On the maximal directional hilbert transform in three dimensions

We establish the sharp growth rate, in terms of cardinality, of the Lp norms of the maximal Hilbert transform Hω along finite subsets of a finite order lacunary set of directions ω c R3, answering a question of Parcet and Rogers in dimension n = 3. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of 2D angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.