L^2 bounds for a maximal directional Hilbert transform

Given a direction set in Euclidean space, we consider a maximal function for the directional Hilbert transforms associated with the direction set.  For each finite p>1, it is known that this maximal function is bounded on \$L^p\$ if and only if the direction set is finite. This raises the following quantitative problems; 

1) What is a sharp uniform upper bound on the \$L^p\$-operator norm of the maximal function that depends only on the cardinality of direction sets? 

2) Under which geometric assumptions on direction sets can this uniform bound be improved? 

We will study these problems for the p=2 case using polynomial partitioning tools from discrete geometry and an almost-orthogonality principle for the maximal function. This is a joint work with Malabika Pramanik.