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 $1\le p<\infty$ 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 $(1,1)$.