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

2019/09/23 by Diogo Oliveira e Silva, Silva, Diogo Oliveira e, René Quilodrán +1 · 1 citation
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1909.10220

openalex publication_date 2019/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbS^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 equation a⋅(fσ_d-1)^∗ (q-1)\vert_\mathbbS^d-1=f, a.e. on \mathbbS^d-1, where a∈ C^∞(\mathbbS^d-1), q≥ 2(d+1)/(d-1) is an integer, and the only a priori assumption is f∈ L²(\mathbbS^d-1). We prove that any such solution belongs to the class C^∞(\mathbbS^d-1). In particular, we show that all critical points associated to the sharp form of the corresponding adjoint Fourier restriction inequality on \mathbbS^d-1 are C^∞-smooth. This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.

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• Smoothness of extremizers for certain inequalities of the Radon transform

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