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

TL;DR

This paper establishes sharp $L^p$-improving bounds for a maximal function along planar curves, extending understanding of such operators under general smoothness and curvature conditions.

Contribution

It provides new $L^p$ to $L^q$ estimates for the maximal function along curves, covering a broad class of curves with specific curvature and smoothness assumptions.

Findings

Established $L^p$ to $L^q$ bounds for the maximal function along curves.

Identified the precise range of exponents where bounds hold, including sharpness results.

Extended previous results to more general curves with curvature conditions.

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 $$M_{\gamma}f(x_1,x_2):=\sup_{u\in [1,2]}\left|\int_{0}^{1}f(x_1-ut,x_2-u \gamma(t))\,\textrm{d}t\right|,$$ and $\gamma$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain $M_{\gamma} : L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ 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 $\Delta:=\{(\frac{1}{p},\frac{1}{q}):\ \frac{1}{2p}<\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \}$ and $\omega:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$. 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.