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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 maximalfunction $\mathcal{H}^U \!f= \sup\{|H^{(u)}\! f|: u\in U\}$. 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 transformsalong more general nonflat homogeneous curves.

A maximal function for families of Hilbert transforms along homogeneous curves