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Counterexamples to Hardy--Littlewood maximal inequalities on solvable groups
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
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