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The weak-type (1,1) bound for the Hardy--Littlewood maximal function is $O(\sqrt{n} \log n)$
We prove a weak-type $(1,1)$ estimate for the centered Hardy--Littlewood maximal function with respect to Euclidean balls with dimensional dependence $O(\sqrt{n} \log n)$. This improves the order of growth in the classical $O(n)$ estimate of Stein and Strömberg. The proof goes through a pointwise bo
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