Jongchon Kim

Title: $L^2$ bounds for the maximal directional Hilbert transform

Abstract: We give essentially sharp bounds for the $L^2$ operator norm of the maximal directional Hilbert transform associated with N arbitrary directions in dimensions larger than 2. The proof uses an almost-orthogonality principle and polynomial partitioning tools from incidence geometry. In dimension 2, we provide a sharp $L^2$ bound for a certain direction set which are neither lacunary nor uniformly distributed. This is a joint work with Malabika Pramanik.