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

中文翻译：

---

多重线性强极大函数的加权和未加权 Solyanik 估计

令 $$\displaystyle \omega $$ 为 $$A^*_\infty $$ 中的权重，并令 $$\displaystyle \mathcal {M}^m_n(\mathbf {f})$$ 为多重线性强极大值$$\displaystyle \mathbf {f}=\left( f_1,\ldots ,f_m\right) $$ 的函数，其中 $$f_1,\ldots ,f_m$$ 是 $$\mathbb R^n$$ 上的函数. 在本文中，我们考虑 $$\displaystyle \mathcal {M}^m_n$$ 的分布函数的渐近估计。我们证明，对于 $$\displaystyle \lambda \in (0,1)$$ ，如果 $$\displaystyle \lambda \rightarrow 1^-$$ ，则多线性 Tauberian 常数 $$\displaystyle \mathcal C^m_n $$ 和加权 Tauberian 常数 $$\displaystyle \mathcal C^m_{n, 与 $$\displaystyle \mathcal M^m_n$$ 相关的 \omega }$$ 享有 $$\begin{aligned} \displaystyle \mathcal C^m_n(\lambda )-1\simeq m\left( 1- \lambda \right) ^{\frac{1}{n}}\quad \hbox {and }\quad \mathcal C^m_{n,\omega }(\lambda )-1\lesssim m(1-\lambda )^{\left( 4n[\omega ]_{A^*_\infty }\right) ^{-1}}。\end{对齐}$$