Analysis and Mathematical Physics

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 any $p> 2d/(d-1)$.  We propose to study this conjecture using Bourgain-Demeter decoupling theorem and incidence estimates for tubes. 

In this talk, we will describe a geometric conjecture, the two-ends Furstenberg conjecture, that would imply Stein's restriction conjecture. We prove it in $R^2$ and obtain a partial result in $R^3$ that implies the restriction estimate in $R^3$ for any $p> 3+1/7$. This restriction estimate implies Wolff's hairbrush Kakeya bound: any Kakeya set in $R^3$ has Hausdorff dimension at least $5/2$.