L2 bounds for a maximal directional Hilbert transform

Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for $L^2$ operator norm dimensions 3 and higher. The ingredients proof consist polynomial partitioning tools from incidence geometry almost-orthogonality principle $H_{\Omega}$. latter can also be used analyze special sets $\Omega$, derive estimates corresponding that are typically stronger than bound mentioned above. A number such examples discussed.