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

Let 𝕊^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)∣_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 ∈ L²(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].