Smoothness of solutions of a convolution equation of restricted-type on the sphere

Diogo Oliveira E Silva, René Quilodrán

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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].
Original languageEnglish
Article numbere12
JournalForum of Mathematics, Sigma
Publication statusPublished - 7 Apr 2021


  • 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


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