Bourgain's condition, sticky Kakeya, and new examples

2025/11/14 by Arian Nadjimzadah, Nadjimzadah, Arian
Mathematics · #Advanced Harmonic Analysis Research #Holomorphic and Operator Theory #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.2511.10918

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.

Citations

• Introduction to the proof of the Kakeya conjecture
• Curved Kakeya sets and Nikodym problems on manifolds
• Spectral Dimension Flow, Localized Non-Orientable Horizon Defects, and Emergent Time in Black Holes: A Geometry-Driven Effective Framework
• Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities
• Oscillatory integral operators and variable Schrödinger propagators: beyond the universal estimates
• Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems
• Sticky Kakeya sets and the sticky Kakeya conjecture
• A dichotomy for Hörmander-type oscillatory integral operators
• The Bochner-Riesz problem: an old approach revisited
• Sharp L^p estimates for oscillatory integral operators of arbitrary signature
• Improved bounds for the Kakeya maximal conjecture in higher dimensions
• Sharp estimates for oscillatory integral operators via polynomial\n partitioning
• New bounds on Kakeya problems

Related

• A dichotomy for Hörmander-type oscillatory integral operators
• Curved Kakeya problems and the projective geometry of paths
• Curved Kakeya sets and Nikodym problems on manifolds
• Curved Kakeya sets for generic phases in odd dimensions
• Bounds on oscillatory integral operators based on multilinear estimates