Bourgain's condition, sticky Kakeya, and new examples

TL;DR

This paper establishes a connection between Bourgain's condition and the sticky Kakeya conjecture for oscillatory integral operators, introduces a geometric characterization, and provides new examples that challenge existing assumptions.

Contribution

It offers a new geometric perspective on Bourgain's condition, reduces sticky Kakeya problems to classical cases, and constructs novel examples in higher dimensions.

Findings

Bourgain's condition reduces sticky Kakeya to classical Kakeya in all dimensions ≥ 3.

Constructed examples show Bourgain's condition does not persist in larger tubes.

First operators satisfying Bourgain's condition with no diffeomorphism to lines.

Abstract

We prove that in all dimensions at least 3 and for any H\"ormander-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 $\delta$-tubes in a $\delta^{1/2}$-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.