A note on hypoellipticity of degenerate elliptic operators (Q1187176)

This paper deals with the \(C^ \infty\) hypoellipticity of the following class of linear partial differential operators: \(L = D_ t^{2m} + f(t) D_ x^{2m} + g(t) D_ y^{2m}\), \(m \geq 1\), \(f(t)>0\), \(g(t)>0\) for \(t \neq 0\). Under an assumption on the behaviour of \(\log f(t)\), \(\log g(t)\) for \(t \to 0\) a necessary and a sufficient condition for hypoellipticity are proved. For a model example a necessary and sufficient condition for hypoellipticity of operator having flat coefficients at 0 is obtained. The proofs given here are similar to those proposed by T. Hoshiro in some of his papers.