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Maximal growth of the Stein-Wainger oscillatory integral

Cheng Zhang, Zhifei Zhu

Abstract

We establish a precise hierarchy for the maximal growth of the Stein-Wainger oscillatory integral as the regularity of the phase varies over Denjoy-Carleman classes, such as the Gevrey classes and their generalizations. In particular, we resolve a problem posed by Wang--Zhang, motivated by eigenfunction restriction estimates on curves, and also provide a new proof of a theorem of Nagel--Wainger on the Hilbert transform along curves. A key ingredient is the sharp estimate on the growth of a phase near a flat point.

Maximal growth of the Stein-Wainger oscillatory integral

Abstract

We establish a precise hierarchy for the maximal growth of the Stein-Wainger oscillatory integral as the regularity of the phase varies over Denjoy-Carleman classes, such as the Gevrey classes and their generalizations. In particular, we resolve a problem posed by Wang--Zhang, motivated by eigenfunction restriction estimates on curves, and also provide a new proof of a theorem of Nagel--Wainger on the Hilbert transform along curves. A key ingredient is the sharp estimate on the growth of a phase near a flat point.
Paper Structure (27 sections, 24 theorems, 258 equations)

This paper contains 27 sections, 24 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Proof Outline.
  5. Paper Structure.
  6. Notation.
  7. Acknowledgments.
  8. Preliminaries
  9. Proof of Proposition \ref{['prop0']}
  10. General smooth functions
  11. Sharpness
  12. Denjoy--Carleman class
  13. Proof of Theorem \ref{['thm:main3']}
  14. Gevrey class
  15. Sharpness
  16. ...and 12 more sections

Key Result

Proposition 1.1

If $\psi\in C^\alpha(I)$ for some integer $\alpha\ge1$, then $m(\lambda)=O(\log\lambda)$. This bound is sharp: there exists a real-valued $\psi\in C^\alpha(I)$ such that $|m(\lambda)|\approx \log\lambda$.

Theorems & Definitions (41)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2: Bang
  • proof
  • ...and 31 more