On selection of infinitely differentiable solutions of a class of partially hypoelliptic equations

In this paper the existence of a constant $\kappa_0>0$ is proved such that all solutions of a class of regular partially hypoelliptic (with respect to the hyperplane $x''=(x_2,\dots,x_n)=0$ of the space $E^n$) equations $P(D)u=0$ in the strip $\Omega_\kappa=\{(x_1,x'')=(x_1,x_2,\dots,x_n)\in E^n;\, |x_1|<\kappa\}$ are infinitely differentiable when $\kappa\ge\kappa_0$ and $D^\alpha u\in L_2(\Omega_\kappa)$ for all multi-indices $\alpha=(0,\alpha'')=(0,\alpha_2,\dots,\alpha_n)$ in the Newton polyhedron of the operator $P(D)\cdot{}$.