\(L^p\)-Improving Bounds of Maximal Functions Along Planar Curves

In this paper, we study the \(L^p({\mathbb {R}}^2)\) $L^p({\mathbb{R}}^2)$ -improving bounds, i.e., \(L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2)\) $L^p({\mathbb{R}}^2)\to L^q({\mathbb{R}}^2)$ estimates, of the maximal function \(M_{\gamma }\) $M_{\gamma }$ along a plane curve \((t,\gamma (t))\) $(t,\gamma (t))$ , where \(\begin{aligned} M_{\gamma }f(x_1,x_2):=\sup _{u\in [1,2]}\left| \int _{0}^{1}f(x_1-ut,x_2-u \gamma (t))\,\text {d}t\right| , \end{aligned}\) $$\begin{array}{c}\hfill M_{\gamma }f(x_1,x_2):=\underset{u\in [1,2]}{sup}\left|{\int }_0^1,f,(x_1-ut,x_2-u\gamma (t)),,\text{d},t\right|,\end{array}$$ and \(\gamma \) $\gamma$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain \(M_{\gamma }: L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2)\) $M_{\gamma }:L^p({\mathbb{R}}^2)\to L^q({\mathbb{R}}^2)$ if \(\left( \frac{1}{p},\frac{1}{q}\right) \in \Delta \cup \{(0,0)\}\) $\left(\frac{1}{p},,,\frac{1}{q}\right)\in \mathrm{\Delta }\cup \{(0,0)\}$ and \(\left( \frac{1}{p},\frac{1}{q}\right) \) $\left(\frac{1}{p},,,\frac{1}{q}\right)$ satisfying \(1+(1 +\omega )\left( \frac{1}{q}-\frac{1}{p}\right) >0\) $1+(1+\omega )\left(\frac{1}{q},-,\frac{1}{p}\right)>0$ , where \(\Delta :=\left\{ \left( \frac{1}{p},\frac{1}{q}\right) : \frac{1}{2p}<\frac{1}{q}\le \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \right\} \) $\mathrm{\Delta }:=\left\{\left(\frac{1}{p},,,\frac{1}{q}\right),:,\frac{1}{2p},<,\frac{1}{q},\le ,\frac{1}{p},,,\frac{1}{q},>,\frac{3}{p},-,1\right\}$ and \(\omega :=\limsup _{t\rightarrow 0^{+}}\frac{\ln |\gamma (t)|}{\ln t}\) $\omega :={lim\; sup}_{t\to 0^{+}}\frac{ln|\gamma (t)|}{lnt}$ . This result is sharp except for some borderline cases.