# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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 martingalesHTML articles powered by AMS MathViewer

by Rodrigo Bañuelos and Adam Osękowski
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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• Rodrigo Bañuelos
• Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
• MR Author ID: 30705
• Email: banuelos@math.purdue.edu
• Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
• ORCID: 0000-0002-8905-2418