Weighted and Unweighted Solyanik Estimates for the Multilinear Strong Maximal Function

Abstract

Let $\displaystyle \omega$ be a weight in $A^*_\infty$ and let $\displaystyle \mathcal {M}^m_n(\mathbf {f})$ be the multilinear strong maximal function of $\displaystyle \mathbf {f}=\left( f_1,\ldots ,f_m\right)$, where $f_1,\ldots ,f_m$ are functions on $\mathbb R^n$. In this paper, we consider the asymptotic estimates for the distribution functions of $\displaystyle \mathcal {M}^m_n$. We show that, for $\displaystyle \lambda \in (0,1)$, if $\displaystyle \lambda \rightarrow 1^-$, then the multilinear Tauberian constant $\displaystyle \mathcal C^m_n$ and the weighted Tauberian constant $\displaystyle \mathcal C^m_{n,\omega }$ associated with $\displaystyle \mathcal M^m_n$ enjoy the properties that

$$\begin{aligned} \displaystyle \mathcal C^m_n(\lambda )-1\simeq m\left( 1-\lambda \right) ^{\frac{1}{n}}\quad \hbox {and }\quad \mathcal C^m_{n,\omega }(\lambda )-1\lesssim m(1-\lambda )^{\left( 4n[\omega ]_{A^*_\infty }\right) ^{-1}}. \end{aligned}$$