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 \(\square :=\{(\frac{1}{p},\frac{1}{q}):\ \frac{2}{p}-1\le \frac{1}{q}\le \frac{1}{p}, \frac{1}{q}>\frac{1}{3p}, \frac{1}{q}>\frac{1}{p}-\frac{1}{3}\}\) 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 \(u\in [1,2].\) The local smoothing phenomenon essentially suggests that integrating over a compact time interval leads to gains in regularity. Since the \(L^q\) integral about u on [1, 2] extends the region of \((\frac{1}{p},\frac{1}{q})\) in the uniform estimate \(T: L_x^p({\mathbb {R}}^2)\rightarrow L_{x}^q({\mathbb {R}}^2),\) we conclude that the operator T has the so called local smoothing phenomenon.
