Sublevel Set Estimates over Global Domains

TL;DR

This paper extends the analysis of oscillatory integrals and sub-level-set estimates from small neighborhoods to global domains like  space, using Newton polyhedra associated with the phase and domain.

Contribution

It introduces a new framework for Newton polyhedra associated with both the phase function and the domain, establishing global analogues of Varchenko's theorem under non-degeneracy conditions.

Findings

Established global sub-level-set estimates for  domains

Developed Newton polyhedra framework for phase and domain

Proved analogues of Varchenko's theorem in global settings

Abstract

Since Varchenko's seminal paper, the asymptotics of oscillatory integrals and related problems have been elucidated through the Newton polyhedra associated with the phase $P$. The supports of those integrals are concentrated on sufficiently small neighborhoods. The aim of this paper is to investigate the estimates of sub-level-sets and oscillatory integrals whose supports are global domains $D$. A basic model of $D$ is $ \mathbb{R}^d$. For this purpose, we define the Newton polyhedra associated with $(P,D)$ and establish analogues of Varchenko's theorem in global domains $D$, under non-degeneracy conditions of $P$.