Year: 2026
Type: preprint
Abstract: We prove that the centered Hardy--Littlewood maximal operator is not of weak type $(p,p)$ for any $1\le p<\infty$ in three solvable settings: the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ with the switch--walk--switch word metric, the Baumslag--Solitar groups $BS(1,n)$, $n\ge2$, 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 $(1,1)$.
Source: arXiv (Cornell University)
Author: Fujiwara, Koji
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: 離散群に潜む新しい構造の発見と解明
OpenAlex ID: W7221011818