论文标题
一种在非振荡积分方面界限振荡积分的方法
A method for bounding oscillatory integrals in terms of non-oscillatory integrals
论文作者
论文摘要
我们描述了一种基本方法，用于在相关的非振荡性积分方面界定一维振荡的积分。从适当的意义上讲，获得的界限是有效的，在阶段的扰动下表现良好。结果，对于$ n $维振荡的积分，起源于临界点，我们可以在径向方向上应用一维估计值，然后整合结果，从而在与阶段相关的超级式设置的度量方面获得$ n $ n $二维振荡性积分的自然界限。为了说明，我们提供了几类示例，包括相位函数具有无限顺序的关键点的情况。
We describe an elementary method for bounding a one-dimensional oscillatory integral in terms of an associated non-oscillatory integral. The bounds obtained are efficient in an appropriate sense and behave well under perturbations of the phase. As a consequence, for an $n$-dimensional oscillatory integral with a critical point at the origin, we may apply the one-dimensional estimates in the radial direction and then integrate the result, thereby obtaining natural bounds for the $n$-dimensional oscillatory integral in terms of the measures of the sublevel sets associated with the phase. To illustrate, we provide several classes of examples, including situations where the phase function has a critical point at which it vanishes to infinite order.
