Non-hypoellipticity for degenerate elliptic operators (Q1821254)

It is proved that for the partial differential operator \(L=D^ 2_ x+(\Phi (x))^ 2 D^ 2_ y+D^ 2_ t\) where \(\Phi \in C^{\infty}(R),\Phi (0)=0\), \(\Phi (x)>0\), \(\Phi (x)=\Phi (-x)\), \(\Phi\) nondecreasing on R, the condition: \(\lim_{x\downarrow 0} x \log \Phi (x)=0\) is necessary for hypoellipticity. This result has been previously proved by Kusuoka via the Malliavin calculus but the proof given here is simpler.