$$L^p$$ -Improving Bounds of Maximal Functions Along Planar Curves

Abstract

In this paper, we study the \(L^p({\mathbb {R}}^2)\)-improving bounds, i.e., \(L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2)\) estimates, of the maximal function \(M_{\gamma }\) along a plane curve \((t,\gamma (t))\), where

$$\begin{aligned} M_{\gamma }f(x_1,x_2):=\sup _{u\in [1,2]}\left| \int _{0}^{1}f(x_1-ut,x_2-u \gamma (t))\,\text {d}t\right| , \end{aligned}$$

and \(\gamma \) is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain \(M_{\gamma }: L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2)\) if \(\left( \frac{1}{p},\frac{1}{q}\right) \in \Delta \cup \{(0,0)\}\) and \(\left( \frac{1}{p},\frac{1}{q}\right) \) satisfying \(1+(1 +\omega )\left( \frac{1}{q}-\frac{1}{p}\right) >0\), where \(\Delta :=\left\{ \left( \frac{1}{p},\frac{1}{q}\right) : \frac{1}{2p}<\frac{1}{q}\le \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \right\} \) and \(\omega :=\limsup _{t\rightarrow 0^{+}}\frac{\ln |\gamma (t)|}{\ln t}\). This result is sharp except for some borderline cases.