A Sharp Estimate for the Hilbert Transform along Finite Order Lacunary Sets of Directions

Let D be a nonnegative integer and Θ ⊂ S1 be a lacunary set of directions of order D. We show that the Lp norms, 1 < p < ∞, of the maximal directional Hilbert transform in the plane HΘ f (x ) B sup v ∈Θ p.v. ∫ R f (x + tv ) dt t , x ∈ R 2, are comparable to (log #Θ) 2 . For vector elds vD with range in a lacunary set of of order D and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector eld vD , HvD,1 f (x ) B p.v. ∫ |t | ≤1 f (x + tvD (x )) dt t , is Lp -bounded for all 1 < p < ∞. These results extend previous bounds of the rst author with Demeter, and of Guo and Thiele.