Weighted mixed-norm inequality on Doob’s maximal operator and John–Nirenberg inequalities in Banach function spaces

Abstract

We prove a weighted mixed-norm inequality for the Doob maximal operator on a filtered measure space. We also give some characterizations of martingale BMO spaces in the setting of Banach function spaces. The main method is based on the technique of extrapolation on martingale Banach spaces.

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Acknowledgement

The authors thank the referee for careful reading and useful comments.

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Correspondence to D. Zhou.

Additional information

Wei Chen is supported by the National Natural Science Foundation of China (11771379), the Natural Science Foundation of Jiangsu Province (BK20161326), the Jiangsu Government Scholarship for Overseas Studies (JS-2017-228) and the School Foundation of Yangzhou University (2016CXJ001).

Dejian Zhou is supported by the National Natural Science Foundation of China (11801573).

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Chen, W., Ho, K., Jiao, Y. et al. Weighted mixed-norm inequality on Doob’s maximal operator and John–Nirenberg inequalities in Banach function spaces. Acta Math. Hungar. 157, 408–433 (2019). https://doi.org/10.1007/s10474-018-0889-5

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Key words and phrases

  • mixed-norm inequality
  • weight
  • Doob’s maximal operator
  • extrapolation
  • BMO

Mathematics Subject Classification

  • primary 60G46
  • secondary 60G42