On a certain theorem of Colin de Verdière
Let \(f: \mathbb{R}^ k \to \mathbb{R}\) be a smooth function. Any smooth function \(\varphi: \mathbb{R}^ k \times \mathbb{R}^ n \to \mathbb{R}\) with \(\varphi(x,0) \equiv f(x)\) is called the deformation of \(f\). We denote by \(C^ \infty_ 0(\mathbb{R}^ k \times \mathbb{R}^ n)\) the set of infinitely smooth functions with compact supports. By the oscillatory integral with phase \(\varphi\) and amplitude \(\alpha \in C^ \infty_ 0(\mathbb{R}^ k \times \mathbb{R}^ n)\) we mean the integral of the form \[ J(t,\varphi,a) = \int_{\mathbb{R}^ k} \alpha(x,s) e^{it \varphi(x,s)} dx. \] In Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, No. 4, 559-576 (1977; Zbl 0409.58011) \textit{Y. Colin de Verdière} formulated the following proposition. Theorem. Let \(\varphi(x,\xi) \in C^ \infty(\mathbb{R}^ k \times \mathbb{R}^ n)\) and with singularities with respect to \(x\) only of simple type or of parabolic one. If \(J(t,\xi)\) is the oscillatory integral with phase \(\varphi\) and amplitude \(\alpha\), then the following inequality holds \[ | J(t,\xi)| \leq c| t|^{-k/2} \sum_{(x,\xi) \in C_ \varphi \cap \text{supp }\alpha} | \text{det }\varphi_{xx}(x,\xi)|^{-1/2}. \] We show that the statement of Colin de Verdière is not true and give the proof of an improved version of Colin de Verdière's estimate for the case where the phase function is a deformation of elliptic singularities.
scientific article; zbMATH DE number 640192
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