二维奇异点的奇点的分解与积分的稳定性
Resolution of Singularities in Two Dimensions and the Stability of Integrals
摘要：
对于以R^2中原点为中心的小圆盘D，D上的光滑实值函数S（x，y），以及正整数，我们考虑D中点的测度，其中|S（x，y）|以及类似的振荡积分。具体地说，我们考虑扰动S（x，y）对这些量的影响。除了固有的兴趣，这些问题是重要的表面支持措施的傅里叶变换的分析。这些问题的复杂和更高维度的类似物也与代数和复几何中的各种问题有关。对于实解析的S（x，y），这个问题已经被研究过，例如Karpushkin，使用双变形理论，以及Phong-Stein-Sturm，他开发了一种通常被称为代数估计方法的方法。在本文中，我们将展示如何使用的奇点的分解算法在两个层面上，随着一些一维范德科普特型引理，提供了另一种方法来处理这样的问题。因此，我们证明了新的估计和定理，这些和相关的数量。此外，由于这些算法适用于所有的光滑函数，定理将适用于所有的光滑函数，而不是早期的实解析结果。
英文摘要：
For a small disk D centered at the origin in R^2, a smooth real-valued function S(x,y) on D, and a positive epsilon, we consider the measure of the points in D where |S(x,y)| < epsilon, as well as oscillatory integral analogues. Specifically, we consider the effect of perturbing S(x,y) on these quantities. Besides being of intrinsic interest, these questions are important in the analysis of Fourier transforms of surface-supported measures. Complex and higher-dimensional analogues of these questions are also connected to various issues in algebraic and complex geometry. For real-analytic S(x,y), this question has been investigated for example by Karpushkin, using versal deformation theory, and by Phong-Stein-Sturm, who developed a method often referred to as the method of algebraic estimates. In this paper, we show how the use of resolution of singularities algorithms in two dimensions, along with some one-dimensional Van der Corput-type lemmas, provides another method for dealing with such questions. As a result, we prove new estimates and theorems for these and related quantities. Furthermore, since these algorithms apply to all smooth functions, the theorems will hold for all smooth functions as opposed to the earlier real-analytic results.
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