Parametrices for pseudodifferential operators with multiple characteristics (Q5892132)

The author considers the following second-order operator in \(\mathbb{R}^3\) \[ P= D^2_1+ x^{2h}_1(D^2_2+ D^2_3)_\lambda x^{h-1}_1 D_2+ g(x_2) x^{h-1}_1 D_3, \] for an integer \(h\geq 1\). The interesting case is when \(g(x_2)\), at some \(x^0_2\in\mathbb{R}\), coincides with one of the eigenvalues \(\rho\) of the ordinary differential operator associated to \(P\) by the Fourier transform. In this case, one is led to consider the order \(k\geq 1\) of the zero of \(g(x_2)-\rho\) at \(x^0_2\). Very precise results of hypoellipticity are given by the author, depending on the values of \(h\) and \(k\). In particular, a large loss of derivatives is observed in the case \(g(x^0_2)=\rho\). The case \(h=k=1\) was already studied by \textit{P. R. Popivanov} [Rend. Semin. Mat., Univ. Politec. Torino 66, No. 4, 321--338 (2008; Zbl 1180.35176)].