Remark on extensions of the Watermelon theorem
In a recent work [Ann. Inst. Fourier 43, No. 5, 1223-1251 (1993; Zbl 0804.35004)] \textit{L. Hörmander} has given some refinements of Kashiwara's Watermelon Theorem. The purpose of this note is to give some slight improvements of one of the results of Hörmander. Let \(u\) be a distribution (or a hyperfunction) on \(\mathbb{R}^n\), \(n \geq 2\), with compact support \(K\). Let \(y_0 \in K\) and let \(\eta_0 \in \mathbb{R}^n\), \(|\eta_0 |= 1\), be an exterior normal vector of \(K\) at \(y_0\) in the \(C^2\) sense. In other words, there exists a real \(C^2\) function, defined in a neighborhood of \(y_0\) such that \(h(y_0) = 0\), \(dh (y_0) = \eta_0\), and \(h(y) \leq 0\) for all \(y\) in \(K\) which are in a neighborhood of \(y_0\). After an analytic change of variables (a convexification) and a truncation away from \(y_0\), we may assume that \((y - y_0) \cdot \eta_0 |_K \leq 0\). Kashiwara's Watermelon theorem then states that if \(\xi_0 \in \mathbb{R}^n \backslash \{0\}\), \(\lambda \in \mathbb{R}\), \((y_0, \xi_0) \notin WF_a (u)\), then \((y_0, \xi_0 + \lambda \eta_0) \notin WF_a\). Here \(WF_a\) is the analytic wavefront set. The following result is a slight refinement of the first half of Corollary 2.7 of [loc. cit.]. Theorem. Assume that there is a sequence \(\eta_j \in \mathbb{R}^n\), \(j = 1,2, \dots\), with \(\eta_j \to \eta_0\), such that \[ \begin{aligned} {\eta_j - \eta_0 \over |\eta_j - \eta_0 |} \to \nu, \quad & j \to \infty, \\ (y - y_0) \cdot \eta_j |_K - \delta_j \leq 0, \quad & \text{where } \delta_j > 0, \;\delta_j \to 0, \\ {\delta_j \over |\eta_j - \eta_0 |^2} \to 0, \quad & j \to \infty. \end{aligned} \] Then if \(\xi_0 \neq 0\), \((y_0, \xi_0) \notin WF_a (u)\), we have \((y_0, \xi_0 + \lambda \nu) \notin WF_a(u)\), for all \(\lambda \in \mathbb{R}\).
scientific article; zbMATH DE number 817697
scientific article