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Martingale transforms of the Rademacher sequence in rearrangement invariant spaces


Author: S. V. Astashkin
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 2.
Journal: St. Petersburg Math. J. 27 (2016), 191-206
MSC (2010): Primary 46E30
Published electronically: January 29, 2016
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Abstract: Let $ v_k=c_k\chi _{\{\tau \ge k\}}$, where $ \tau $ is a stopping time with respect to the Rademacher system $ \{r_k\}$ and $ c_k\in \mathbb{R}$, $ k=1,2,\dots $. Then $ \big \Vert\sum _{k=1}^n v_kr_k\big \Vert _X\asymp \big \Vert(\sum _{k=1}^n v_k^2)^{1/2}\big \Vert _X$ if and only if the rearrangement invariant Banach function space $ X$ has nontrivial Boyd indices. If the $ v_k$ are the vectors $ \sum _{i=0}^{k-1} a_k^ir_i$, $ k=1,2,\dots $, the same relation is fulfilled if and only if $ X$ contains the closure of $ L_\infty $ in the Orlicz space $ \exp L_1$. In the second part of the paper, a new unconditionality criterion for the Haar system in a rearrangement invariant space is obtained in terms of a decoupling version of the transforms $ f_n=\sum _{k=1}^n v_kr_k$.


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Additional Information

S. V. Astashkin
Affiliation: Samara State University, Academician Pavlov st., 443011 Samara, Russia
Email: astash@samsu.ru

DOI: https://doi.org/10.1090/spmj/1383
Keywords: Rearrangement invariant space, Orlicz space, martingale transform, Rademacher functions, Paley function, Haar functions, Boyd indices, stopping time
Received by editor(s): May 23, 2014
Published electronically: January 29, 2016
Additional Notes: Supported by the Ministry of Education and Science of Russia within the basic state financing
Article copyright: © Copyright 2016 American Mathematical Society