Regularity of a class of subLaplacians on the 3-dimensional torus (Q860793)

The authors of the very interesting paper under review present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a 3-dimensional torus. Precisely, let \[ P=-\partial^2_{t_1}-\big(\partial_{t_2}+a(t_1,t_2)\partial_x\big)^2, \] where \((t_1,t_2,x)\in\mathbb T^3\) and \(a\in C^\infty(\mathbb T^2)\) is real-valued. Set \[ a_0(t_1)={1\over{2\pi}}\int_{\mathbb T} a(t_1,s)\,ds. \] Then the operator \(P\) is globally hypoelliptic on \(\mathbb T^3\) if and only if either the range of \(a_0(t_1)\) contains an irrational non-Liouville number (that is, a Diophantine condition) or there exists a point \(p\in\mathbb T^3\) of finite type for the vector fields \(X_1=\partial_{t_1}\) and \(X_2=\partial_{t_2}+a(t_t,t_2)\partial_x.\) The proofs are based on \(L^2\)-estimates and microlocal analysis. Analytic and Gevrey versions of the result are also described.