Lifting curved Kakeya sets to linear Kakeya sets

Abstract

We prove that many curved Kakeya problems, originally motivated by Hörmander's oscillatory integral problem, lift to the classical Kakeya problem in higher dimensions. As a consequence, if the classical Kakeya set conjecture were true in all dimensions, the families of curves whose curved Kakeya sets have full dimension are dense among Hörmander-type families. This is in contrast to the nowhere dense Bourgain's condition, which is a necessary condition for best-case curved Kakeya maximal function estimates. Furthermore the linear Kakeya set conjecture would imply a fairly complete understanding of Kakeya sets of quadratic curves, as first studied systematically by Wisewell. These results give a better understanding of a question posed by Guo--Guth--Nadjimzadah--Shen--Zhang, and they more generally show that the classical Kakeya conjecture in high dimensions rests on a broad range of curved Kakeya problems in lower dimensions.