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The maximal function and conditional square function control the variation: An elementary proof


Authors: Kevin Hughes, Ben Krause and Bartosz Trojan
Journal: Proc. Amer. Math. Soc. 144 (2016), 3583-3588
MSC (2010): Primary 60G42, 60E15; Secondary 47B38, 46N30
DOI: https://doi.org/10.1090/proc/12866
Published electronically: April 14, 2016
MathSciNet review: 3503727
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Abstract: In this note we prove the following good-$ \lambda $ inequality, for $ r>2$, all $ \lambda > 0$, $ \delta \in \big (0, \frac {1}{2} \big )$,

$\displaystyle C \cdot \nu \big \{ V_r(f) > 3 \lambda ; \mathcal {M}(f) < \delta... ...da \} + \frac {\delta ^2}{(r-2)^2} \cdot \nu \big \{ V_r(f) > \lambda \big \}, $

where $ \mathcal {M}(f)$ is the martingale maximal function, $ s(f)$ is the conditional martingale square function, and $ C > 0$ is (absolute) constant. This immediately proves that $ V_r(f)$ is bounded on $ L^p$, $ 1 < p <\infty $, and moreover is integrable when the maximal function and the conditional square function are.

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Additional Information

Kevin Hughes
Affiliation: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom
Email: khughes.math@gmail.com

Ben Krause
Affiliation: Department of Mathematics, University of California Los Angeles, Math Sciences Building, Los Angeles, California 90095-1555
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, V6T1Z2 Canada
Email: benkrause@math.ubc.ca

Bartosz Trojan
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocławi, Poland
Address at time of publication: WydziałMatematyki, Politechnika Wrocławska, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: bartosz.trojan@pwr.edu.pl

DOI: https://doi.org/10.1090/proc/12866
Received by editor(s): October 30, 2014
Received by editor(s) in revised form: June 5, 2015
Published electronically: April 14, 2016
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society