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The behaviour of square functions from ergodic theory in $ L^{\infty}$


Author: Guixiang Hong
Journal: Proc. Amer. Math. Soc. 143 (2015), 4797-4802
MSC (2010): Primary 42B25; Secondary 47G10
DOI: https://doi.org/10.1090/proc12737
Published electronically: April 29, 2015
MathSciNet review: 3391037
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Abstract: In this paper, we analyze carefully the behaviour in $ L^\infty (\mathbb{R})$ of the square functions $ S$ and $ S_{\mathcal {I}}$, originating from ergodic theory. First, we show that we can find some function $ f\in L^\infty (\mathbb{R})$, such that $ Sf$ equals infinity on a nonzero measurable set. Second, we can find compact supported function $ f\in L^\infty (\mathbb{R})$ and $ \mathcal {I}$ such that $ S_{\mathcal {I}} f$ does not belong to $ BMO$ space. Finally, we show that $ S$ is bounded from $ L^{\infty }_c$, the space of compactly supported $ L^\infty (\mathbb{R})$ functions, to $ BMO$ space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl (2000). That is, $ S_{\mathcal {I}}$ are uniformly bounded in $ L^p(\mathbb{R})$ with respect to $ \mathcal {I}$ for $ 2<p<\infty $.


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Guixiang Hong
Affiliation: Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/Nicolás Cabrera 13-15, 28049, Madrid, Spain
Email: guixiang.hong@icmat.es

DOI: https://doi.org/10.1090/proc12737
Keywords: Square function, behaviour in $L^{\infty}$
Received by editor(s): April 28, 2014
Received by editor(s) in revised form: July 29, 2014
Published electronically: April 29, 2015
Additional Notes: The author was supported by MINECO: ICMAT Severo Ochoa project SEV-2011-0087 and ERC Grant StG-256997-CZOSQP (EU)
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society