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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A weighted maximal inequality for differentially subordinate martingales
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by Rodrigo Bañuelos and Adam Osękowski PDF
Proc. Amer. Math. Soc. 146 (2018), 2263-2275 Request permission

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
  • 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
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 2263-2275
  • MSC (2010): Primary 60G44; Secondary 42B25
  • DOI: https://doi.org/10.1090/proc/13912
  • MathSciNet review: 3767376