On the Gevrey index for some hypoelliptic operators (Q1114848)

Let P be a partial differential operator with analytic coefficients in an open set of \(R^ d\). Then, the author defines the Gevrey index \(\gamma_ x(P)\) for P as \[ \gamma_ x(P)=\inf \{s\quad in\quad R;\quad P\quad is\quad \gamma^{(s)}-hypoelliptic\quad in\quad an\quad n.b.d.\quad of\quad x\}. \] The author treats the operator \[ P=\sum^{n-1}_{j=0}(-i\partial /\partial x_ j)^ 2+ix^ k_ 0(- i\partial /\partial x_ n) \] and proves the following four theorems. Theorem 1: \(\gamma_ 0(P)\leq k+2\). Theorem 2: When \(n=1\), \(\gamma_ 0(P)\leq 2\). Theorem 3: When \(k=1\), \(\gamma_ 0(P)\leq 2\). Theorem 4: When \(n\geq 2\) and either \(k=3\) or even number, \(\gamma_ 0(P)\geq k+2\). By these theorems we obtain \(\gamma_ 0(P)=2\) when \(n=1\) or \(k=1\) and \(\gamma_ 0(P)=k+2\) when \(n\geq 2\) and either \(k=3\) or even.