$L^p$函数的分布函数在端点处的极限性质

Limiting Property of the Distribution Function of $L^p$ Function at Endpoints

中文关键词:  Hardy-Littlewood极大函数  极限行为  分布函数

英文关键词:Hardy-Littlewood maximal function  limiting behavior  distribution function

中文摘要:

      本文考虑了$L^p$函数的分布函数在$0$和$\infty$这两个端点处的极限性质,同时证明了如下两个等式$$\lim_{\lambda \to +\infty}\lambda^p m(\{x:|f(x)|>\lambda\})=0,~~\lim_{\lambda \to 0^+}\lambda^p m(\{x:|f(x)|>\lambda\})=0$$对于任何$f\in L^p(\mathbb{R}^n)$和$1\leq p<\infty$成立. 本结果可自然推广到许多$(p,q)$ 型算子.

英文摘要:

      We consider the limiting property of the distribution function of $L^p$ function at endpoints $0$ and $\infty$ and prove that for $\lambda>0$ the following two equations $$\lim_{\lambda \to +\infty}\lambda^p m(\{x:|f(x)|>\lambda\})=0,~~\lim_{\lambda \to 0^+}\lambda^p m(\{x:|f(x)|>\lambda\})=0$$ hold for $f\in L^p(\mathbb{R}^n)$ with $1\leq p<\infty$. This result is naturally applied to many operators of type $(p,q)$ as well.