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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
DOI: https://doi.org/10.1090/proc/13912
Published electronically: January 12, 2018
MathSciNet review: 3767376
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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 $Mf=\sup _{n\geq 0}|f_n|$, $Mw=\sup _{n\geq 0}w_n$ the maximal functions of $f$ and $w$, we prove the weighted inequality \begin{equation*} ||g||_{L^1(w)}\leq C||Mf||_{L^1(Mw)}, \end{equation*} 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
MR Author ID: 30705
Email: banuelos@math.purdue.edu

Adam Osękowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
ORCID: 0000-0002-8905-2418
Email: ados@mimuw.edu.pl

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