  Let $M^{(u)}$, $H^{(u)}$ be the maximal operator and Hilbert transform along
the parabola $(t, ut^2) $. For $U\subset(0,\infty)$ we consider $L^p$ estimates
for the maximal functions $\sup_{u\in U}|M^{(u)} f|$ and $\sup_{u\in U}|H^{(u)}
f|$, when $1<p\le 2$. The parabolae can be replaced by more general non-flat
homogeneous curves.
