The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators (Q1376545)

At the beginning of the paper, the authors give rather a wide review of the results obtained by different authors in the investigation of the hypoellipticity of second-order infinitely degenerate elliptic operators and indicate the equivalence of various conditions. They prove that if in a segment of the finite interval in \(\mathbb{R}_x\) the functions are \(f>0\) and \(g>0\), then the operator \[ P= D^2_x+ f(x)D^2_t+ g(x)D^2y\quad\text{in }\mathbb{R}^3_{x,t,y} \] is hypoelliptic if and only if \(f\) and \(g\) satisfy a certain reduction condition. An analogous result is proved for a multivariable operator \(P\), where \(x\in\mathbb{R}^n_x\), \(t\in \mathbb{R}^k_t\), \(y\in\mathbb{R}^n_y\). In addition, some estimates are obtained that are applied to value estimates for Schrödinger operators.