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 $(ut, uγ(t))$, defined by \[ Tf(x, u) := \int_{0}^{1} f(x_1 - ut, x_2 - uγ(t)) \, dt, \] where $ x := (x_1, x_2) $ and $γ$ 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 $$1 + (1 + ω)\left(\frac{1}{q} - \frac{1}{p}\right) > 0,$$ where $ r \in [1, \infty] $ and $ ω:= \limsup_{t \rightarrow 0^+} \frac{\ln|γ(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 $p\in [1, \frac{3}{2}]\cup [4, \infty]$.