$L^2$ Bounds for a maximal directional Hilbert transform

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