Singular Oscillatory Integrals on R^n

TL;DR

This paper establishes new bounds on the measure of polynomial sublevel sets and applies these to derive sharp estimates for singular oscillatory integrals on R^n, advancing understanding in harmonic analysis.

Contribution

It introduces a novel estimate for polynomial sublevel sets in one dimension and uses it to obtain sharp bounds for singular oscillatory integrals in multiple dimensions.

Findings

New estimate for polynomial sublevel set measure in R^1

Sharp bounds for singular oscillatory integrals on R^n

Improved understanding of polynomial behavior in harmonic analysis

Abstract

Let Pd,n denote the space of all real polynomials of degree at most d on R^n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P in Pd,1. Using this estimate, we prove a sharp estimate for a singular oscillatory integral on R^n.