the wellposedness of Goursat's problem.