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Counterexamples to Hardy--Littlewood maximal inequalities on solvable groups
Abstract
We prove that the centered Hardy--Littlewood maximal operator is not of weak type (p,p) for any 1≤ p<∞ in three solvable settings: the lamplighter group (ℤ/2ℤ)wrℤ with the switch--walk--switch word metric, the Baumslag--Solitar groups BS(1,n), n≥2, with their digit word metrics, and the three-dimensional Sol group with any left-invariant Riemannian metric and its Riemannian volume. To the best of our knowledge, we give the first examples of finitely generated groups, equipped with word metrics and counting measure, for which the centered Hardy--Littlewood maximal operator fails to be of weak type (1,1).
Related
- Sharp Hausdorff--Young inequalities for solvable and semisimple Lie groups
- Winning property of counterexamples to Uniform Littlewood's Conjecture
- The norm of the centered Hardy--Littlewood maximal operator
- The weak-type (1,1) bound for the Hardy--Littlewood maximal function is O(√n log n)
- Muckenhoupt weights for the Hardy-Littlewood maximal operator on hyperbolic spaces
BibTeX
@misc{fujiwara2026counterexamples,
title = {{Counterexamples to Hardy--Littlewood maximal inequalities on solvable groups}},
author = {Koji Fujiwara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08121},
eprint = {2610.08121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08121},
}
RIS
TY - UNPB
TI - Counterexamples to Hardy--Littlewood maximal inequalities on solvable groups
AU - Koji Fujiwara
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08121
UR - https://arxiv.org/abs/2610.08121
AB - We prove that the centered Hardy--Littlewood maximal operator is not of weak type (p,p) for any 1≤ p<∞ in three solvable settings: the lamplighter group (ℤ/2ℤ)wrℤ with the switch--walk--switch word metric, the Baumslag--Solitar groups BS(1,n), n≥2, with their digit word metrics, and the three-dimensional Sol group with any left-invariant Riemannian metric and its Riemannian volume. To the best of our knowledge, we give the first examples of finitely generated groups, equipped with word metrics and counting measure, for which the centered Hardy--Littlewood maximal operator fails to be of weak type (1,1).
ER -