## Abstract

Let 𝕊

a⋅(fσ

where a ∈ C

^{d−1}denote the unit sphere in Euclidean space ℝ^{d}, d≥2, equipped with surface measure σ_{d−1}. An instance of our main result concerns the regularity of solutions of the convolution equationa⋅(fσ

_{d−1})^{∗(q−1)}∣_{𝕊d−1 }= f, a.e. on 𝕊^{d−1},where a ∈ C

^{∞}(𝕊^{d−1}), q≥2(d+1)/(d−1) is an integer, and the only a priori assumption is f ∈ L^{2}(𝕊^{d−1}). We prove that any such solution belongs to the class C^{∞}(𝕊^{d−1}). In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on 𝕊^{d−1}are C^{∞}-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].Original language | English |
---|---|

Article number | e12 |

Number of pages | 40 |

Journal | Forum of Mathematics, Sigma |

Volume | 9 |

DOIs | |

Publication status | Published - 7 Apr 2021 |

### Bibliographical note

Publisher Copyright:© The Author(s), 2021. Published by Cambridge University Press.

## Keywords

- 2020 Mathematics Subject Classification
- 35B38
- 42B37
- 49N60

## ASJC Scopus subject areas

- Analysis
- Theoretical Computer Science
- Algebra and Number Theory
- Statistics and Probability
- Mathematical Physics
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Mathematics