Sharp mixed norm spherical restriction

Let $d\ge 2$ be an integer and let $2d\mathrm{/}(d-1)<q\le \mathrm{\infty }$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big|\widehat{f\sigma}\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}\sigma)}, \end{equation*} established by L. Vega in 1988. Letting ${\mathcal{A}}_d\subset (2d\mathrm{/}(d-1),\mathrm{\infty }]$ 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 $\mathrm{\infty }$; (ii) ${\mathcal{A}}_d$ is an open set in the extended topology; (iii) ${\mathcal{A}}_d$ contains a neighborhood of infinity $(q_0(d),\mathrm{\infty }]$ with $q_0(d)\le (\frac{1}{2}+o(1))d\log d$. In low dimensions we show that $q_0(2) \leq 6.76,;,q_0(3) \leq 5.45 ,;, q_0(4) \leq 5.53 ,;, q_0(5) \leq 6.07$. 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.