Restriction implies Bochner–Riesz for paraboloids

Abstract

Let Σ ⊆ ℝn be a (compact) hypersurface with non-vanishing Gaussian curvature, with suitable parameterizations, also called Σ: U → ℝn (U open patches in ℝn−1). The restriction problem for Σ is the question of the a priori estimate (for f ∈ S(ℝ))(^denoting the Fourier transform). The Bochner-Riesz problem for Σ is the question of whether the functionsdefine Lp-bounded Fourier multiplier operators on ℝn in the range.