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Smoothness of solutions of a convolution equation of restricted type on the sphere
Published online by Cambridge University Press: 07 April 2021
Abstract
Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space
$\mathbb {R}^d$,
$d\geq 2$, equipped with surface measure
$\sigma _{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation
$a\in C^\infty (\mathbb {S}^{d-1})$,
$q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is
$f\in L^2(\mathbb {S}^{d-1})$. We prove that any such solution belongs to the class
$C^\infty (\mathbb {S}^{d-1})$. In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on
$\mathbb {S}^{d-1}$ are
$C^\infty $-smooth. This extends previous work of Christ and Shao [4] to arbitrary dimensions and general even exponents and plays a key role in the companion paper [24].
MSC classification
- Type
- Analysis
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
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