Fourier transform and regularity of characteristic functions

Ko, Hyerim; Lee, Sanghyuk

doi:10.1090/proc/13435

Fourier transform and regularity of characteristic functions
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by Hyerim Ko and Sanghyuk Lee PDF

Proc. Amer. Math. Soc. 145 (2017), 1097-1107 Request permission

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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