A remark on the \(C^ \infty\)-Goursat problem. II (Q1328876)

The author continues his research from [J. Math. Kyoto Univ. 28, No. 1, 37-51 (1988; Zbl 0701.35050)] on the \(C^ \infty\)-Goursat problem for partial differential operators with constant coefficients of the type \[ L= \sum_{i+j+|\alpha|\leq m} a_{ij\alpha} \partial^ i_ t \partial^ j_ x \partial^ \alpha_ y,\quad t\geq 0,\quad x\in\mathbb{R},\quad y\in \mathbb{R}^ n \] with \[ a_{ij\alpha}=0\quad\text{ for } i+ j+|\alpha|= m,\;i>m- r:=s\quad\text{ and }\sum_{j+| \alpha|= r} a_{sj\alpha} \xi^ j \mu^ \alpha\neq 0. \] The main result says: If the Goursat problem is \(\varepsilon\)-well posed for \(t\geq 0\) and for \(t\leq 0\) then \[ L= \sum_{i\leq m-r} a_{ij\alpha} \partial^ i_ t \partial^ j_ x \partial^ \alpha_ y\quad\text{with } a_{sr0}\neq 0. \]