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Failure of the linear A2 bound for the strong maximal operator
Abstract
We show that the strong maximal operator on ℝ2 does not satisfy a linear weighted L2 estimate in terms of the rectangular A2 characteristic. More precisely, we construct weights with arbitrarily large characteristic for which the operator norm is bounded below by [w]A2str√(log [w]A2str). The proof is elementary. It uses an explicit two-parameter mass recurrence on a product of geometric interval partitions, together with a direct comparison of arbitrary rectangular averages with anchored averages.
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BibTeX
@misc{lerner2026failure,
title = {{Failure of the linear $A_2$ bound for the strong maximal operator}},
author = {Andrei K. Lerner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14008},
eprint = {2609.14008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14008},
}
RIS
TY - UNPB
TI - Failure of the linear A_2 bound for the strong maximal operator
AU - Andrei K. Lerner
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14008
UR - https://arxiv.org/abs/2609.14008
AB - We show that the strong maximal operator on ℝ^2 does not satisfy a linear weighted L^2 estimate in terms of the rectangular A_2 characteristic. More precisely, we construct weights with arbitrarily large characteristic for which the operator norm is bounded below by [w]_A_2^str√(log [w]_A_2^str). The proof is elementary. It uses an explicit two-parameter mass recurrence on a product of geometric interval partitions, together with a direct comparison of arbitrary rectangular averages with anchored averages.
ER -