The weak-type (1,1) bound for the Hardy--Littlewood maximal function is O(√n log n)

@misc{spector2026weaktype,
  title = {{The weak-type (1,1) bound for the Hardy--Littlewood maximal function is $O(\sqrtn \log n)$}},
  author = {Daniel Spector and Cody B. Stockdale},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05377},
  eprint = {2609.05377},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05377},
}

TY  - UNPB
TI  - The weak-type (1,1) bound for the Hardy--Littlewood maximal function is O(√n log n)
AU  - Daniel Spector
AU  - Cody B. Stockdale
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05377
UR  - https://arxiv.org/abs/2609.05377
AB  - We prove a weak-type (1,1) estimate for the centered Hardy--Littlewood maximal function with respect to Euclidean balls with dimensional dependence O(√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 √n loss. The key technical aspect of our result is an improvement of the weak-type bound for the heat maximal operator from O(√n) to O(log n).
ER  -