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Convolution of a measure with itself
and a restriction theorem


Authors: Jong-Guk Bak and David McMichael
Journal: Proc. Amer. Math. Soc. 125 (1997), 463-470
MSC (1991): Primary 42B10
DOI: https://doi.org/10.1090/S0002-9939-97-03569-7
MathSciNet review: 1350932
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Abstract: Let $S_{k}=\left \{ (y,|y|^{k})\colon y \in \mathbf {R}^{n-1} \right \} \subset \mathbf {R}^{n}$ and $\sigma $ be the measure defined by $\langle \sigma , \phi \rangle = \int _{\mathbf {R}^{n-1}}\phi (y, |y|^{k}) dy$. Let $\sigma _{P}$ denote the measure obtained by restricting $\sigma $ to the set $P=[0,\infty )^{n-1}$. We prove estimates on $\sigma _{P}*\sigma _{P}$. As a corollary we obtain results on the restriction to $S_{k} \subset \mathbf {R}^{3}$ of the Fourier transform of functions on $\mathbf {R}^{3}$ for $k\in \mathbf {R}$, $2<k<6$.


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Additional Information

Jong-Guk Bak
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea
Email: bak@euclid.postech.ac.kr

David McMichael
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306

DOI: https://doi.org/10.1090/S0002-9939-97-03569-7
Received by editor(s): April 13, 1995
Received by editor(s) in revised form: August 10, 1995
Additional Notes: The first author was supported in part by a grant from TGRC–KOSEF of Korea.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society

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