Hypoellipticity for elliptic operators with infinite degeneracy (Q2759790)

This paper deals with the hypoellipticity for infinitely degenerate second order elliptic operators in \(\mathbb{R}^3\). The authors assume that the average \(f_I\), \(g_I\) of the coefficients \(f(x),g(x)\), \(x\in\mathbb{R}^1\) is positive on any interval \(I\subset I_0\), \(I_0\) fixed, and they define a logarithmic weight type condition \((M,f,g)\) imposed on \(f_I,g_I\). Under the condition \(f,g>0\) on \(\partial I_0\) it is proved that the operators under consideration are \(C^\infty\) (micro) hypoelliptic if and only if the functions \(f,g\) satisfy the conditions \((M,f,g)\) and \((M,g,f)\) in \(I_0\). To do this logarithmic regularity estimates are established and Wick calculus of pseudodifferential operators is used.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].