Resolution of Singularities in Two Dimensions and the Stability of Integrals

TL;DR

This paper introduces a new method using resolution of singularities in two dimensions to analyze the stability of integrals involving smooth functions, extending previous real-analytic results to all smooth functions.

Contribution

The paper develops a novel approach employing resolution of singularities algorithms combined with Van der Corput lemmas for analyzing integrals and measures related to smooth functions.

Findings

Proves new estimates for measure and oscillatory integrals.

Extends results from real-analytic to all smooth functions.

Provides a versatile method applicable in algebraic and geometric contexts.

Abstract

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.