Let \(M_w\) be the uncentered Hardy-Littlewood maximal operator with respect to a measure \(\mu\), that is \[ M_w f(x):= \sup_{x\in Q} \int_Q |f(y)| w(y) \;dy, \;x\in \mathbb{R}^n, \] where the supremum is taken over all cubes \(Q\) in \(\mathbb{R}^n\) containing the point \(x\) with sides parallel to the axis.
The paper deals with the asymptotic behavior, as \(\alpha \rightarrow 1^{-}\), of the so called sharp Tauberian constant defined as \[ C_w(\alpha)=\sup_{E:0<w(E)<\infty} w(E)^{-1} w(\{x\in \mathbb{R}^n: \;M_w\chi_{E}(x)>\alpha \}). \] In particular, it is investigated whether \[ \lim_{\alpha\to 1^{-}}C_w(\alpha)=1. \] The motivation is due to a result of A. A. Solyanik, who showed that in the case \(w\equiv 1\), which corresponds to the Lebesgue measure, \(C(\alpha)-1\simeq (1-\alpha)^{1/n}\) when \(\alpha \to 1^{-}\). The authors proved that the answer is also positive for the weighted maximal operator \(M_w\).
Also, the behavior of the sharp Tauberian constant for \(M\), the maximal uncentered Hardy-Littlewood operator with respect to the Lebesgue measure, is studied. More precisely, it is shown that \(\displaystyle{\lim_{\alpha\to 1^{-}} C^w(\alpha)=1}\) if and only \(w\) belongs to the Muckenhoupt class of weights \(A_{\infty}=\displaystyle{\bigcup_{p\geq 1} A_p}\). In this case, \(C^w\) is defined in the same way as \(C_w\) but replacing the uncentered weighted maximal operator \(M_w\) by \(M\). The study of the Tauberian constant \(C^w\) makes it possible to obtain a quantitative result about the \(A_{\infty}\)-constant of the weight that, roughly speaking, establishes that this constant is equivalent to the smallest number \(c>1\) such that \[ C^w(\alpha)-1 \lesssim (1-\alpha)^{1/c} \;\;\text{as} \;\;\alpha\to 1^{-}. \]