A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space
Abstract
:1. Introduction
- 1
- Motivated by the proof of [4] (Theorem B), we get
- 2
- Using the construction of principal sets [2] and the conditional sparsity [5], we have where are the constants in the construction of principal sets (Appendix A).
- 3
- Long [1] [Theorem 6.6.3] qualitatively evaluated . Modifying Long’s proof, we have which is the same as 1.
2. Preliminaries
2.1. Filtered Measure Space
2.2. Stopping Times
2.3. Operators and Weights
3. Approaches of Theorem 1
4. Comparison of and
- 1
- We claim that the function is decreasing on Writing and we will show that and are both decreasing on Combining this with and we obtain that is decreasing on We now check that and are both decreasing.For with it is clear thatThus, is decreasing onFor with considerIt is clear thatUsing the mean value theorem, we have
- 2
- We claim that the function is decreasing on It suffices to show that We haveIt is clear that if and only if Let with Because of on the function is strictly increasing on It follows from that on That is, with Thus, is decreasing on
5. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Appendix A. Construction of Principal Sets
- The sets where are disjoint and
- on
- on
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Chen, W.; Cui, J. A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space. Mathematics 2021, 9, 2953. https://doi.org/10.3390/math9222953
Chen W, Cui J. A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space. Mathematics. 2021; 9(22):2953. https://doi.org/10.3390/math9222953
Chicago/Turabian StyleChen, Wei, and Jingya Cui. 2021. "A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space" Mathematics 9, no. 22: 2953. https://doi.org/10.3390/math9222953
APA StyleChen, W., & Cui, J. (2021). A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space. Mathematics, 9(22), 2953. https://doi.org/10.3390/math9222953