Hypoellipticity for infinitely degenerate elliptic and parabolic operators of second order (Q583483)

Let f(t) and g(t) be positive functions except at \(t=0\) and vanish at \(t=0\) with infinite order. Then, the author investigates whether the operator \(L_ 1=D_ t^ 2+f(t)D^ 2_ x+g(t)D^ 2_ y\) is hypoelliptic or not, and also proves that the operator \(L_ 2=D_ t^ 2+f(t)D^ 2_ x+ig(t)Dy\) is hypoelliptic under some conditions. When g(t)\(\equiv 1\), S. Kusuoka, D. Strook and Y. Morimoto investigate whether \(L_ 1\) is hypoelliptic or not. In this paper, the author extends these results to the case where g(t) vanishes at \(t=0\) with infinite order. The method of proving the hypoellipticity is the \(\alpha\)-\(\beta\) method due to S. Mizohata.