A sharp estimate for the Hilbert transform along finite order lacunary sets of directions

Let Θ ⊂ S1 be a lacunary set of directions of order D. We show that the maximal directional Hilbert transform HΘf(x):=supv∈Θ|p.v∫Rf(x+tv)dtt| obeys the bounds ||HΘ||Lp→Lp ≃p,D (log#Θ) 1/2, for all 1 < p < ∞. For vector fields vD with range in a lacunary set of order D and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field vD, HvD,1f(x):=p.v.∫|t|≤1f(x+tvD(x))dtt, satisfies the bounds ||HvD,1||Lp→Lp ≲p,D 1 for all 1 < p < ∞. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele