$L^p$$ -沿平面曲线最大函数边界的改进

Naijia Liu, Haixia Yu
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摘要

在本文中,我们研究了 \(L^p({\mathbb {R}}^2)\)-improving bounds,即、\沿着平面曲线 \((t,\gamma (t))\) 的最大函数 \(M_{\gamma }\) 的 L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2) 估计值,其中 $$\begin{aligned}M_{\gamma }f(x_1,x_2):=\sup _{u\in [1,2]}\left| \int _{0}^{1}f(x_1-ut,x_2-u \gamma (t))\,\text {d}t\right| , \end{aligned}$$ 而 \(\gamma \) 是一条一般的平面曲线,满足一些合适的光滑度和曲率条件。我们得到 \(M_\{gamma }:L^p({\mathbb {R}}^2)\rightarrow L^q({\mathbb {R}}^2)\) if ((left(\frac{1}{p},\frac{1}{q}\right) \in \Delta \cup \{(0、0)和(left(\frac{1}{p},\frac{1}{q}/right))满足(1+(1 +\omega)left(\frac{1}{q}-\frac{1}{p}/right) >;0), where \(\Delta :=\left\{ \left( \frac{1}{p},\frac{1}{q}\right) :\frac{1}{2p}<\frac{1}{q}\le \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \right\}\)和 (omega :=limsup _{t\rightarrow 0^{+}}\frac{ln |\gamma (t)|}\{ln t}/)。除了一些边缘情况,这个结果是尖锐的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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

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.

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