Uniform estimates of some oscillating integrals. (Russian) Zbl 0679.58047

Karpushkin, V. N.

Sib. Mat. Zh. 30, No. 2(174), 90-101 (1989).

Consider the oscillatory integral \(\int_{{\mathbb{R}}^ n}e^{i\tau f}\phi dx\), where \(\phi\) is concentrated in a neighborhood of the origin and f: \({\mathbb{R}}^ n\to {\mathbb{R}}\) is analytic in a neighborhood of the origin with a critical point there. As \(\tau\) \(\to \infty\), the integral is of order \(\tau^{\beta}(\ln \tau)^ p\), where \(\beta\leq 0\) and p is a natural number. The author has shown previously [J. Sov. Math. 35, 2809- 2826 (1986); translation from Tr. Semin. Im. I. G. Petrovskogo 10, 150- 169 (1984; Zbl 0569.58033)] that when the dimension n equals 2, this estimate is uniform over small analytic perturbations of f in a neighborhood of the origin. This paper sharpens the result in the special case that \(p=0\).

Reviewer: H.Boas

MSC:

58J40	Pseudodifferential and Fourier integral operators on manifolds
58C25	Differentiable maps on manifolds
58K99	Theory of singularities and catastrophe theory

Keywords:

uniform estimate; oscillatory integral; analytic perturbations