Mixed norm estimates for dilated averages over planar curves

Abstract

In this paper, we investigate the mixed norm estimates for the operator $T$associated with a dilated plane curve $(ut,u{\hat{I}}^3(t))$, defined by \[ Tf(x, u) := \int_{0}^{1} f(x_1 - ut, x_2 - uγ(t)) \, dt, \] where $x:=(x_1,x_2)$ and ${\hat{I}}^3$ 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 \[ \max\left\{0, \frac{1}{2p} - \frac{1}{2r}, \frac{3}{p} - \frac{r+2}{r}\right\} < \frac{1}{q} \leq \frac{1}{p} < \frac{r+1}{2r} \] and $$1+(1+\ddot{I}‰)\left(\frac{1}{q}-\frac{1}{p}\right)>0,$$ where $r\in [1,\infty ]$ and $\ddot{I}‰:={\mathrm{limsup}}_{t\to 0^{+}}\frac{\ln |{\hat{I}}^3(t)|}{\ln t}$. 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 $p\in [1,\frac{3}{2}]\cup [4,\infty ]$.