  In this paper, we consider the $L_x^p(\mathbb{R}^2)\rightarrow
L_{x,u}^q(\mathbb{R}^2\times [1,2])$ estimate for the operator $T$ along a
dilated plane curve $(ut,u\gamma(t))$, where
  $$Tf(x,u):=\int_{0}^{1}f(x_1-ut,x_2-u \gamma(t))\,\textrm{d}t,$$
  $x:=(x_1,x_2)$ and $\gamma$ is a general plane curve satisfying some suitable
smoothness and curvature conditions. We show that $T$ is $L_x^p(\mathbb{R}^2)$
to $L_{x,u}^q(\mathbb{R}^2\times [1,2])$ bounded whenever
$(\frac{1}{p},\frac{1}{q})\in \square \cup \{(0,0)\}\cup
\{(\frac{2}{3},\frac{1}{3})\}$ and $1+(1 +\omega)(\frac{1}{q}-\frac{1}{p})>0$,
where the trapezium $\square:=\{(\frac{1}{p},\frac{1}{q}):\
\frac{2}{p}-1\leq\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{1}{3p},
\frac{1}{q}>\frac{1}{p}-\frac{1}{3}\}$ and $\omega:=\limsup_{t\rightarrow
0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$. This result is sharp except for some
borderline cases. On the other hand, in a smaller $(\frac{1}{p},\frac{1}{q})$
region, we also obtain the almost sharp estimate $T :
L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$ uniformly for $u\in
[1,2]$. These results imply that the operator $T$ has the so called local
smoothing phenomenon, i.e., the $L^q$ integral about $u$ on $[1,2]$ extends the
region of $(\frac{1}{p},\frac{1}{q})$ in uniform estimate $T :
L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$.
