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A note on bi-linear multipliers


Author: Saurabh Shrivastava
Journal: Proc. Amer. Math. Soc. 143 (2015), 3055-3061
MSC (2010): Primary 42A45, 42B15; Secondary 42B25
DOI: https://doi.org/10.1090/S0002-9939-2015-12679-2
Published electronically: March 18, 2015
MathSciNet review: 3336630
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Abstract: In this paper we prove that if $ \chi _{_E}(\xi -\eta )$ - the indicator function of a measurable set $ E\subseteq \mathbb{R}^d$ - is a bi-linear multiplier symbol for exponents $ p,q,r$ satisfying the Hölder's condition $ \frac {1}{p}+\frac {1}{q}=\frac {1}{r}$ and exactly one of $ p,q,$ or $ r'=\frac {r}{r-1}$ is less than $ 2,$ then $ E$ is equivalent to an open subset of $ \mathbb{R}^d.$


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Additional Information

Saurabh Shrivastava
Affiliation: Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Indore By-pass road Bhauri, Bhopal-462066, India
Email: saurabhk@iiserb.ac.in

DOI: https://doi.org/10.1090/S0002-9939-2015-12679-2
Keywords: Fourier multipliers, Littlewood-Paley operators, bi-linear multipliers, transference methods
Received by editor(s): March 18, 2014
Published electronically: March 18, 2015
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society