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Bourgain's condition, sticky Kakeya, and new examples

Arian Nadjimzadah

TL;DR

The paper develops a geometric framework for Bourgain's condition in Hörmander-type oscillatory integrals, proving a new invariant, diffeomorphism-based characterization that reduces sticky curved Kakeya problems to the classical sticky Kakeya problem in dimension $n$ when Bourgain's condition holds. It introduces a discretization scheme for sticky $\phi$-Kakeya sets and proves that discretized stickiness implies Hausdorff-dimension bounds, enabling a 3D sticky reduction result for the curved setting. A key contribution is the construction of the $\tan$-example, which satisfies Bourgain's condition yet cannot be diffeomorphically straightened to lines across certain scales, revealing intrinsic obstructions to a general sticky reduction and signaling the need for new ideas beyond the Wang–Zahl paradigm. Together, these results support the Guo–Wang–Zhang conjecture in the sticky regime and illuminate the geometry of wavepackets behind oscillatory integral bounds, with implications for restriction and Bochner–Riesz phenomena in higher dimensions.

Abstract

We prove that in all dimensions at least 3 and for any Hörmander-type oscillatory integral operator satisfying Bourgain's condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture. This supports a conjecture of Guo-Wang-Zhang, that an operator satisfies the same $L^p$ bounds as in the restriction conjecture exactly when it satisfies Bourgain's condition. Our result follows from a new geometric characterization of Bourgain's condition based on the structure of curved $δ$-tubes in a $δ^{1/2}$-tube. We find examples in all dimensions at least 3 which show this property does not persist in a larger tube, and in particular these are the first operators satisfying Bourgain's condition for which there is no diffeomorphism taking the corresponding families of curves to lines. This suggests that a general to sticky reduction in the spirit of Wang-Zahl needs substantial new ideas. We expect these examples to provide a good starting point.

Bourgain's condition, sticky Kakeya, and new examples

TL;DR

The paper develops a geometric framework for Bourgain's condition in Hörmander-type oscillatory integrals, proving a new invariant, diffeomorphism-based characterization that reduces sticky curved Kakeya problems to the classical sticky Kakeya problem in dimension when Bourgain's condition holds. It introduces a discretization scheme for sticky -Kakeya sets and proves that discretized stickiness implies Hausdorff-dimension bounds, enabling a 3D sticky reduction result for the curved setting. A key contribution is the construction of the -example, which satisfies Bourgain's condition yet cannot be diffeomorphically straightened to lines across certain scales, revealing intrinsic obstructions to a general sticky reduction and signaling the need for new ideas beyond the Wang–Zahl paradigm. Together, these results support the Guo–Wang–Zhang conjecture in the sticky regime and illuminate the geometry of wavepackets behind oscillatory integral bounds, with implications for restriction and Bochner–Riesz phenomena in higher dimensions.

Abstract

We prove that in all dimensions at least 3 and for any Hörmander-type oscillatory integral operator satisfying Bourgain's condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture. This supports a conjecture of Guo-Wang-Zhang, that an operator satisfies the same bounds as in the restriction conjecture exactly when it satisfies Bourgain's condition. Our result follows from a new geometric characterization of Bourgain's condition based on the structure of curved -tubes in a -tube. We find examples in all dimensions at least 3 which show this property does not persist in a larger tube, and in particular these are the first operators satisfying Bourgain's condition for which there is no diffeomorphism taking the corresponding families of curves to lines. This suggests that a general to sticky reduction in the spirit of Wang-Zahl needs substantial new ideas. We expect these examples to provide a good starting point.

Paper Structure

This paper contains 21 sections, 10 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The wavepacket geometry of $T^\lambda_\phi$ and Kakeya sets of curves
  3. Geometric characterization of Bourgain's condition
  4. Sticky $\phi$-Kakeya sets, and a reduction to classical sticky Kakeya sets under Bourgain's condition
  5. Sharpness examples for Theorem \ref{['thm: geometric characterization of Bourgain']} and barriers to a general to sticky reduction
  6. Organization of article
  7. Discussion and future work
  8. Thanks
  9. Proof of Proposition \ref{['prop: mild reformulation Bourgain']}
  10. Forward implication
  11. Reverse implication
  12. Proof of Theorem \ref{['thm: geometric characterization of Bourgain']}
  13. Forward implication
  14. Reverse implication
  15. Discretizing sticky $\phi$-Kakeya sets and Proposition \ref{['prop: discretized implies dimension']}
  16. ...and 6 more sections

Key Result

Theorem 1.8

A Hörmander-type phase function $\phi$ satisfies Bourgain's condition if and only if there exist smooth maps $F_{\xi_0,v_0} : M \to \mathbb{R}^n$, $\Xi_{\xi_0,v_0} : \Sigma \to \mathbb{R}^{n-1}$, and $V_{\xi_0,v_0} : \Sigma \times \mathcal{V} \to \mathbb{R}^{n-1}$ depending smoothly on $(\xi_0,v_0)

Figures (2)

  • Figure 1.1: Cartoon of Theorem \ref{['thm: geometric characterization of Bourgain']}.
  • Figure 1.2: Cartoon of Proposition \ref{['prop: tan example curved']}.

Theorems & Definitions (31)

  • Definition 1.2: Bourgain's condition
  • Conjecture 1.3: Hörmander dichotomy conjecture
  • Definition 1.4: $\phi$-curves and $(\delta,\phi)$-tubes
  • Definition 1.5: $\phi$-Kakeya set
  • Conjecture 1.6: Curved Kakeya conjecture for Bourgain's condition
  • Remark 1.7
  • Theorem 1.8: Geometric characterization of Bourgain's condition
  • Remark 1.9
  • Proposition 1.10: Mild reformulation of Bourgain's condition
  • Example 1.11: Proposition \ref{['prop: mild reformulation Bourgain']} for the restriction and Bochner--Riesz examples
  • ...and 21 more