Hypoellipticity of a certain operator with double characteristic (Q1607211)

This paper concerns differential operators of the type \[ L= D^2_1+ f(x_1) D^2_2+ g(x_1) D^2_3+ (g(x_1)- 1)D_4 \] in \(\mathbb{R}^4\). The functions \(f\) and \(g\) are supposed to be \(C^\infty\) on an interval \(I\), and necessary and sufficient are states for \(L\) to be hypoelliptic on \(I\times \mathbb{R}^3\).