Abstract
Grafakos systematically proved that \(A_\infty \) weights have different characterizations for cubes in Euclidean spaces in his classical text book. Very recently, Duoandikoetxea, Martín-Reyes, Ombrosi and Kosz discussed several characterizations of the \(A_{\infty }\) weights in the setting of general bases. By conditional expectations, we study \(A_\infty \) weights in martingale spaces. Because conditional expectations are Radon–Nikodým derivatives with respect to sub\(\hbox {-}\sigma \hbox {-}\)fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the \(A_{\infty }\) weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations.
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The authors thank the referees for many valuable comments and suggestions. These greatly improved the presentation of our results.
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W. Chen is supported by the National Natural Science Foundation of China (11971419, 12271469). J. Ju is supported by the Jiangsu Students’ Platform for Innovation and Entrepreneurship Training Program (202211117018Z). C. Zhang is supported by the National Natural Science Foundation of China (11971431) and the Natural Science Foundation of Zhejiang Province (LY22A010011).
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Ju, J., Chen, W., Cui, J. et al. Characterizations of 
Weights in Martingale Spaces.
J Geom Anal 34, 224 (2024). https://doi.org/10.1007/s12220-024-01674-x
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DOI: https://doi.org/10.1007/s12220-024-01674-x