Hypoellipticity for semi-elliptic operators which degenerate on hypersurface
This paper studies the hypoellipticity of operators of the form \[ L=a(x,y,D_ x)+g(x)b(x,y,D_ y), \] where: (1) \(g(0)=0\), \(g(x)>0\) for \(x\neq 0\), (2) \(\text{Re} a(x,y,\xi)\geq c_ 1|\xi|^{2\ell}\), \(|\xi|\geq c\), (3) \(\text{Re} b(x,y,\eta)\geq c_ 2|\eta|^{2m}\), \(|\eta|\geq c\). Operators satisfying (1), (2) and (3) are hypoelliptic. This extends previous results of \textit{K. Taniguchi} [Math. Japonicae 20, 301-320 (1975; Zbl 0341.35029)] and \textit{Y. Morimoto} [J. Math. Soc. Japan 30, 327-358 (1978; Zbl 0369.35016)]. Next the hypoellipticity of the following operator is studied: \[ L=D_ t^{2k}+D_ x^{2\ell}+g(x)D_ y^{2m}. \]
scientific article; zbMATH DE number 59707