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

Zbl 0537.42018
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References:

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