Weak-type estimates for martingale maximal functions

Adam Osękowski; Mateusz Wojtas

doi:10.1214/22-ECP494

2022 Weak-type estimates for martingale maximal functions

Adam Osękowski, Mateusz Wojtas

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Electron. Commun. Probab. 27: 1-11 (2022). DOI: 10.1214/22-ECP494

Abstract

The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given $1\textless p\textless \mathrm{\infty }$ and a pair $(x,y)$ of nonnegative numbers satisfying ${x^{p}}\le y$ , we identify the optimal upper bounds for $\| |{\sup _{n}}{f_{n}}|{\| _{p,\mathrm{\infty }}}$ , for nonnegative martingales $f={({f_{n}})_{n\ge 0}}$ satisfying $\| f{\| _{1}}=x$ and $\| f{\| _{p}^{p}}=y$ .

Acknowledgments

The authors would like to express their gratitude to the anonymous Referees for the careful reading of the first version of the paper, many helpful comments and suggestions.

Citation

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Adam Osękowski. Mateusz Wojtas. "Weak-type estimates for martingale maximal functions." Electron. Commun. Probab. 27 1 - 11, 2022. https://doi.org/10.1214/22-ECP494