The APDE seminar on Monday, 11/18, will be given by Arian Nadjimzadah (UCLA) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Improved bounds for intermediate curved Kakeya sets in R^3.

Abstract: A central problem in harmonic analysis is to understand the L^p bounds of oscillatory integral operators. Bourgain showed that if many wavepackets can be arranged to cluster tightly in space, then the operator has poor L^p bounds. The classical Kakeya Conjecture says that this clustering cannot happen for the extension operator for the paraboloid. At the other extreme, there are operators for which the full length of the wavepackets can cluster near a surface.

In this talk we discuss the intermediate case, where the wavepackets through a small ball can cluster near a surface. Most operators exhibit this behavior. For a large class of such operators, we show improved bounds for the corresponding curved Kakeya problems.

The main tools are Wolff’s hairbrush argument, the multilinear Kakeya inequality of Bennett-Carbery-Tao, and a variant of Wolff’s circular maximal function theorem due to Pramanik-Yang-Zahl. The geometric conditions we will call “coniness’’ and “twistiness’’ play a central role.