Geometry Seminar

Suppose f is a function with Fourier transform supported on the unit sphere in R^d, Elias Stein conjectured in the 1960s that the L^p norm of f is bounded by the L^p norm of its Fourier transform, for p> 2d/(d-1).  We propose to study this conjecture using Bourgain-Demeter decoupling theorems and incidences estimates for tubes.  

 

In this talk, we will describe a geometric conjecture on the number of incidences for tubes that would imply Stein's restriction conjecture. We prove this geometric conjecture in R² and use it to prove a restriction estimate in R³ for p> 3+ 1/7, which implies Wolff's hairbrush Kakeya estimate (i.e. any Kakeya set in R³ has Hausdorff dimension at least 5/2).