  In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the
Hilbert transform $H^{\gamma}$ along a variable plane curve $(t,u(x_1,
x_2)\gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm,
and $\gamma$ is a general curve satisfying some suitable smoothness and
curvature conditions.
