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

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

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Acknowledgements

The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.

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Correspondence to Qingying Xue.

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Q. Xue was supported partly by NSFC (Nos. 11671039, 11871101) and NSFC-DFG (No. 11761131002).

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Qin, M., Xue, Q. Weighted and Unweighted Solyanik Estimates for the Multilinear Strong Maximal Function. Results Math 76, 6 (2021). https://doi.org/10.1007/s00025-020-01317-x

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Keywords

  • Solyanik estimates
  • Tauberian constant
  • multilinear strong maximal function

Mathematics Subject Classification

  • Primary 42B20
  • Secondary 42B35