L² bounds for a maximal directional Hilbert transform

Given any finite direction set Ω of cardinality N in Euclidean space, we consider the maximal directional Hilbert transform H_Ω associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on N, for the L² operator norm of H_Ω in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for H_Ω. The latter principle can also be used to analyze special direction sets Ω and derive sharp L² estimates for the corresponding operator H_Ω that are typically stronger than the uniform L² bound mentioned above. A number of such examples are discussed.