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The weak type \((1,1)\) bounds for the maximal function associated to cubes grow to infinity with the dimension. (English) Zbl 1230.42025

The author shows that if \(M_df(x)=\sup_{r>0}\frac{1}{|Q(x,r)|}\int_{Q(x,r)}|f(y)|dy\) is the centered Hardy-Littlewood maximal function associated to cubes in \(\mathbb R^d\) and one denotes by \(c_d\) the best constant satisfying the weak type inequality \[ \alpha |\{M_d f\geq \alpha\}|\leq c_d \|f\|_1 \] then \(\lim_{d\to \infty} c_d=\infty\). This answers in the negative a longstanding open question by E. M. Stein and J. O. Stromberg [Ark. Mat. 21, 259–269 (1983; Zbl 0537.42018)]. It was previously known by Stein and Stromberg that for maximal functions associated to arbitrary balls this constant \(c_d\) grows at most like \(O(d \log d)\) and grows like \(O(d)\) for euclidean balls. They asked if some uniform bounds can be found. The result shows that this is not the case for square balls.

MSC:

42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0537.42018
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References:

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