Hypoellipticity for a class of degenerate elliptic operators of second order (Q1202803)

The paper under consideration is devoted mainly to the investigation of some microhypoelliptic properties for the following class of degenerate elliptic partial differential operators of second order \(p(x,D) = D^ 2_ 1 + \alpha (x)D^ 2_ 2 + \beta (x) D_ 2\), \(x \in \mathbb{R}^ 2\), \(\alpha (x) \geq 0\), \(\alpha (0) = 0\). Under several assumptions for subordination of \(\text{Re} \beta\), \(\text{Im} \beta\) to \(\alpha\), respectively for the geometrical structure of the set \(S = \{x | \alpha (x) = 0\}\) microhypoellipticity is proved. To illustrate the results here obtained the next example will be proposed. Example. Assume \(S = \{x | f(x) = 0\}\), where \(df(0) \neq 0\), \((\partial f/ \partial x_ 1)(0) = 0\), \((\partial f/ \partial x_ 1)(x) \neq 0\) if \(x \neq 0\). Moreover, let \((\text{Re} \beta)^ 2 + (\text{Im} \beta)^{2\ell} \leq C \alpha\) for some \(C>0\) and \(\ell \in \mathbb{N}\). Then \(P\) is microhypoelliptic in \(\mathbb{R}^ 2\). The main theorems of the author are a further development and generalization of some papers of Hörmander, Fedii, Morimoto, Kajitani and Wakabayashi.