Hypoellipticity of second order operators in \(\mathbb{R}^ 2\) of the form \(fX^ 2+Y+g\) (Q1356385)

Let \(\Omega\) be an open set in \(\mathbb{R}^2\) and \(L\) be a second-order differential operator defined in \(\Omega\) of the form \(L= fX^2+Y+g\) with \[ X:= a{\partial\over\partial x}+ b{\partial\over\partial y},\quad Y:=c{\partial\over\partial x}+d{\partial\over\partial y}; \] \(f\), \(a\), \(b\), \(c\), \(d\), \(g\) are real-valued analytic functions defined in \(\Omega\), \(ad-bc\neq 0\) in \(\Omega\). Then \(L\) is hypoelliptic in \(\Omega\) if and only if \(Xf(x,y)=0\) for any \((x,y)\in\Omega\) such that \(f(x,y)=0\), \(f\) does not vanish identically on any integral curve of \(Y\), and \(f\) does not change sign from plus to minus along any integral curve of \(Y\), where \(Y\) is considered as a vector field in \(\Omega\).