Hypoelliptic operators of principal type with infinite degeneracy (Q1904470)

The author considers the problem of hypoellipticity for pseudodifferential operators of principal type with infinite degeneracy. It is well-known that \[ P_0= D_t+ i(t^s D_{x_1}+ t^k x_1^m |D_x|),\quad (t, x)\in \mathbb{R}_t\times \mathbb{R}^n_x, \] where \(s\), \(k\), \(m\) are nonnegative integers, is a first order pseudodifferential operator of Egorov type. Let \(s\), \(m\) even, \(k\) odd, then \(P_0\) is subelliptic with loss of \({r\over r+ 1}\) derivatives \((r= k+ m(s+ 1))\) and hence hypoelliptic. If \(t^s\), \(t^k\), \(x_1^m\) are replaced by functions infinitely vanishing then the hypoellipticity of \(P_0\) is unknown. The author studies this question for special cases.