Weighted estimates for one-sided martingale transforms

Author:

Chen Wei,Han Rui,Lacey Michael

Abstract

Let T f = I ε I f , h I + h I Tf =\sum _{I} \varepsilon _I \langle f,h_{I^+}\rangle h_{I^-} . Here, | ε I | = 1 \lvert \varepsilon _I\rvert =1 , and h J h_J is the Haar function defined on dyadic interval J J . We show that, for instance, T L 2 ( w ) L 2 ( w ) [ w ] A 2 + . \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 A_2 characteristic for the weight w w . This is an instance of a one-sided A 2 A_2 conjecture. Our proof of this fact is difficult, as the very quick known proofs of the A 2 A_2 theorem do not seem to apply in the one-sided setting.

Funder

National Natural Science Foundation of China

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference19 articles.

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4. A counting problem in ergodic theory and extrapolation for one-sided weights;Carro, María J.;J. Anal. Math.,2018

5. Weighted inequalities for singular integral operators on the half-line;Chill, Ralph;Studia Math.,2018

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1. A Study of One-Sided Singular Integral and Function Space via Reproducing Formula;The Journal of Geometric Analysis;2023-06-29

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