Sharp mixed norm spherical restriction

Abstract

Let $d\ge 2$ be an integer and let $2d/(d-1)<q\le \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{fσ}\big\|_{L^q_{\rm rad}L^2_{\rm ang}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}σ)}, \end{equation*} established by L. Vega in 1988. Letting ${\mathcal{A}}_d\subset (2d/(d-1),\infty ]$ be the set of exponents for which the constant functions on ${\mathbb{S}}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) ${\mathcal{A}}_d$ contains the even integers and $\infty$; (ii) ${\mathcal{A}}_d$ is an open set in the extended topology; (iii) ${\mathcal{A}}_d$ contains a neighborhood of infinity $(q_0(d),\infty ]$ with $q_0(d)\le \left(\frac{1}{2}+o(1)\right)d\log d$. In low dimensions we show that $q_0(2)\le 6.76;q_0(3)\le 5.45;q_0(4)\le 5.53;q_0(5)\le 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions.

21 pages

References in corpus (3)

• A sharp trilinear inequality related to Fourier restriction on the circle
• Extremizers for Fourier restriction on hyperboloids
• Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

Cited by in corpus (3)

• Local maximizers of adjoint Fourier restriction estimates for the cone, paraboloid and sphere
• Dimension-free Fourier restriction inequalities
• On profile decomposition for Airy type equation