A maximal function for families of Hilbert transforms along homogeneous curves

Let $H^{(u)}$ be the Hilbert transform along the parabola $(t, ut^2)$ where $u\in \mathbb R$. For a set $U$ of positive numbers consider the maximal function $\mathcal{H}^U \!f= \sup\{|H^{(u)}\! f|: u\in U\}$. We obtain an (essentially) optimal result for the $L^p$ operator norm of $\mathcal{H}^U$ when $2<p<\infty$. The results are proved for families of Hilbert transforms along more general nonflat homogeneous curves.