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arXiv · 2609.05377

The weak-type (1,1) bound for the Hardy--Littlewood maximal function is $O(\sqrt{n} \log n)$

Daniel Spector·Cody B. Stockdale
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Abstract

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 Str\"omberg. The proof goes through a pointwise bound of the Hardy--Littlewood maximal operator by the heat maximal operator with $\sqrt{n}$ loss. The key technical aspect of our result is an improvement of the weak-type bound for the heat maximal operator from $O(\sqrt{n})$ to $O(\log n)$.

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Daniel Spector, Cody B. Stockdale. 2026-09-04. The weak-type (1,1) bound for the Hardy--Littlewood maximal function is $O(\sqrt{n} \log n)$. https://arxiv.org/abs/2609.05377

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