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

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.