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Oscillatory integrals with polynomial phase and regularity of distributions

Egor Kosov

TL;DR

This work establishes dimension-free, sharp bounds for the regularity of densities of polynomial images under $s$-concave and product measures, resolving the Carbery–Wright conjecture on oscillatory integrals with polynomial phase over convex domains. The authors develop a framework that connects oscillatory integrals to image-measure regularity via the modulus of continuity functional $\sigma$, leveraging one-dimensional van der Corput-type results, localization, and properties of $s$-concave and log-concave measures. They provide bounds for general polynomials on the unit cube and extend to product and high-dimensional settings, including dimension-free estimates that depend only on polynomial degree and structure, not on ambient dimension. Consequently, the results yield dimension-free total-variation and Kantorovich-distance bounds in convergence problems and invariance principles for polynomial mappings of random vectors, with broad applicability to probabilistic and harmonic-analytic questions involving polynomial phases.

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.

Oscillatory integrals with polynomial phase and regularity of distributions

TL;DR

This work establishes dimension-free, sharp bounds for the regularity of densities of polynomial images under -concave and product measures, resolving the Carbery–Wright conjecture on oscillatory integrals with polynomial phase over convex domains. The authors develop a framework that connects oscillatory integrals to image-measure regularity via the modulus of continuity functional , leveraging one-dimensional van der Corput-type results, localization, and properties of -concave and log-concave measures. They provide bounds for general polynomials on the unit cube and extend to product and high-dimensional settings, including dimension-free estimates that depend only on polynomial degree and structure, not on ambient dimension. Consequently, the results yield dimension-free total-variation and Kantorovich-distance bounds in convergence problems and invariance principles for polynomial mappings of random vectors, with broad applicability to probabilistic and harmonic-analytic questions involving polynomial phases.

Abstract

We obtain dimension-free estimates for the modulus of continuity of densities of polynomial images of -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.

Paper Structure

This paper contains 19 sections, 26 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Oscillatory integrals and van der Corput lemma
  3. Connection to fractional regularity of image measures
  4. The case of product measures
  5. Convergence of random variables
  6. Preliminaries and notation
  7. Notation for constants
  8. Measures and functions of bounded variation
  9. Log-concave and $s$-concave measures
  10. Localization lemma with several constraints
  11. Isotropic constant and isotropic measures
  12. Poincaré inequality for an isotropic log-concave measure
  13. Derivatives of log-concave functions
  14. One-dimensional van der Corput type theorem for fractional regularity
  15. Polynomial images of $s$-concave measures
  16. ...and 4 more sections

Key Result

Theorem 1.2

There exists an absolute constant $C>0$ such that for all $n, d\in \mathbb{N}$, for every convex body $K\subset \mathbb{R}^n$ of volume $1$, and for every polynomial $f\in \mathcal{P}_d(\mathbb{R}^n)$, one has where $\mu_K$ denotes the restriction of the Lebesgue measure to $K$ and In other words, when $f$ is a non-constant polynomial of degree at most $d$, the distribution density $\varrho_f$ o

Theorems & Definitions (53)

  • Conjecture 1.1: see CarWr
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: see GM22
  • Theorem 2.1: see Kos or Kos-MS
  • Definition 2.2: see BGVV
  • ...and 43 more