Oscillatory integrals with polynomial phase and regularity of distributions

TL;DR

This paper establishes sharp bounds for oscillatory integrals with polynomial phase functions over convex sets, and provides dimension-free regularity estimates for densities of polynomial images of certain measures.

Contribution

It proves a conjecture by Carbery and Wright by deriving optimal bounds for oscillatory integrals with polynomial phase over convex sets.

Findings

Dimension-free estimates for densities of polynomial images of measures

Resolution of Carbery and Wright's conjecture on oscillatory integrals

Sharp upper bounds for oscillatory integrals with polynomial phase

Abstract

We obtain dimension-free estimates for the modulus of continuity of densities of polynomial images of $s$-concave and product measures. As a consequence, we settle a conjecture of A. Carbery and J. Wright (2001) on sharp upper bounds for oscillatory integrals over convex sets with polynomial phase.