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Multipliers for noncommutative Walsh-Fourier series


Author: Lian Wu
Journal: Proc. Amer. Math. Soc. 144 (2016), 1073-1085
MSC (2010): Primary 46L52, 46L53, 47A30; Secondary 60G42
DOI: https://doi.org/10.1090/proc/12831
Published electronically: August 5, 2015
MathSciNet review: 3447661
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Abstract: We consider multipliers for noncommutative Walsh-Fourier series. Let $ \mathcal {R}$ be the type $ II_1$ hyperfinite factor. For $ x \in L_1(\mathcal {R})$, $ 0<\alpha <1$, the multiplier transformation of $ x$ is defined by setting

$\displaystyle J^\alpha x = \sum _{\gamma \in \mathcal {F}}2^{-\gamma (1)\alpha }\hat {x}(\gamma )w_\gamma ,$

where $ (w_\gamma )_{\gamma \in \mathcal {F}}$ is the noncommutative Walsh system in $ \mathcal {R}$ and $ \sum _{\gamma \in \mathcal {F}}\hat {x}(\gamma )w_\gamma $ is the Walsh-Fourier series of $ x$. It is shown that

$\displaystyle \Vert J^\alpha x\Vert _{L_{1/(1-\alpha ),\infty }(\mathcal {R})}\leq c \Vert x\Vert _{L_1(\mathcal {R})},$

where $ c$ is a constant depending only on $ \alpha $. Via interpolations, we deduce that $ J^\alpha $ is bounded from $ L_{p}(\mathcal {R})$ into $ L_{q}(\mathcal {R})$ where $ 1<p<q$ and $ \alpha =1/p-1/q$, thus providing a noncommutative analogue of a classical result of Watari.

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Additional Information

Lian Wu
Affiliation: School of Mathematics and Statistics, Central South University, Changsha 410083, People’s Republic of China
Address at time of publication: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: wul5@miamioh.edu, wulian567@126.com

DOI: https://doi.org/10.1090/proc/12831
Keywords: Noncommutative martingales, multipliers, Walsh-Fourier series
Received by editor(s): April 10, 2014
Received by editor(s) in revised form: February 5, 2015
Published electronically: August 5, 2015
Additional Notes: The author was partially supported by NSFC (No. 11471337) and the Fundamental Research Funds for the Central Universities of Central South University: 2013zzts007.
Communicated by: Marius Junge
Article copyright: © Copyright 2015 American Mathematical Society