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Fourier transform and regularity of characteristic functions


Authors: Hyerim Ko and Sanghyuk Lee
Journal: Proc. Amer. Math. Soc. 145 (2017), 1097-1107
MSC (2010): Primary 42B25; Secondary 42B15
DOI: https://doi.org/10.1090/proc/13435
Published electronically: November 21, 2016
MathSciNet review: 3589310
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Abstract: Let $ E$ be a bounded domain in $ \mathbb{R}^d$. We study regularity property of $ \chi _E$ and integrability of $ \widehat {\chi _E }$ when its boundary $ \partial E$ satisfies some conditions. At the critical case these properties are generally known to fail. By making use of Lorentz and Lorentz-Sobolev spaces we obtain the endpoint cases of the previous known results. Our results are based on a refined version of Littlewood-Paley inequality, which makes it possible to exploit cancellation effectively.


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Additional Information

Hyerim Ko
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151–747, Republic of Korea
Email: kohr@snu.ac.kr

Sanghyuk Lee
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151–747, Republic of Korea
Email: shklee@snu.ac.kr

DOI: https://doi.org/10.1090/proc/13435
Received by editor(s): June 25, 2015
Published electronically: November 21, 2016
Additional Notes: The authors were supported in part by NRF grant No.2009-0083521 and NRF grant No. 2015R1A4A1041675 (Republic of Korea).
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society