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A weighted maximal inequality for differentially subordinate martingales

Authors: Rodrigo Bañuelos and Adam Osękowski
Journal: Proc. Amer. Math. Soc. 146 (2018), 2263-2275
MSC (2010): Primary 60G44; Secondary 42B25
Published electronically: January 12, 2018
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Abstract: The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that $ f=(f_n)_{n\geq 0}$, $ g=(g_n)_{n\geq 0}$ are martingales such that $ g$ is differentially subordinate to $ f$, and let $ w=(w_n)_{n\geq 0}$ be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by $ Mf=\sup _{n\geq 0}\vert f_n\vert$, $ Mw=\sup _{n\geq 0}w_n$ the maximal functions of $ f$ and $ w$, we prove the weighted inequality

$\displaystyle \vert\vert g\vert\vert _{L^1(w)}\leq C\vert\vert Mf\vert\vert _{L^1(Mw)},$    

where $ C=3+\sqrt {2}+4\ln 2=7.186802\ldots $. The proof rests on the existence of a special function enjoying appropriate majorization and concavity.

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Additional Information

Rodrigo Bañuelos
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Adam Osękowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Keywords: Martingale, differential subordination, weight
Received by editor(s): January 4, 2017
Received by editor(s) in revised form: July 30, 2017
Published electronically: January 12, 2018
Additional Notes: The first author was supported in part by NSF Grant #0603701-DMS
The second author was supported in part by the NCN grant DEC-2014/14/E/ST1/00532.
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2018 American Mathematical Society

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