  In this paper, for general plane curves $\gamma$ satisfying some suitable
smoothness and curvature conditions, we obtain the single annulus
$L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,\gamma}$
along the variable plane curves $(t,U(x_1, x_2)\gamma(t))$ and the
$L^p(\mathbb{R}^2)$-boundedness of the corresponding maximal functions
$M^\infty_{U,\gamma}$, where $p>2$ and $U$ is a measurable function. The range
on $p$ is sharp. Furthermore, for $1<p\leq 2$, under the additional conditions
that $U$ is Lipschitz and making a $\varepsilon_0$-truncation with $\gamma(2
\varepsilon_0)\leq 1/4\|U\|_{\textrm{Lip}}$, we also obtain similar boundedness
for these two operators $H^{\varepsilon_0}_{U,\gamma}$ and
$M^{\varepsilon_0}_{U,\gamma}$.
