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

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.

MSC:

42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 1015.42015
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