Hypoellipticity of some degenerate subelliptic operators (Q1273948)

The author studies the following problem: Let \(L_1\) and \(L_2\) be two second-order differential operators, respectively in \(\mathbb{R}^n_x\) and \(\mathbb{R}^m_y\), which satisfy some subelliptic estimate, and \(\lambda=\lambda(x)\) a nonnegative function with a zero of infinite order at the origin, but with all other zeros of finite order. Put \(L=L_1+ \lambda(x)L_2\). Then: Theorem: Let \(u\) be a distribution in \(\mathbb{R}^n_x\times \mathbb{R}^m_y\), such that \(\varphi Lu\) is in \(H^s\), \((s\) given), for all \(C^\infty\)-functions \(\varphi\) with compact support in \(\Omega\) (some given open set in \(\mathbb{R}^n_x \times\mathbb{R}^m_y)\). Then \(\varphi u\) is in \(H^s\), for all \(C^\infty\)-functions \(\varphi\) with compact support in \(\Omega\). This problem is motivated by the study of \(\overline \partial_b\) on some model domains in \(\mathbb{C}^2\), with infinite type points. The main part in the proof is the Section 2, where the author proves a series of lemmas, giving precise a-priori estimates. In section 3, the proof is finished with the use of suitable partially smoothing operators.