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 $T : L_x^p(\mathbb{ R}^2)\rightarrow L_{ x}^q(\mathbb{ R}^2)$.
Categories: math.CA