Hypoellipticity for operators of infinitely degenerate Egorov type (Q1961889)

The authors prove the hypoellipticity of the operator \(P= D_t+ i\alpha(t) b(t,x,D_x)\) in \(\mathbb{R}_t\times \mathbb{R}^n_x\), under the assumptions: 1) \(\alpha\in C^\infty(\mathbb{R})\), \(\int_I \alpha(t) dt> 0\) for any interval \(I\subset\mathbb{R}\). 2) \(b(t,x,\xi)\in C^\infty(\mathbb{R}_t, S^1_{1,0}(\mathbb{R}^n_x))\), and the principal symbol \(b_1\) of \(b\) is real-valued. 3) \((\partial_tb_1)(t,x,\xi)\geq 0\) for \((t,x,\xi)\in \mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n\). 4) \((\tau,b_1(t,x,\xi))\) satisfies the Hörmander's bracket condition (or \(b_1(t,x,D)\) verifies a logarithmic regularity estimate).