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Title: Weighted estimates for one-sided martingale transforms
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, $$\displaystyle \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} .$$ 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.  more » « less
Award ID(s):
1800689
PAR ID:
10105667
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Proceedings of the American Mathematical Society
ISSN:
0002-9939
Page Range / eLocation ID:
1
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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