$L^p$-improving bounds of maximal functions along planar curves

Abstract

In this paper, we study the $L^p({\mathbb{R}}^2)$-improving bounds, i.e., $L^p({\mathbb{R}}^2)\to L^q({\mathbb{R}}^2)$ estimates, of the maximal function $M_{\gamma }$ along a plane curve $(t,\gamma (t))$, where $$M_{\gamma }f(x_1,x_2):=\underset{u\in [1,2]}{\sup }\left|{\int }_0^{1}f(x_1-ut,x_2-u\gamma (t))\text{d}t\right|,$$ and $\gamma$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain $M_{\gamma }:L^p({\mathbb{R}}^2)\to L^q({\mathbb{R}}^2)$ if $(\frac{1}{p},\frac{1}{q})\in \Delta \cup \{(0,0)\}$ and $(\frac{1}{p},\frac{1}{q})$ satisfying $1+(1+\omega )(\frac{1}{q}-\frac{1}{p})>0$, where $\Delta :=\{(\frac{1}{p},\frac{1}{q}):\frac{1}{2p}<\frac{1}{q}\le \frac{1}{p},\frac{1}{q}>\frac{3}{p}-1\}$ and $\omega :={\mathrm{limsup}}_{t\to 0^{+}}\frac{\ln |\gamma (t)|}{\ln t}$. This result is sharp except for some borderline cases. As Hickman stated in [J. Funct. Anal. 270 (2016), pp. 560--608], this is a very different situation.