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 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 $O(\log n)$.