Bourgain's condition, sticky Kakeya, and new examples

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.