Weighted estimates for one-sided martingale transforms
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- by Wei Chen, Rui Han and Michael T. Lacey PDF
- Proc. Amer. Math. Soc. 148 (2020), 235-245 Request permission
Abstract:
Let $Tf =\sum _{I} \varepsilon _I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $\lvert \varepsilon _I\rvert =1$, and $h_J$ is the Haar function defined on dyadic interval $J$. We show that, for instance, \begin{equation*} \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} . \end{equation*} Above, we use the one-sided $A_2$ characteristic for the weight $w$. This is an instance of a one-sided $A_2$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $A_2$ theorem do not seem to apply in the one-sided setting.References
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Additional Information
- Wei Chen
- Affiliation: School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China; School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- Email: weichen@yzu.edu.cn
- Rui Han
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- Email: rui.han@math.gatech.edu
- Michael T. Lacey
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- MR Author ID: 109040
- Email: lacey@math.gatech.edu
- Received by editor(s): November 5, 2018
- Received by editor(s) in revised form: April 12, 2019
- Published electronically: July 8, 2019
- Additional Notes: The first author was supported by the National Natural Science Foundation of China (11771379), the Natural Science Foundation of Jiangsu Province (BK20161326), and the Jiangsu Government Scholarship for Overseas Studies (JS-2017-228)
The research of the second author was supported in part by a grant from the US National Science Foundation, DMS-1800689
The research of the third author was supported in part by a grant from the US National Science Foundation, DMS-1600693, and the Australian Research Council, ARC DP160100153 - Communicated by: Svitlana Mayboroda
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 235-245
- MSC (2010): Primary 42B25
- DOI: https://doi.org/10.1090/proc/14665
- MathSciNet review: 4042846