Remarks on the Hardy-Littlewood maximal function. (English) Zbl 0892.42010

Aldaz, J. M.

Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 1, 1-9 (1998).

Let \(Mf(x)\equiv \sup_{h>0} {1\over 2h} \int^{x+h}_{x-h}| f(y)| dy\) be the centred Hardy-Littlewood maximal function on \(\mathbb{R}\). Let \(C\) be the constant such that for every \(f\in L^1(\mathbb{R})\) and every \(\lambda>0\), \(\lambda| \{x\in\mathbb{R}: Mf(x)>\lambda\}| \leq C\| f\| _{L^1(\mathbb{R})}\). The author showed that the \(C\in [37/24, (9+\sqrt{41})/8]\).

Reviewer: Yang Dachun (Beijing)

MSC:

42B25	Maximal functions, Littlewood-Paley theory
47J20	Variational and other types of inequalities involving nonlinear operators (general)

Keywords:

centred Hardy-Littlewood maximal function; weak type (1,1); best constant

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