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 $L_x^p(\mathbb{R}^2) \rightarrow L_x^q L_u^r(\mathbb{R}^2 \times [1, 2])$ (space-time) estimates for $T$, whenever $(\frac{1}{p},\frac{1}{q})$ satisfy $$\max\left\{0, \frac{1}{2p} - \frac{1}{2r}, \frac{3}{p} - \frac{r+2}{r}\right\} < \frac{1}{q} \leq \frac{1}{p} < \frac{r+1}{2r}$$ and $$1 + (1 + \omega)\left(\frac{1}{q} - \frac{1}{p}\right) > 0,$$ where $r \in [1, \infty]$ and $\omega := \limsup_{t \rightarrow 0^+} \frac{\ln|\gamma(t)|}{\ln t}$. These results are sharp, except for certain borderline cases. Additionally, we examine the $L_x^p(\mathbb{R}^2) \rightarrow L_u^r L_x^q(\mathbb{R}^2 \times [1, 2])$ (time-space) estimates for $T$, which are especially almost sharp when $p=2$ or $p\in [1, \frac{3}{2}]\cup [4, \infty]$.