The best constant for the centered Hardy-Littlewood maximal inequality. (English) Zbl 1055.42013

Melas, Antonios D.

Ann. Math. (2) 157, No. 2, 647-688 (2003).

Consider the one-dimensional Hardy-Littlewood maximal function \[ Mf(x)=\sup_{h>0}\frac{1}{2h}\int_{x-h}^{x+h}| f(t)| \,dt. \] The author proves the weak type \((1,1)\) inequality \[ | \{x:Mf(x)>\lambda\}| \leq \frac{\sqrt{61}+11}{12\lambda} \| f\| _1. \] As shown by the author in [Trans. Am. Math. Soc. 354, No. 8, 3263–3273 (2002; Zbl 1015.42015)], the constant in this inequality is optimal.

Reviewer: Mehmet Burak Erdogan (Urbana)

MSC:

42B25

Maximal functions, Littlewood-Paley theory

Keywords:

Hardy-Littlewood maximal function

Citations:

Zbl 1015.42015