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

Diogo Oliveira E Silva, René Quilodrán

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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].
Original languageEnglish
Article numbere12
Number of pages40
JournalForum of Mathematics, Sigma
Volume9
DOIs
Publication statusPublished - 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

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