On a certain theorem of Colin de Verdière
type: Item
instance of: scholarly article
title: On a certain theorem of Colin de Verdière
zbMATH Open document ID: 0805.58015
published in: Russian Mathematics
publication date: 1994-10-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
zbMATH DE Number: 640192
zbMATH Keywords: Colin de Verdière's Theorem
zbMATH Keywords: deformation
zbMATH Keywords: smooth functions
zbMATH Keywords: oscillatory integral
author: Isroil A. Ikromov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 640192