Hardy-Littlewood maximal theorem

There is a constant $K>0$ such that for each Lebesgue integrable function $f\in L^1({ℝ}^n)$, and each $t>0$,

$$m(\{x:Mf(x)>t\})\le \frac{K}{t}{\parallel f\parallel }_1=\frac{K}{t}{\int }_{{ℝ}^n}|f(x)|𝑑x,$$

where $Mf$ is the Hardy-Littlewood maximal function of $f$.

Remark. The theorem holds for the constant $K=3^n$.

Title	Hardy-Littlewood maximal theorem
Canonical name	HardyLittlewoodMaximalTheorem
Date of creation	2013-03-22 13:27:33
Last modified on	2013-03-22 13:27:33
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	4
Author	Koro (127)
Entry type	Theorem
Classification	msc 28A15
Classification	msc 28A25
Related topic	HardyLittlewoodMaximalOperator
Defines	Hardy-Littlewood theorem