Limiting Property of the Distribution Function of $L^p$ Function at Endpoints
Received:March 15, 2015  Revised:July 08, 2015
Key Words: Hardy-Littlewood maximal function   limiting behavior   distribution function  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11471309; 11271162; 11561062).
Author NameAffiliation
Shaozhen XU School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China 
Dunyan YAN School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China 
Hits: 2760
Download times: 2516
Abstract:
      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.
Citation:
DOI:10.3770/j.issn:2095-2651.2016.02.006
View Full Text  View/Add Comment