Germ hypoellipticity and loss of derivatives (Q2845057)

Let \(\mathcal{P}\) be the pseudodifferential operator on \(\mathbb{R}^3 = \{(t,x_1,x_2)\}\) defined by NEWLINE\[NEWLINE \mathcal{P} = L_q \overline{L}_q + \overline{L}_q t^{2k} L_q + Q^2, NEWLINE\]NEWLINE where \(L_q = D_t + i t^{q-1} \sqrt{-\Delta_x}\) with \(\Delta_x\) the Laplacian in the \(x\) variables, \(q\) is an even integer, and \(Q = x_1 D_2 - x_2 D_1\). In this paper, the author shows that the operator \(\mathcal{P}\) is not hypoelliptic (that is, there exist distributions \(u \notin C^\infty\) such that \(\mathcal{P} u \in C^\infty\)), but it is hypoelliptic in the sense of germs. Specifically, let \(K\) be a disk of the form \(\{(0, x_1, x_2) : x_1^2 + x_2^2 \leq c^2\}\) for some \(c \geq 0\). It is shown that for each open neighborhood \(U\) of \(K\), there exists a smaller open neighborhood \(V\) such that, if \(u\) is any distribution on \(U\) such that \(\mathcal{P}u \in C^\infty(U)\), then we must have \(u \in C^\infty(V)\).NEWLINENEWLINEThe proof is based on ideas from [\textit{A. Bove} et al., Math. Res. Lett. 13, No. 5--6, 683--701 (2006; Zbl 1220.35021)] and proceeds via obtaining a priori estimates.