Mixed norm estimates for dilated averages over planar curves

In this paper, we investigate the mixed norm estimates for the operator $T$associated with a dilated plane curve $(ut,u\gamma (t))$, defined by $$Tf(x,u):={\int }_0^{1}f(x_1-ut,x_2-u\gamma (t))dt,$$ where $x:=(x_1,x_2)$ and $\gamma$ is a general plane curve satisfying appropriate smoothness and curvature conditions. More precisely, we establish the $L_x^p({\mathbb{R}}^2)\to L_x^q{L}_u^r({\mathbb{R}}^2\times [1,2])$ (space-time) estimates for $T$, whenever $(\frac{1}{p},\frac{1}{q})$ satisfy $$\max \{0,\frac{1}{2p}-\frac{1}{2r},\frac{3}{p}-\frac{r+2}{r}\}<\frac{1}{q}\le \frac{1}{p}<\frac{r+1}{2r}$$ and $$1+(1+\omega )(\frac{1}{q}-\frac{1}{p})>0,$$ where $r\in [1,\mathrm{\infty }]$ and $\omega :={\mathrm{lim\hspace{0.17em}sup}}_{t\to 0^{+}}\frac{\ln \mathrm{\mid }\gamma (t)\mathrm{\mid }}{\ln t}$. These results are sharp, except for certain borderline cases. Additionally, we examine the $L_x^p({\mathbb{R}}^2)\to L_u^r{L}_x^q({\mathbb{R}}^2\times [1,2])$ (time-space) estimates for $T$, which are especially almost sharp when $p=2$ or $p\in [1,\frac{3}{2}]\cup [4,\mathrm{\infty }]$.