On existence of an adapted coordinate system (Q2742939)

Let \(f(x)\) be an infinitely differentiable real function (phase) and \(a\in C_0^\infty(\mathbb R^n)\). The following oscillatory trigonometric integral NEWLINE\[NEWLINEI(t)=\int_{\mathbb R^n}a(x)e^{itf(x)} dx,NEWLINE\]NEWLINE where \( t \gg 1\), is considered. The notion of adapted coordinate system for a smooth phase function \(f(x)\) is given. The existence of adapted coordinate system for an infinite differentiable real phase \(f(x)\) is the main result of the article.