## Abstract

Let Sd−1 denote the unit sphere in Euclidean space Rd, d≥2, equipped with surface measure σd−1. An instance of our main result concerns the regularity of solutions of the convolution equation

a⋅(fσd−1)∗(q−1)∣∣Sd−1=f, a.e. on Sd−1,

where a∈C∞(Sd−1), q≥2(d+1)/(d−1) is an integer, and the only a priori assumption is f∈L2(Sd−1). We prove that any such solution belongs to the class C∞(Sd−1). In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on Sd−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].

a⋅(fσd−1)∗(q−1)∣∣Sd−1=f, a.e. on Sd−1,

where a∈C∞(Sd−1), q≥2(d+1)/(d−1) is an integer, and the only a priori assumption is f∈L2(Sd−1). We prove that any such solution belongs to the class C∞(Sd−1). In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on Sd−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 |
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Article number | e12 |

Journal | Forum of Mathematics, Sigma |

Volume | 9 |

DOIs | |

Publication status | Published - 7 Apr 2021 |

## 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