A remark on the Cauchy problem for the differntial operator of principal type with polynomial coefficients. II (Q1848691)

Put \(x=(x_0,x')\), \(x'=(x_1,\cdots, x_n)\) where \(x_0 \in \mathbb C\), \(x'\in \mathbb C^n\). Let \(a(x,D) = \sum_{|\alpha|\leq m}a_{\alpha}(x)D^{\alpha}\), where \(D=(\frac {\partial}{\partial x_0},\dots, \frac {\partial}{\partial x_n})\), be a partial differential operator of order \(m\) with polynomial coefficients \(a_{\alpha}\). Consider the Cauchy problem \[ a(x,D)u(x)=0,\quad \Biggl( \frac {\partial}{\partial x_0}\Biggr)^ku(0,x')=w_k(x'),\quad 0\leq k\leq m-1, \] where \(w_k\) is a polynomial. This note is complementary to author's earlier paper [Tôhoku Math. J. (2) 50, 133-138 (1998; Zbl 0912.35034)] by presenting examples of Cauchy problems such that the domains of holomorphy of the solutions are one-sheeted in \(\mathbb C^{n+1}\) and have exterior points.