带缓和因子的平坦平面曲线的极大函数
Maximal functions associated to flat plane curves with Mitigating factors
摘要：
我们研究了最大算子$\mathbb{M}_σ$的有界性问题与缓和因子的平面曲线，由$$\mathbb{M}_σf(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{0}^{1} f(x-tΓ(s)) \, (κ(s))^σ \, ds\right|,$$定义，其中$κ(s)$表示$\mathbb{R}^2$中的曲线$Γ(s)=(s, g(s)+1), ~g(s) \in C^5[0,1]$的曲率。设$\triangle$是以$P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).$为顶点的闭三角形，本文证明了对于$ (\frac{1}{p}, \frac{1}{q}) \in \left[(\frac{1}{p}, \frac{1}{q}) :(\frac{1}{p}, \frac{1}{q}) \in \triangle \setminus \{P, Q\} \right] \cap \left[(\frac{1}{p}, \frac{1}{q}) :q > max\{σ^{-1},2\} \right]$，存在一个常数$B$，使得$ \|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}. $
英文摘要：
We study the boundedness problem for maximal operators $\mathbb{M}_σ$ associated to flat plane curves with Mitigating factors, defined by $$\mathbb{M}_σf(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{0}^{1} f(x-tΓ(s)) \, (κ(s))^σ \, ds\right|,$$ where $κ(s)$ denotes the curvature of the curve $Γ(s)=(s, g(s)+1), ~g(s) \in C^5[0,1]$ in $\mathbb{R}^2$. Let $\triangle$ be the closed triangle with vertices $P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).$ In this paper, we prove that for $ (\frac{1}{p}, \frac{1}{q}) \in \left[(\frac{1}{p}, \frac{1}{q}) :(\frac{1}{p}, \frac{1}{q}) \in \triangle \setminus \{P, Q\} \right] \cap \left[(\frac{1}{p}, \frac{1}{q}) :q > max\{σ^{-1},2\} \right]$, there is a constant $B$ such that $ \|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}. $
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