Bounds for volumes of sub-level sets of polynomials and applications

TL;DR

This paper provides explicit bounds on the volumes of polynomial sub-level sets and explores their applications in analyzing the decay of oscillatory integrals and the convergence of singular integrals.

Contribution

It introduces explicit exponents for volume estimates of polynomial sub-level sets and applies these to problems in oscillatory and singular integral analysis.

Findings

Derived explicit volume bounds for polynomial sub-level sets.

Applied bounds to analyze decay rates of oscillatory integrals.

Demonstrated convergence criteria for singular integrals.

Abstract

In this paper, we present some explicit exponents in the estimates for the volumes of sub-level sets of polynomials on bounded sets, and applications to the decay of oscillatory integrals and the convergent of singular integrals.