$L_x^p\to L_{x,u}^q$ estimates for dilated averages over planar curves

In this paper, we consider the $L_x^p({\mathbb{R}}^2)\to L_{x,u}^q({\mathbb{R}}^2\times [1,2])$ estimate for the operator $T$ along a dilated plane curve $(ut,u\gamma (t))$, where $$Tf(x,u):={\int }_0^{1}f(x_1-ut,x_2-u\gamma (t))\text{d}t,$$ $x:=(x_1,x_2)$ 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 $(\frac{1}{p},\frac{1}{q})\in \mathrm{\square }\cup \{(0,0)\}\cup \{(\frac{2}{3},\frac{1}{3})\}$ and $1+(1+\omega )(\frac{1}{q}-\frac{1}{p})>0$, where the trapezium $\mathrm{\square }:=\{(\frac{1}{p},\frac{1}{q}):\frac{2}{p}-1\le \frac{1}{q}\le \frac{1}{p},\frac{1}{q}>\frac{1}{3p},\frac{1}{q}>\frac{1}{p}-\frac{1}{3}\}$ and $\omega :={\mathrm{lim\hspace{0.17em}sup}}_{t\to 0^{+}}\frac{\ln \mathrm{\mid }\gamma (t)\mathrm{\mid }}{\ln t}$. 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 $T:L_x^p({\mathbb{R}}^2)\to L_x^q({\mathbb{R}}^2)$ 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 $T:L_x^p({\mathbb{R}}^2)\to L_x^q({\mathbb{R}}^2)$.